text:
\section{Introduction and description of the main results}

The object of study in the present paper is a class of entire solutions of the system
\beq\label{Euler-Lagrange equation}
\Delta u-W_u(u)=0,\quad u:\BR^n\ri\BR^m,
\eeq
$n,m\in\mathbb{N}^+$, where $W:\BR^m\ri\BR$ is a phase transition potential that is nonnegative and vanishes only on a finite set $\{W=0\}=:A=\{a_1,...a_N\}$ for some distinct points $a_1,...,a_N\in\BR^m$ that represent the phases of a substance which can exist in $N\geq 2$ different equally preferred phases.

The system \eqref{Euler-Lagrange equation} is the Euler-Lagrange equation corresponding to the Allen-Cahn free energy functional
\beq\label{AC functional}
J_D(v)=\int_D \left( \f12|\na v|^2+W(v) \right)\,dx.
\eeq

We restrict ourselves to maps $u\in W_{loc}^{1,2}(\BR^n,\BR^m)\cap L^\infty(\BR^n,\BR^m)$, which minimize $J$ subject to their Dirichlet data,
\beq\label{def of minimizer}
J_D(u+v)\geq J_D(u),\quad \forall v\in W_0^{1,2}(D,\BR^m)\cap L^\infty(D,\BR^m)
\eeq
for any open, bounded Lipschitz set $D\subset \BR^n$. We call such maps \emph{entire minimizers} of $J$, and note that they clearly satisfy \eqref{Euler-Lagrange equation}, under appropriate regularity hypothesis on $W$.

In relation to the phase transition interpretation, an interesting and difficult problem is the existence of multi-phase solutions, that are solutions with the following geometrical properties:

\emph{There is an $\hat{N}\in \mathbb{N}$ with $2\leq \hat{N}\leq N$, $\hat{a}_1,...,\hat{a}_{\hat{N}}\in A$, a small $\g>0$ and open sets $\Om_1,...,\Om_{\hat{N}}$ such that}
\beq\label{partition by various phases}
\BR^n=I\cup\left( \bigcup\limits_{j=1}^{\hat{N}}\Om_j \right)
\eeq
\emph{with $I$ being a set of thickness $O(1)$ and}
\beq\label{in each phase}
|u(x)-\hat{a}_j|\leq \g,\quad\forall x\in \Om_j, \text{ for some small }\g
\eeq
The set $I$ plays the role of a diffuse interface that separates the coexisting phases. Understanding the geometrical structure of this diffuse interface is a major point in the study of such solutions \cite{F}.

In the scalar case $m=1$, for $W\in C^2(\BR^m,\BR)$, $a_i$ nondegenerate (i.e. $\f{\pa^2 W}{\pa u^2}>0$ at the minima), and for $N=2$ which is the natural choice, there is a rich literature and many important results. We list some of these works and organize them in two groups: papers that address various general aspects (see \cite{ft,ks,gurtin,sternberg,cc,cc2,bcn,modica1,modica2,farina}); and papers that are motivated by a celebrated conjecture of De Giorgi (see \cite{degiorgi,mm,gg,ac,s,wei,ct,dkw0}) where a relationship of $I$ with minimal surfaces, and in particular with hyperplanes in low dimensions, is established. The reader could also consult the expository papers \cite{fv,savin,s0,dkw}.

In the vector case $m\geq 2$ and $N\geq 3$, the structure of $I$ is not expected to be planar in any dimension $n\geq 2$ but rather linked to the minimal cones\footnote{The complete classification of the minimal cones in $\BR^n$ is known only for n=3 \cite{taylor}. We refer to Section 7 in the expository paper of David \cite{david} and references therein.} in $\BR^n$. This kind of solutions are called \emph{junctions} and have been shown to exist for $n=m=2$ and $n=m=3$ with $\hat{N}=N=3$ and $\hat{N}=N=4$ respectively, but so far only under symmetry hypotheses. Specifically, the existence is established for:
\begin{itemize}
\item $n=2$ with respect to the symmetries of the equilateral triangle \cite{bgs},
\item $n=3$ with respect to the symmetries of the tetrahedron \cite{gs},
\end{itemize}
and with \eqref{def of minimizer} verified only in their respective equivariance classes. We refer to \cite{book-afs} where the symmetric case is covered in detail for general reflection point groups and also for lattices.

For subquadratic potentials that behave like $|u-a|^\al$ near $a\in A$, $\al\in(0,2)$, the interface is less diffuse as we explain below. More precisely we consider potentials that satisfy

\begin{enumerate}
\itemsep0.6em
   \item[H1.] $W\in C(\BR^m, [0,\infty))\cap C^2_{loc}(\BR^m\backslash A)$ with $\{W=0\}=A=\{a_1,\dots,a_N\}$, $N\geq 2$.

   \item[H2.] Set
   \beq\label{def of r0}
   r_0:=\f12\min\limits_{1\leq i\neq j\leq N}|a_i-a_j|.
   \eeq
 For any $a_i\in A$, $W(u)$ can be written as
       \beq\label{formula for W near ai}
       W(u)=|u-a_i|^\al g_i(u) \;\;\text{ in }u\in B_{r_0}(a_i)
       \eeq
       for some function $g_i(u)\in C^2(B_{r_0}(a_i), [c_i,\infty))$ where $c_i$ is a positive constant.
\end{enumerate}

For such subquadratic potentials, \eqref{in each phase} is replaced by
\beq\label{in each phase free bdy}
u(x)=\hat{a}_j,\quad\forall  x\in \Om_j.
\eeq
Indeed, in this case the entire minimizer possesses a free boundary and the phases $\hat{a}_j$ are attained \cite{agz}, while for $\al=2$, the solution converges exponentially at infinity to the phases. The subquadratic assumption can be thought as a reduction which simplifies without changing the essential features of this type of solution. In a suitable limit $\al\ri0$, \eqref{AC functional} becomes the Alt-Caffarelli functional.

We now define the appropriate analog of the set $I$ in \eqref{partition by various phases} for vector analog of the subquadratic potentials.
\begin{definition}\label{diffuse interface}
Let $0<\g_0<\f12\min\limits_{i\neq j}|a_i-a_j|$ fixed. Set
\beq\label{def of delta}
\d(x)=\dist(u(x),\{W=0\})
\eeq
where $\dist$ stands for the Euclidean distance. Let $0<\g<
\g_0$ and assume $\g_0<\sup\limits_{\BR^n} \d(x)$. We define the set
\beq\label{def of diffuse interface}
I_\g=\{x\in\BR^n:\d(x)\geq \g\}.
\eeq
\end{definition}

For entire minimizers satisfying $\|u\|_{L^\infty(\BR^n,\BR^m)}<\infty$, $\|\na u\|_{L^\infty}(\BR^n,\BR^m)<\infty$, and $0<\al\leq 2$, the following estimate is known (see \cite[Lemma 5.5]{book-afs})
\beq\label{est of I_g}
c_1(\g)r^{n-1}\leq \cl(I_\g\cap B_r(x_0))\leq c_2(\g)r^{n-1},\quad r\geq r(x_0),
\eeq
where $x_0\in\BR^n$ is arbitrarily chosen, $r(x_0)$ is a positive constant depending on $x_0$, and the constants $c_i(\g)>0 \,(i=1,2)$ is independent of $x_0,\,r$.

Clearly $I_{\g_1}\subset I_{\g_2}$ if $\g_1>\g_2$ and we define
\beq\label{I_0}
\lim\limits_{\g\ri 0} I_\g=I_0=\{x\in\BR^n: \d(x)>0\}=: \text{Diffuse Interface}.
\eeq

The constants $c_1(\g),\, c_2(\g)$ in \eqref{est of I_g} degenerate as $\g\ri 0$ (see \cite{book-afs}) and so no useful information can be obtained for $I_0$ out of \eqref{est of I_g}.

Our main result in the present paper concerns certain global facts on $I_0$ that can be summarized in the following theorem.

\begin{theorem}\label{main}
Let $\al\in(0,2)$, $u:\BR^n\ri\BR^m$ be a bounded entire minimizer with $I_{\g_0}\neq \varnothing$ (i.e. $u\not\equiv a_i$), and assume H1, H2 above. Then there exists a radius $r_0>0$ and positive constants $c,\,c_1,\,c_2$, which only depend on $u$ but not on $r$, such that for $r\geq r_0$ the following estimates hold:
\begin{align}
\label{measure of I_0} &\cl^n(I_0\cap B_r(0^n))\leq cr^{n-1},\\
\label{free bdy lower bdd}  &\ch^{n-1}(\pa^* I_0\cap B_r(0^n))\geq c_1r^{n-1},
\end{align}
where $\pa^*$ denotes the De Giorgi reduced boundary.

If furthermore $\al=1$ we have the upper bound estimate
\beq\label{est free bdy}
\ch^{n-1}(\pa^* I_0\cap B_r(0^n))\leq c_2r^{n-1}.
\eeq
\end{theorem}

The analogs of the ``minimal cone" solutions in $\BR^2$ and $\BR^3$ mentioned above as well as the cylindrical triple junction cone in $\BR^3$  have been shown to exist also for $0\leq \al<2$ and possess free boundary (see \cite[Theorem 1 and Proposition 4]{agz})\footnote{For $\al=0$, $J(u)=\int\f12|\na u|^2+\chi_{\{u\in S_A\}}$, where $S_A$ denotes the interior of the convex hull of $A=\{a_1,...,a_N\}$. It is the vectorial analog of the Alt-Caffarelli functional. }. Combining Theorem \ref{main} above with that result we obtain the following
\begin{corol}\label{corol with symmetry}
Under H1 and H2, $0\leq \al <2$, for the equivariant triangle junction in $\BR^2$ and the equivariant quadruple junction in $\BR^3$ the estimates \eqref{measure of I_0}, \eqref{free bdy lower bdd} and \eqref{est free bdy} hold for any $r\geq r_0>0$.
 \end{corol}

\begin{Remark}
We note that the assumption on the existence of minimizers as above is not restrictive as such nonconstant entire minimizers $U:\BR^1\ri\BR^m$ have been shown to exist in \cite{ms} under the hypothesis of continuity on $W$ together with the mild condition
\beq\label{mild condition}
\sqrt{W(u)}\geq f(|u|),\text{ for some nonnegative }f:(0,+\infty)\ri\BR\text{ s.t. }\int_0^{+\infty}f(r)\,dr=+\infty,
\eeq
that allows even decay to zero of the potential at infinity.

A more convenient sufficient condition for the existence of such nonconstant entire minimizers (called \emph{connections}) is (see \cite{fgn})
\beq\label{condition at infty}
\liminf\limits_{|u|\ri\infty} W(u)>0.
\eeq
\end{Remark}

In \cite{cc}, for the scalar two phase problem, the entire range of potentials $F_0=\chi_{\{|u|<1\}}$, $F(u)=(1-u^2)^{\al/2}$ ($0<\al\leq 1$) was already introduced. As $\al\ri 0$, the minimizers get increasingly localized. In particular for $\al=0$ the connections are affine functions. For these reasons we expect the construction of entire minimizers mentioned in Corollary \ref{corol with symmetry} above, under no symmetry hypotheses, to be more accessible for singular potentials.

From the point of view of regularity, our problem can be regarded as a generalization of the more simplified model which studies the minimization problem of the functional
\beq\label{simplified model}
\int_{\Om}\left( \f12|\na u|^2+|u|^\al\right)\,dx,\quad u:\BR^n\ri\BR^m, \; \al\in(0,2).
\eeq
In the scalar case, i.e. $m=1$, one recovers the two phase free boundary problem
\beqo
\D u= \al(u^+)^{\al-1}-\al(u^-)^{\al-1}.
\eeqo
which has been extensively studied under various conditions and settings, see for example \cite{fs} for the case $\al\in(1,2)$ and \cite{lp} for the 2D problem with $\al\in (0,1)$. One can also refer to \cite{lqt,ls} for the optimal regularity theory for a functional with a more general form of the potential $W$. When $\al=1$, the problem becomes the so-called two-phase obstacle-type problem. The regularity of the solution as well as the free boundary regularity has been summarised in detail in \cite{book-psu}.

For the vector-valued case, i.e. $m\geq 2$, the minimization problem \eqref{simplified model} was investigated in \cite{asuw} for $\al=1$ and \cite{fsw} for $\al\in(1,2)$, where the authors studied the regularity of the minimizers and the asymptotic behavior for the minimizer near the ``regular" point of the free boundary. Also we would like to mention the works \cite{csy,mtv} studying the vector-valued Bernoulli free boundary problem, where the potential function $W(u)$ takes the form $Q^2(x)\chi_{\{|u|>0\}}$. Such a problem is quite close to the $\al=0$ case of the functional \eqref{simplified model}. All these works mainly focus on the behavior and regularity of the free boundary

The biggest difference between our functional \eqref{AC functional} and the simplified one \eqref{simplified model} is that the potential $W(u)$ possesses more than one global minimum. Furthermore, our emphasis is not so much on the local behavior of the free boundary, but on its asymptotic behavior on the large domain $B_R(0^n)$ as $R\ri \infty$
We note that usually the existence of the free boundary is forced by the Dirichlet boundary condition given on $\pa\Om$. However, in this paper we do not assume any boundary condition and the existence of the free boundary results from the assumption that $u$ is an entire bounded non-constant minimizer.

To prove the main Theorem \ref{main}, we first show in Section \ref{section 2} that the Euler-Lagrange equation is satisfied by the minimizer $u$ via the regularity of the solution. Then using the regularity property and a non-degeneracy lemma, we prove the first part of the theorem (upper bound estimate for the $\cl^n$ measure of $I_0$) by introducing a method of dividing $B_R$ into identical smaller sub-cubes and classifying all the cubes according to how much measure of the ``contact set" they contain. This is done in Theorem \ref{first part main theorem} from Section \ref{section 3}. We also demonstrate in Theorem \ref{two phase existence} the coexistence of at least two phases for the minimizer at each large scale. In Section \ref{growth est} we derive a Weiss-type formula and then give a growth estimate for the minimizer near the free boundary. These results from Section \ref{growth est} are used in Section \ref{section 5}, where we estimate the $\mathcal{H}^{n-1}$ measure of the free boundary when $\al=1$ and prove the second part of Theorem \ref{main}. Note that currently our method of estimating the $\ch^{n-1}$ measure of the free boundary from above cannot be generalized for arbitrary $\al\in (0,2)$, and we hope to solve this problem in the future.

\section{Regularity of the minimizer}\label{section 2}

We first prove the optimal regularity of the entire minimizer $u$ for $\al\in(0,1)$.

\begin{proposition}\label{reg 0<al<=1}
Suppose $0<\al<1$ and let $u$ be an entire minimizer of the energy functional $J$ satisfying $|u(x)|\leq M$. Assuming H1, H2. Then we have
\beqo
u\in C_{loc}^{1,\beta}(\BR^n,\,\BR^m),
\eeqo
where $\beta$ is a constant defined by $\beta=\f{\al}{2-\al}$. In particular, this $C^{1,\b}$ regularity is sharp.
\end{proposition}

\begin{rmk}
We would like to note that in the following proof, unless being specifically stated, all the constants denoted by $C$ only depend on the upper bound $M$, the potential function $W$ and the dimensions $m,\, n$.
\end{rmk}

\begin{proof}
For any ball $B_R(x)\subset \BR^n$, let $v$ be the harmonic function (which means each $v_i$ is harmonic) in $B_R(x)$ such that $u=v$ on $\pa B_R$. Since $v$ is harmonic, then $\na v$ satisfies a Campanato type growth condition
\beq\label{campanato}
\int_{B_\rho}|\na v-(\na v)_\rho|^2\,dx\leq C(\f{\r}{R})^{n+2}\int_{B_R}|\na v-(\na v)_R|^2\,dx
\eeq

For $\r<R$, we deduce
\beq\label{decay of nabla u}
\begin{split}
\int_{B_\r}|\na u-(\na u)_\r|^2\,dx&\leq C\int_{B_\r}|\na v-(\na v)_\r|^2\,dx+C\int_{B_R}|\na u-\na v|^2\,dx\\
&\leq C(\f{\r}{R})^{n+2}\int_{B_R}|\na v-(\na v)_R|^2\,dx+C\int_{B_R}|\na u-\na v|^2\,dx
\end{split}
\eeq
For the second term, we use the minimization property to obtain
\beq\label{mini}
\int_{B_R}|\na u-\na v|^2\,dx= \int_{B_R}\left(|\na u|^2-|\na v|^2\right)\,dx
\leq\int_{B_R}2\left(W(v)-W(u)\right)\,dx.
\eeq

By hypothesises H1, H2 and $\|u\|_{L^\infty}\leq M$, we can easily verify that $W$ can be written as
\beq\label{formula for W}
W(u)=\left(\prod\limits_{i=1}^N |u-a_i|^\al \right)g(u),
\eeq
where $g$ is a function such that
\beq\label{condition on g}
g\in C^2(\overline{B}_M), \quad g(u)\geq C \text{ for some constant }C>0.
\eeq

When $\al\in(0,1)$, since both $v$ and $u$ are uniformly bounded, we compute
\beq\label{diff w alpha}
\begin{split}
& W(v)-W(u)\\
=&\left(\prod\limits_{i=1}^N|v-a_i|^\al\right) g(v)-\left(\prod\limits_{i=1}^N|u-a_i|^\al\right) g(u)\\
\leq&\left(\prod\limits_{i=1}^N(|u-a_i|^\al+|u-v|^\al)\right)\left(g(u)+C|u-v|\right)-\left(\prod\limits_{i=1}^N|u-a_i|^\al\right) g(u)\\
\leq & C|u-v|^\al,
\end{split}
\eeq
where $C$ is a constant only depending on $W, M, n, m$.
By H\"{o}lder, Poincar\'{e} and Young, we have
\begin{align}
\non&\int_{B_R}|u-v|^\al\,dx \\
\non\leq & C(\int_{B_R} |u-v|^2\,dx)^{\f{\al}{2}} R^{n(1-\f{\al}{2})}\\
\label{thirdline}\leq & C(\int_{B_R}|\na(u-v)|^2\,dx)^{\f{\al}{2}}R^{n(1-\f{\al}{2})+\al}\\
\non\leq& \delta \int_{B_R}|\na(u-v)|^2\,dx+C(\delta)R^{n+\f{2\al}{2-\al}}
\end{align}
where $\delta$ is a suitably chosen small number. Combining \eqref{mini}, \eqref{diff w alpha} and \eqref{thirdline}, we obtain
\beq\label{decay of nabla u-v}
\int_{B_R}|\na u-\na v|^2\,dx\leq CR^{n+2\beta},\quad \beta:=\f{\al}{2-\al}
\eeq
Therefore from \eqref{campanato}, \eqref{decay of nabla u} and \eqref{decay of nabla u-v} we get the following Campanato type decay estimate for the minimizer $u$
\beq\label{ucampanato}
\int_{B_\rho}|\na u-(\na u)_\r|^2\,dx\leq c(\f{\r}{R})^{n+2}\int_{B_R}|\na u-(\na u)_R|^2\,dx+CR^{n+2\beta}
\eeq
By a standard iteration argument we conclude that there exists a constant $C$ such that
\beq\label{decayrate}
\int_{B_\rho}|\na u-(\na u)_\r|^2\,dx\leq C\r^{n+2\beta},
\eeq
which by the Morrey-Campanato theory implies $u\in C^{1,\beta}$.

To show that this $C^{1,\beta}$ estimate is sharp, we need some results that will be proved later. Firstly by Theorem \ref{first part main theorem} and its proof, we know there exists $a_i\in A$ such that $\cl^n(\{u(x)=a_i\})>0$ and $\{u(x)=a_i\}$ contains interior points. Let $x_1\in\{u(x)=a_i\}$ be an interior point, and define
\beqo
r_1:=\sup \{r>0:  B_r(x_1)\subset \{u(x)=a_i\}\}.
\eeqo
When $u\not\equiv a_i$, we know that $r_1\in(0,\infty)$ and there is a point $x_2\in \pa B_{r_1}(x_1)$ such that $u(x_2)=a_i$ and $x_2\in \overline{\{|u-a_i|>0\}}$. For $r$ sufficiently small, by Lemma \ref{nondeg} we have
\beq\label{nondeg optimal reg}
\sup\limits_{B_r(x_2)}|u-a_i|\geq cr^{1+\beta}.
\eeq
Also we claim that $\na u(x_2)=0$. Firstly, by $u(x)\equiv a_i$ for any $x\in B_r(x_2)\cap B_{r_1}(x_1)$, we can easily check that $\pa_\nu u(x_2)=0$ where $\nu$ denotes the normal vector on $\pa B_{r_1}(x_1)$. On the other hand, take any unit vector $\mu$ which points to one of the tangential directions at $x_2$ on $\pa B_{r_1}(x_1)$. For any $|t|\leq r_1$,
\beqo
x_2+t\mu+(\sqrt{r_1^2-t^2}-r_1)\nu\in\pa B_{r_1}(x_1),\quad u(x_2+t\mu+(\sqrt{r_1^2-t^2}-r_1)\nu)=a_i.
\eeqo
Thus for any $i\in\{1,2,...,m\}$ we have
\beqo
0=\f{d}{dt}\bigg|_{t=0}u_i(x_2+t\mu+(\sqrt{r_1^2-t^2}-r_1)\nu)=\na u_i(x_2)\cdot \mu.
\eeqo
Since $\mu$ is an arbitrary tangential direction, the claim $\na u(x_2)=0$ is proved.

Finally, the vanishing of $\na u(x_2)$ together with \eqref{nondeg optimal reg} imply the sharpness of the $C^{1,\beta}$ regularity.

\end{proof}

\begin{rmk}\label{c1,alpha remark}
For $1\leq \al<2$, by exactly the same argument we can also prove $u\in C^{1,\g}_{loc}$ for any $\g\in(0,1)$. Just notice that we need to replace \eqref{diff w alpha} with $W(u)-W(v)\leq C|u-v|$. When $\al=1$, in contrast to the scalar case problem (see \cite{sha}), it is still open whether $u\in C^{1,1}_{loc}$.
\end{rmk}

Now with the regularity result we can identify the Euler-Lagrange equation for the entire minimizer $u$ in the following lemma.

\begin{lemma}\label{EL}
Let $u$ be an entire minimizer of the functional $J$ satisfying $|u(x)|\leq M$. Assume H1, H2. Then we have

\begin{enumerate}
\itemsep0.5em
\item $1<\al<2$, $u$ is a strong solution of
\beq\label{eleq for 1<al<2}
\D u=W_u(u),\qquad \forall x\in\BR^n.
\eeq
\item $\al=1$, $u$ is a strong solution of
\beq\label{eleq for al=1}
\D u=W_u(u)\chi_{\{d(u,A)>0\}}.
\eeq
\item $0<\al<1$, in the open set $\{x: d(u(x),A)>0\}$, $u$ solves the equation \eqref{eleq for 1<al<2}.
\end{enumerate}
Here $W_u(u)$ denotes the derivative of $W$ with respect to $u$, $\chi$ is the characteristic function and $d(x,A):=\dist(x,A)$.

\end{lemma}

\begin{proof}
Take $D\in\BR^n$ to be an arbitrary bounded Lipschitz domain, and for $\phi\in C_0^\infty(D,\BR^m)$, we compute the first variation of the energy $J_D(u)$
\beq\label{first variation}
\begin{split}
0&\leq \int_{D} \left(\f12|\na (u+t\phi)|^2-\f12|\na u|^2+W(u+t\phi)-W(u)\right)\,dx\\
& = t\int_{D}\na u\cdot \na \phi\,dx+\f{t^2}{2}\int_{D}|\na\phi|^2\,dx+\int_{D}\left(W(u+t\phi)-W(u)\right)\,dx.
\end{split}
\eeq

When $1<\al<2$, $W$ can be written as \eqref{formula for W} with the condition \eqref{condition on g}. One can directly compute
\beq\label{formula W_u(u)}
W_u(u)=\left( \al(u-a_i)|u-a_i|^{\al-2}\prod\limits_{k\neq i}|u-a_k|^\al \right)g(u)+\left(\prod\limits_{i=1}^N |u-a_i|^\al \right)D_ug(u).
\eeq
Since we already proved $u\in C^{1,\g}(D)$, it is obvious that $W_u(u)\in C(D)$. Therefore when we divide \eqref{first variation} by $t$ and take $t\ri 0+$ or $0-$, it follows that $u$ solves \eqref{eleq for 1<al<2} in $D$.

For $\al=1$, dividing \eqref{first variation} by $t$ and letting $t\ri 0$, we can prove that
\beqo
\left|\int_{D}\na u\cdot \na \phi\,dx\right|\leq C\int_{D}|\phi|\,dx,
\eeqo
which implies $\D u\in L^\infty(D,\BR^m)$. Moreover, on any sub-domain $K\subset D\cap \{d(u,A)>0\}$, the equation \eqref{eleq for 1<al<2} holds. Combining the fact that $\D u=0$ a.e. on $\{x:u(x)\in A\}$ (which is due to the fact that weak derivatives for Sobolev functions vanish on level sets a.e. \cite{book-eg}), we have (since $u$ is continuous) that $u$ is a strong solution of \eqref{eleq for al=1} in $D$.

For $0<\al<1$, one can easily check that $u$ solves \eqref{eleq for 1<al<2} in the open set $\{x: d(u,A)>0\}$. However, since $W_u(u)$ blows up as $d(u,A)\ri 0$, the local integrability of $W_u(u)$ is not a priori known, so $u$ may not solve \eqref{eleq for al=1} on the whole space in distributional sense.
\end{proof}

\begin{rmk}
In the case $0<\al<1$, even though we cannot say $u$ is a distributional solution to \eqref{eleq for al=1}, we can deduce another equation for $u$ from the first domain variation. For any $\varphi\in C_0^1(D,\BR^n)$, it holds that
\begin{align*}
0&=\f{d}{dt}J_{D}(u(x+t\varphi(x)))\\
&=\int_D \left((\na u\na\va)\cdot \na u-(\di\va)(\f12|\na u|^2+W(u))\right)\,dx
\end{align*}
This formulation has also been utilized in \cite{lp,Weiss3}. Here we just present this form of equation for completeness and it will not be utilized in the rest of the paper.
\end{rmk}

With the Euler-Lagrange equation \eqref{eleq for 1<al<2} and the formula for $W_u(u)$ \eqref{formula W_u(u)}, we can easily improve the regularity of $u$ when $1<\al<2$.

\begin{proposition}\label{reg al>1}
When $1<\al<2$, the entire minimizer $u\in C^{2,\al-1}_{loc}(\BR^n,\BR^m)$.
\end{proposition}

\begin{proof}
According to Lemma \ref{EL}, $u$ satisfies the equation \eqref{eleq for 1<al<2} when $1<\al<2$. Using the formula \eqref{formula W_u(u)} for $W_u(u)$ and the rough estimate $u\in C^{1,\gamma}_{loc}$, we see that $W_u(u)\in C^{\al-1}_{loc}(\BR^n,\BR^m)$. Then the $C^{2,\al-1}_{loc}$ regularity immediately follows from the classical Schauder estimate.
\end{proof}

\section{Estimate of $\cl^n(I_0)$ and existence of the free boundary}\label{section 3}
In this section we will prove the estimate \eqref{measure of I_0} for any nontrivial entire minimizer $u$ of the functional $J$. Furthermore, we also show that at every sufficiently large scale, the minimizer $u$ must contain at least two different phases.

Take $W$ satisfying the hypothesises H1 $\&$ H2 and assume $u$ is an entire minimizer for the functional $J$. Before stating our new results, we first recall two estimates from \cite{book-afs} and \cite{agz} without proof, which will play an important role in our arguments. Readers can refer to \cite{book-afs} and \cite{agz} for detailed proofs. These estimates for the scalar case $m=1$ are obtained in \cite{cc}.

\begin{proposition}\label{two estimates}
When $0<\al<2$, for any entire minimizer $u$ satisfying $\|u\|_{L^\infty(\BR^n)}<\infty$, the following two estimates hold true:
\begin{enumerate}
\itemsep0.5em
\item \underline{The basic estimate} (see \cite[Lemma 2.2]{agz})
For any $x_0\in\BR^n$, there exists an $r_0$ such that for $r>r_0$,
\beq\label{basic est}
J_{B_r(x_0)}(u)\leq Cr^{n-1},
\eeq
where the constant $C=C(M)$ is independent of $u$. We notice that this $r_0$ can be $0$ when $\al\in[1,2)$.
\item \underline{The density estimate} (see \cite[Theorem 5.2]{book-afs}) Take $a\in A$ to be a minimal point for $W(u)$. If for some $r_0,\,\lam,\,\mu_0>0$,
    \beqo
    \mathcal{L}^n(B_{r_0}(x)\cap \{|u-a|>\lam\})\geq \mu_0,
    \eeqo
    then there exists a constant $C(\mu_0,\lam)>0$ such that
    \beq\label{density est}
    \mathcal{L}^n(B_{r}(x)\cap \{|u-a|>\lam\})\geq C(\mu_0,\lam) r^n, \quad \forall r\geq r_0.
    \eeq
\end{enumerate}
\end{proposition}

Another important component of our arguments is the following non-degeneracy lemma.

\begin{lemma}\label{nondeg}
Assume $0<\al<2$. We take the point $a_1\in A$ and an entire minimizer $u$ for the functional J such that $\|u\|_{L^\infty(\BR^n)}<\infty$. There exists a suitably small number $\theta(W)$ and a constant $c=c(n,W)$, such that if $x_0\in\overline{\{0<|u-a_1|<\theta\}}$ and $B_r(x_0)\subset\overline{\{|u-a_1|<\theta\}}$, then
\beq\label{nondegeneracy}
\sup\limits_{B_r(x_0)}|u-a_1|\geq c(n,W)r^\f{2}{2-\al},
\eeq
where the constant $c(n,W)$ only depends on the dimension $n$ and the potential function $W$. Moreover $c(n,W)\sim O(\alpha)$ for $\alpha<<1$.
\end{lemma}

\begin{proof}
Without loss of generality, suppose $a_1=0^m$. First we require that
\beqo
\theta<\f12\min\limits_{i\neq j}|a_i-a_j|=r_0.
\eeqo
By (H2), when $|u|<r_0$, $W(u)$ can be written as $W(u)=|u|^\al\cdot g(u)$ for some $g(u)\in C^2(B_{r_0}(0^m))$ satisfying \beqo
g(u)\geq C_g
\eeqo
for some constant $C_g>0$.

Assume $|u(x_0)|>0$ (if $|u(x_0)|=0$, then we simply take a sequence of points $\{x_i\}$ converging to $x_0$ and satisfy $|u(x_i)|>0$). Taking $h(x)=|u|^{2-\al}-c|x-x_0|^2$ for some constant $c$ which will be determined later, by direct calculation we have that if $|u(x)|>0$, then
\beq\label{laplaceh}
\D h=(2-\al)(-\al)\f{\left|\na|u|\right|^2}{|u|^\al}+(2-\al)\f{|\na u|^2}{|u|^\al}+(2-\al)u\cdot D_u(g)+\al(2-\al)g(u)-2nc
\eeq
If we take
\beq\label{condition on theta, c}
\theta<\min\{\f{\al C_g}{4\|D_ug(u)\|_{L^\infty(B_{r_0}(0^m))}}, \f12\min\limits_{i\neq j}|a_i-a_j| \},\quad c<\f{\al(2-\al)C_g}{8n},
\eeq
then \eqref{laplaceh} implies that
\beqo
\Delta h\geq -(2-\al)\al \f{\left|\na|u|\right|^2}{|u|^\al}+(2-\al)\f{\left|\na u\right|^2}{|u|^\al}
\eeqo

When $\al\leq 1$, it follows that $\Delta h\geq 0$ in $\{|u(x)|>0\}\cap B_r(x_0)$. Since $h(x_0)>0$ and $h(x)<0$ on $\pa\{|u(x)>0|\}\cap B_r(x_0)$, we must have
\beqo
\max\limits_{x\in\pa B_r(x_0)}h(x)>0,
\eeqo
which implies the lemma.

For $\al\in(1,2)$, combining \eqref{laplaceh} and \eqref{condition on theta, c} we deduce that in $\{|u|>0\}\cap B_{r}(x_0)$
\begin{align*}
&\D h+(2-\al)\al\f{\left|\na|u|\right|^2}{|u|^\al}-(2-\al)\f{\left|\na u\right|^2}{|u|^\al}\geq \f{\al(2-\al)C_g}{4}\\
\Rightarrow & \D h+\f{\al}{2-\al}\f{\na h\cdot \na(|u|^{2-\al}+c|x-x_0|^2)+4c^2|x-x_0|^2}{|u|^{2-\al}}\geq (2-\al)\f{\left|\na u\right|^2}{|u|^\al}+\f{\al(2-\al)C_g}{4}\\
\Rightarrow & \D h+\left( \f{\al\na(h+2c|x-x_0|^2)}{(2-\al)|u|^{2-\al}}  \right)\cdot\na h+\f{4c\al(-h+|u|^{2-\al})}{(2-\al)|u|^{2-\al}}\geq \f{\al(2-\al)C_g}{4}\\
\Rightarrow & \D h+\left( \f{\al\na(h+2c|x-x_0|^2)}{(2-\al)|u|^{2-\al}}  \right)\cdot\na h-\f{4c\al}{(2-\al)|u|^{2-\al}}\cdot h\geq 0.
\end{align*}
Here to derive the last inequality we further require that $c$ satisfies
\beq\label{condition on c}
c\leq \f{(2-\al)^2C_g}{16}.
\eeq
Then the maximum principle argument can be applied again to get
\beqo
\max\limits_{x\in\pa B_r(x_0)} |u(x)|> cr^\f{2}{2-\al}.
\eeqo
This completes the proof.
\end{proof}
\vspace{5mm}

Now we are ready to prove the first part \eqref{measure of I_0} of the Theorem \ref{main}. For the sake of convenience, we rewrite the statement in the following theorem.

\begin{theorem}[First part of Theorem \ref{main}]\label{first part main theorem}
Let $x_0\in\BR^n$, $u: \BR^n\ri \BR^m$ be a bounded nonconstant entire minimizer of the energy $J$. Then there are positive constants $R_0$ and $c$ such that
\beq\label{I_0 est}
\cl^n(B_R(x_0)\cap I_0)\leq cR^{n-1},\quad R>R_0,
\eeq
where $I_0$ is defined in \eqref{I_0}, which is the region where $W(u)>0$. The constant $c$ only depends on the dimension $n$, the potential function $W$ and $\|u\|_{L^\infty(\BR^n)}$.
\end{theorem}

\begin{proof}
Without loss of generality, suppose $x_0=0^n$ and write $B_R=B_R(0^n)$. According to the basic estimate \eqref{basic est} in Proposition \ref{two estimates}, we know that there exist positive constants $C_0, \,r_0$ such that for any $R>r_0$
\begin{equation}
\label{energy density 0}\int_{B_R} \f12|\na u|^2+W(u)\,dx\leq C_0R^{n-1}.
\end{equation}

For the sake of convenience, we use the cubes which are centered at $0^n$ to replace $B_R$. Define
\beqo
\tilde{S}_R:=\{x\in \BR^n: x_i\in(-R,R),\text{ for }i=1,2,...,n\}.
\eeqo
Let $L$ be a constant whose value will be specified later. For any positive integer $k$,  we can divide the cube $\tilde{S}_{kL}$ into $(2k)^n$ identical cubes with the side length $L$. We number all these sub-cubes by $S_1,S_2,...,S_{K}$ where $K:=(2k)^n$. And we take $\theta$ to be the constant $\theta(W)$ in the Lemma \ref{nondeg}. Then we define
\beq\label{def sigma i,j}
\sigma_i^j:=\cl^n(\{|u-a_j|<\f{\th}{2}\}\cap S_i), \quad \text{for }i=1,...K,\; j=1,..., N.
\eeq

Take $\e:=\e(\theta)$ to be a small constant to be specified later, depending only on $\theta$ and $\|u\|_{L^\infty}$. Also we introduce the notion of adjacent sub-cubes: $S_{i_1}$ and $S_{i_2}$ are called adjacent if and only if
\beqo
\overline{S_{i_1}}\cap \overline{S_{i_2}}\neq \varnothing,\quad  1\leq i_1,\,i_2\leq K
\eeqo
We divide $\{S_i\}_1^K$ into the following five non-overlapping classes
\begin{itemize}
\itemsep0.5em
\item[1] Boundary sub-cubes of $\tilde{S}_{kL}$:
\beqo
T_1:=\{S_i:\text{ the number of adjacent cubes of }S_i \text{ is less than }3^n-1\}.
\eeqo
\item[2] Sub-cubes that contain two phases:
\beqo
T_2:=\{S_i: \exists j_0 \text{ s.t. } \s_i^{j_0}=\max\limits_{1\leq j\leq N}\s_i^j\leq(1-2\e)L^n, \;\max\limits_{j\neq j_0} \s_i^j\geq\f{\e}{N-1} L^n\}\backslash T_1.
\eeqo
\item[3] Sub-cubes that contain regions where $u$ stays away from any $a_j$.
\beqo
T_3:=\{S_i: \exists j_0  \text{ s.t. } \s_i^{j_0}=\max\limits_{1\leq j\leq N}\s_i^j\leq(1-2\e)L^n, \;\max\limits_{j\neq j_0} \s_i^j<\f{\e}{N-1} L^n\}\backslash T_1.
\eeqo
\item[4] ``Interior" sub-cubes of the contact set $\{x:u(x)\in A\}$:
    \beqo
    T_4:=\{S_i: \exists j_0 \text{ s.t. } \s_i^{j_0}>(1-2\e)L^n \text{ and }\s_p^{j_0}> (1-2\e)L^n,
     \forall S_p\text{ adjacent to }S_{i}\}\backslash T_1.
     \eeqo
\item[5] Sub-cubes close to the boundary of the contact set:
\beqo
T_5:=\{S_i: \exists j_0 \text{ s.t. } \s_i^{j_0}> (1-2\e)L^n \text{ and }\s_p^{j_0}\leq  (1-2\e)L^n,\text{ for some } S_p\text{ adjacent to }S_{i}\}\backslash T_1.
\eeqo
\end{itemize}

Now we estimate the number of cubes in each class. First note that $S_i\in T_1$ means $S_i$ is one of the boundary cubes of $\tilde{S}_{kL}$, therefore
\beq\label{number of T_1}
|T_1|\leq c_0(n)k^{n-1} \quad \text{ for some dimensional constant }c_0(n).
\eeq
For a cube $S_i\in T_2$, assume $\s_i^{j_0}=\max\limits_{1\leq j\leq N}\s_i^j\leq (1-2\e)L^n$ and $\s_i^{j_1}\geq \f{\e}{N-1}L^n$ for some $j_1\neq j_0$. By the definition of $\s_i^j$ and $\theta<\f12|a_{j_0}-a_{j_1}|$, we can infer that for any $r\in[\f{\th}{2},\,|a_{j_0}-a_{j_1}|-\f{\th}{2}]$, it holds that
\beqo
\cl^n(\{|u-a_{j_0}|<r\}\cap S_i)\geq  \f{\e}{N-1} L^n,\quad \cl^n(S_i\backslash \{|u-a_{j_0}|<r\})\geq \f{\e}{N-1} L^n.
\eeqo
Applying the co-area formula and the relative isoperimetric inequality (see for example \cite{Thomas}), we have
\begin{equation}\label{co-area0}
\begin{split}
&\int_{S_i}|\na u|^2\,dx\\
\geq & \f{1}{L^n} \left(\int_{S_i} |\na(u-a_{j_0})|\,dx\right)^2\\
\geq &\f{1}{L^n} \left( \int_{\th/2}^{|a_{j_0}-a_{j_1}|-\th/2} \ch^{n-1}(\{|u-a_{j_0}|=r\}\cap S_i)\,dr \right)^2\\
\geq & \f{1}{L^n} \left( \int_{\th/2}^{|a_{j_0}-a_{j_1}|-\th/2} C \left(\min\{\cl^n(\{|u-a_{j_0}|<r\}\cap S_i),\cl^n(S_i\backslash \{|u-a_{j_0}|<r\}) \}\right)^{\f{n-1}{n}} \,dr \right)^2\\
\geq &c_1(L,\th,\e)>0
\end{split}
\end{equation}
From the basic estimate \eqref{basic est} we get
\beq\label{number of T_2}
|T_2|\leq \f{C(kL)^{n-1}}{c_1}= c_2(L,\th,\e)k^{n-1},\quad k\geq k_0,
\eeq
where $k_0$ is a constant.

For a cube $S_i\in T_3$,
\beqo
\cl^n\left(\left\{|u-a_j|>\f{\theta}{2},\;\forall 1\leq j\leq N\right\}\cap S_i\right)>\e L^n
\eeqo
By the Hypothesis H1 and H2 on $W$ and the assumption $\|u\|_{L^\infty}<\infty$, there is a constant $c_3$ which depends on $\|u\|_{L^\infty},\theta$ such that
\beqo
W(u)\geq c_3,\quad \text{when }|u-a_j|>\f{\theta}{2},\; \forall 1\leq j\leq N
\eeqo
Thus by \eqref{energy density 0} the number of sub-cubes $T_3$ is bounded by
\beq\label{number of T_3}
|T_3|\leq c_4(L,\th,\e,\|u\|_{L^\infty})k^{n-1},\quad k\geq k_0.
\eeq

From now on we focus on the analysis of cubes in $T_4$ and $T_5$. Take $S_i$ in $T_4$ or $T_5$, then there is a $j_0$ such that $\s_i^{j_0}>(1-2\e)L^n$. In this case, we claim that when $\e$ is suitably chosen, we can assure that
\beqo
\max\limits_{x\in S_i}|u(x)-a_{j_0}|<\theta
\eeqo
If there exists $x_0\in S_i$ such that $|u(x_0)-a_{j_0}|\geq \theta$, then we have that there exists a constant $c_5(\theta,\|\na u\|_{L^\infty})$ such that
\beqo
\cl^n(\{|u-a_{j_0}|>\theta/2\}\cap  S_i)\geq c_5.
\eeqo
We note that the uniform boundedness of $|\na u|$ follows from the $C^{1,\beta}$ regularity (Proposition \ref{reg 0<al<=1} and Proposition \ref{reg al>1}) and the assumption that $|u|$ is uniformly bounded. And $c_5$ doesn't depend on $j_0$. Then the claim follows if we simply take
\beq\label{value of epsilon}
\e<\f{c_5}{2L^n}.
\eeq

\begin{lemma}\label{value of L}
When $L$ is suitably chosen depending on $\theta$, in any cube $S_i\in T_4\cup T_5$, it holds
\beq\label{measureofa_j}
\cl^n(\{u(x)=a_{j_0}\}\cap S_i)\geq \om_n\left(\f{L}{4}\right)^n
\eeq
where $\om_n$ is the volume of the n-dimensional unit ball
\end{lemma}

\begin{proof}
We proceed by contradiction and denote the central point of $S_i$ by $z_i$. So,
\beqo
|\{u(x)=a_{j_0}\}\cap S_i|< \om_n\left(\f{L}{4}\right)^n
\eeqo
Then there must be a point $x_1\in B_R(z_i,\f{L}{4})$ such that $x_1\in \overline{\{0<|u-a_{j_0}|<\theta\}}$. Moreover, we have
\beqo
B_{\f{L}{4}}(x_1)\subset S_i\subset \overline{\{|u-a_{j_0}|<\theta\}}
\eeqo
Therefore we are in the position to apply Lemma \ref{nondeg} to deduce that
\beqo
\sup\limits_{B_{\f{L}{4}}(x_1)}|u-a_{j_0}|\geq c(n,W)\left(\f{L}{4}\right)^{\f{2}{2-\al}}
\eeqo
which contradicts with $\max\limits_{x\in S_i}|u-a_{j_0}|<\theta$ if we choose the constant $L$ at the beginning satisfying $c(n,W)\left(\f{L}{4}\right)^{\f{2}{2-\al}}>2\theta$. This completes the proof of Lemma \ref{value of L}.
\end{proof}

If the cube $S_i\in T_4$, then by definition we have
\beqo
|u(x)-a_{j_0}|<\theta, \quad \forall x\in S_i\cup(\bigcup_{S_p\text{ adjacent to }S_i} S_p)
\eeqo
By the same argument as in the proof of the lemma above, we obtain that
\beqo
u(x)\equiv a_{j_0},\quad x\in S_i.
\eeqo

If $S_i\in T_5$, then there must be at least one adjacent cube of $S_i$, denoted by $S_{p}$, such that
\beq\label{est in Sp}
|\{|u-a_{j_0}|>\f{\theta}{2}\}\cap S_{p}|>\e L^n.
\eeq
We set
\beqo
Q_{S_i}:=S_i\cup(\bigcup_{S_p\text{ adjacent to }S_i} S_p)
\eeqo
Then by \eqref{measureofa_j}, \eqref{est in Sp} and the co-area formula, we can compute similarly as in \eqref{co-area0} to get
\beqo
\int_{Q_{S_i}}|\na u|^2\,dx\geq c_6(L, \th, \e).
\eeqo
Since each point can belong to at most $3^n$ different $Q_{S_i}$, utilizing \eqref{basic est} we conclude
\beqo
C(n)(kL)^{n-1}\geq \sum\limits_{S_i\in T_5}\int_{Q_{S_i}}|\na u|^2\,dx\geq c_6|T_5|,
\eeqo
which implies
\beq\label{number of T_5}
|T_5|\leq c_7(n, L,\th,\e)k^{n-1}.
\eeq

Finally, combining \eqref{number of T_1}, \eqref{number of T_2}, \eqref{number of T_3} and \eqref{number of T_5} we get
\beq
\cl^n(\tilde{S}_{kL}\cap I_0)\leq (|T_1|+|T_2|+|T_3|+|T_5|)L^n\leq c_8(n, L,\th,\e)(kL)^{n-1}.
\eeq
Since $B_{kL}\subset \tilde{S}_{kL}$, we can get \eqref{I_0 est} after taking $k$ to be the smallest integer larger than $\f{R}{L}$ for sufficiently large $R$. Also if we carefully check the definitions of all the constants in the proof we conclude that $c_8$ only depends on the dimension $n$, the potential $W$ and the uniform bound of $|u|$, but not on the specific solution $u$. This completes the proof of Theorem \ref{first part main theorem}.

\end{proof}

Theorem \ref{first part main theorem} implies that a bounded entire minimizer $u(x)$ should satisfy $W(u)=0$ in ``most of the space". Next we further show that at sufficiently large scales, $u$ must possess at least two different phases, each of which contains some definite measure of order $R^n$.

\begin{lemma}\label{lemma ci(th)}
Let $x_0\in\BR^n$, $u:\BR^n\ri\BR^m$ be a bounded entire minimizer of $J$. Assume that $u\not\equiv a_i$ for any $i\in \{1,2,...,N\}$. We take an arbitrary constant $\theta<r_0:=\f12 \min\limits_{1\leq i\neq j\leq N}|a_i-a_j|$, then there exist positive constants $R_0, c(u,\theta)$ such that for any $R\geq R_0$, there are $a_i,a_j\in A$, which depend on $R$, satisfying
\beq\label{ci(th)}
\cl^n(B_R(x_0)\cap \{|u-a_k|<\theta\})\geq cR^n,\quad k=i,j.
\eeq
\end{lemma}

\begin{proof}
Since $u$ is nonconstant, by $C^{1,\beta}$ regularity of $u$ there is some $R_1>0, \,0<\lam<r_0,\,\mu_0>0$ such that
\beqo
\cl^n(B_{R_1}(x_0)\cap \{|u-a_1|>\lam\})\geq \mu_0.
\eeqo
Then by the density estimate \eqref{density est} in Proposition \ref{two estimates}, there exists $\mu_1$ such that
\beq\label{u-a1>lam}
\cl^n(B_{R}(x_0)\cap \{|u-a_1|>\lam\})\geq \mu_1R^n,\quad \forall R\geq R_1
\eeq
Take $\theta<r_0$ to be an arbitrary constant. By our hypothesis on $W$, there is a positive constant $C=C(\lam,\theta, \|u\|_{L^\infty})$ such that
\beqo
W(u)>C, \;\text{ when } |u-a_1|>\lam,\,|u-a_j|\geq \theta \text{ for any }j\neq 1.
\eeqo
Applying the basic estimate \eqref{basic est} in Proposition \ref{two estimates}, for enough large $R$,
\beq\label{u-a_2>theta}
\cl^n(B_R(x_0)\cap \{|u-a_1|>\lam,\,|u-a_j|\geq \theta \text{ for any }j\neq 1\})\leq C_2R^{n-1},
\eeq
for some constant $C_2$. Combining \eqref{u-a1>lam} and \eqref{u-a_2>theta}, we obtain that
\beqo
\begin{split}
&\cl^n(B_{R}(x_0)\cap (\bigcup\limits_{j\neq 1}\{|u-a_j|<\theta\}))\\
\geq & \cl^n(B_R(x_0)\cap \{|u-a_1|>\lam, |u-a_j|<\theta \text{ for some }j\neq 1\})\geq c_1(u,\theta)R^n, \quad \forall R>\tilde{R}_1
\end{split}
\eeqo
for some constants $\tilde{R}_1$ and $c_1$. The same argument also works for the set $B_R(x_0)\cap(\bigcup\limits_{j\neq k}\{|u-a_j|<\theta\})$ for any $k\in \{1,2,...,N\}$, i.e. there exists $\tilde{R}_k, \;c_k>0$ such that
\beqo
\cl^n(B_{R}(x_0)\cap (\bigcup\limits_{j\neq k}\{|u-a_j|<\theta\}))\geq c_k(u,\theta) R^n, \forall R\geq \tilde{R}_k
\eeqo
Finally, we take $R_0=\min\limits_{k} \tilde{R}_k$ and $c=\f{1}{N-1}\min\limits_k c_k$ and the conclusion of the lemma easily follows.
\end{proof}

In the following theorem, we show that in any ball $B_R(x_0)$ with radius $R$ large enough, the sets $\{u=a_i\}$ and $\{u=a_j\}$ ($a_i,\,a_j$ from Lemma \ref{lemma ci(th)}) must contain a set of measure of the order $R^n$.

\begin{theorem}\label{two phase existence}
Let $x_0\in \BR^n$, $u: \BR^n\ri\BR^m$ be a bounded entire minimizer of the energy $J$, and $u\not\equiv a_j$ for any $j\in\{1,2,...,N\}$. Then there are positive constants $R_0$ and $c$ (both depend on $u$) such that for any $R\geq R_0$, there are $a_i,a_j$ depending on $R$ such that
\beq
\min\{\mathcal{L}^n(B_R(x_0)\cap \{u=a_i\}),\mathcal{L}^n(B_R(x_0)\cap \{u=a_j\})\}\geq  cR^n,\quad \forall R>R_0
\eeq
\end{theorem}
\begin{proof}
Without loss of generality, suppose $x_0=0^n$ and write $B_R=B_R(0^n)$. According to the Proposition \ref{two estimates} and Lemma \ref{lemma ci(th)}, we know that for any sufficiently large $R$, there are $a_i,a_j\in A$ such that \eqref{ci(th)} holds.

The proof relies on the same technique as the proof of Theorem \ref{first part main theorem}. So we will only present the main ingredients and omit some technical details. Take $L$ as the same constant in Theorem \ref{first part main theorem} and $k\in\mathbb{N}$. We consider the domain
\beqo
\tilde{S}_{kL}:=\{x\in \BR^n: x_i\in (-kL,kL)\},
\eeqo
and then divide $\tilde{S}_{kL}$ into $K=(2k)^n$ identical sub-cubes $S_1,...,S_K$, each of which has side of length $L$. We also recall the definition of $\s_i^j$ in \eqref{def sigma i,j}. By Lemma \ref{lemma ci(th)}, there are two phases $a_i, a_j$ (for simplicity we assume they are $a_1,a_2$) such that
\beq\label{a_1,a_2 in S_kL}
\cl^n(\tilde{S}_{kL}\cap \{|u-a_j|<\f{\th}{2}\})\geq c(kL)^n,\quad j=1,2.
\eeq

Take $\e:=\e(u,\theta)$ be a small constant such that
\begin{itemize}
\itemsep0.5em
\item[a.] \eqref{value of epsilon} holds. As a result, if $\s_i^j>(1-2\e)L^n$, then $|u(x)-a_j|<\theta$ for any $x\in S_i$.
\item[b.] $\e\leq \f{c}{2^{n+3}}$ where $c$ is the constant in \eqref{a_1,a_2 in S_kL}.
\end{itemize}

Then we divide $\{S_i\}_1^K$ into the following two classes
\begin{itemize}
\itemsep0.5em
\item[1] $U_1:=\{S_i: \exists j_0 \text{ s.t. } \s_i^{j_0}=\max\limits_{1\leq j\leq N}S_i^j\leq(1-2\e)L^n\}.$
\item[2] $U_2:=\{S_i: \exists j_0 \text{ s.t. } \s_i^{j_0}=\max\limits_{1\leq j\leq N}S_i^j>(1-2\e)L^n.\}$
\end{itemize}

From the proof of Theorem \ref{first part main theorem}, we have
\beqo
|U_1|\leq c_0(L,\theta,\e)k^{n-1}.
\eeqo

Let $K_1$ denote the number of sub-cubes $S_i$ satisfying $\s_i^1>(1-2\e)L^n$. We obtain from \eqref{a_1,a_2 in S_kL}
\beq
\begin{split}
c(kL)^n&\leq \sum\limits_{1\leq i\leq (2k)^n}\s_i^1\\
&\leq |U_1|L^n+K_1 L^n+ \left((2k)^n-|U_1|-K_1 \right)(2\e L^n)\\
&\leq c_0k^{n-1}L^n +K_1L^n+\f{c}{4} (kL)^n\quad (\text{Property b of }\e),
\end{split}
\eeq
which immediately implies that $K_1\geq \f{c}{2}k^n$ whenever $k$ is large enough. Together with Lemma \ref{value of L} we have
\beq\label{estimate two phase}
\cl^n (\tilde{S}_{kL}\cap \{u=a_1\})\geq \f{c}{2}k^n\om_n (\f{L}{4})^n\geq c_1(kL)^n,
\eeq
for some constant $c_1=c_1(W,u)$. For $\{u=a_2\}$ the estimate \eqref{estimate two phase} still holds. One can easily check that \eqref{estimate two phase} implies the statement of Theorem \ref{two phase existence}.

\end{proof}

\section{Weiss' Monotonicity formula and a growth estimate in the case $\al=1$}\label{growth est}

Thanks to Theorem \ref{two phase existence}, we know for a uniformly bounded entire minimizer $u$, the free boundary $\pa\{|u-a_i|>0\}$ $(i=1,2)$ must exist. In this section we will derive a growth rate estimate for $|u-a_i|$ away from the free boundary in the case $\underline{\al=1}$.

From now on we fix $\al=1$ and assume $a_1=0^m$. By the hypothesis H2, $W(u)$ has the form $W(u)=g(u)|u|$ for some $g\in C^2(B_\theta)$. Here $\th$ is the constant in Lemma \ref{nondeg}. Since $u$ is a local minimizer, it satisfies the Euler-Lagrange equation near the free boundary point,

\beq\label{ELeq}
\D u= g(u)\f{u}{|u|}+|u| D_u g(u).
\eeq

Also there exists a positive constant $C>0$ such that $g(u)>C$ when $|u|\leq \theta$. We use the notation
\beq\label{notations}
\Om(u):=\{|u(x)|>0\},\quad \G(u):=\pa^*\Om(u).
\eeq
Here $\pa^*$ denotes De Giorgi's reduced boundary. An easy observation is that for any point $x\in\G(u)$, we must have $|u(x)|=|\na u(x)|=0$. The proof is straightforward: if at some point $x_0\in \G(u)$, $|\na u|>0$, then by continuity of $\na u$ we have that in a small neighborhood $B_{r}(x_0)$, $|\na u_i|\geq c$ for some $1\leq i\leq m$ and $c>0$. The inverse function theorem implies that in $B_r(x_0)$, $\{u_i=0\}$ is a $(n-1)$-dimensional hypersurface, which further gives $x_0\not\in\pa^e\Om(u)$, where $\pa^e$ denotes the measure theoretic boundary. Finally we arrive at a contradiction thanks to the well-known result $\pa^* E\subset \pa^e E$ for any set $E$ of locally finite perimeter. For the definitions of the reduced boundary and the measure theoretic boundary, as well as their relationship, we refer to \cite[Chapter 5.7\&5.8]{book-eg} for details.

We first establish an almost monotonicity formula for $|u|<\th$. The proof closely follows the classical arguments of Weiss (see \cite{Weiss1,Weiss2})

\begin{lemma}\label{weiss lemma}
Let $u$ be a solution of \eqref{ELeq} in $B_r(x_0)$ such that $|u|<\th$ in $B_r(x_0)$, and set
\beq\label{def weiss}
W(u,x_0,r)=\f{1}{r^{n+2}}\int_{B_r(x_0)}\left(\f12|\na u|^2+g(u)|u|\right)\,dx-\f{1}{r^{n+3}}\int_{\pa B_r(x_0)}|u|^2 \,d\mathcal{H}^{n-1}.
\eeq

Then $W(u,x_0,r)$ satisfies
\beq\label{Weiss}
\f{d}{dr} W(u,x_0,r) =r\int_{\pa B_1}|\f{du_r}{dr}|^2\,d\mathcal{H}^{n-1}+2r\int_{B_1}D_ug\cdot u_r|u_r|\,dx
\eeq
where
\beqo
u_r(x):=\f{u(x_0+rx)}{r^2}.
\eeqo
\end{lemma}

\begin{proof}
First we write $W(u,x_0,r)$ as
\beqo
W(u,x_0,r)=\int_{B_1}\left(\f12|\na u_r|^2+g(r^2 u_r)|u_r|\right)\,dx-\int_{\pa B_1}|u_r|^2\,d\mathcal{H}^{n-1}.
\eeqo
Then by direct calculation we have
\begin{align*}\
&\f{d}{dr}W(u,x_0,r)\\
=&\int_{B_1}\left( \na u_r\cdot\f{d}{dr}(\na u_r)+D_u g(r^2 u_r)\cdot \f{d}{dr}(r^2 u_r)|u_r|+ g(r^2 u_r)|u_r|^{-1}u_r\cdot \f{d}{dr}u_r \right)\,dx\\
& -2\int_{\pa B_1}u_r\cdot\f{d}{dr}u_r\,d\mathcal{H}^{n-1}\\
=&\int_{B_1}\left(-\Delta u_r\cdot \f{d}{dr}u_r+D_u g(r^2 u_r)\cdot \f{d}{dr}(r^2 u_r)|u_r|+ g(r^2 u_r)|u_r|^{-1}u_r\cdot \f{d}{dr}u_r \right)\,dx\\
& -2\int_{\pa B_1}u_r\cdot\f{d}{dr}u_r\,d\mathcal{H}^{n-1}+\int_{\pa B_1}(x\cdot \na u_r)\cdot \f{d}{dr}u_r\,d\mathcal{H}^{n-1}.\\
=&\int_{B_1}\left(-\Delta u_r\cdot \f{d}{dr}u_r+D_u g(r^2 u_r)\cdot \f{d}{dr}(r^2 u_r)|u_r|+ g(r^2 u_r)|u_r|^{-1}u_r\cdot \f{d}{dr}u_r \right)\,dx\\
&+\int_{\pa B_1}r|\f{d}{dr}u_r|^2\,\mathcal{H}^{n-1}.
\end{align*}
Here we have used integration by parts in the second step and the formula $\f{d}{dr}u_r=\f{1}{r}(x\cdot \na u_r-2u_r)$ in the last step. Since $u$ satisfies the equation \eqref{ELeq}, direct computation implies that
\beq\label{ELu_r}
\Delta u_r=\left( g(r^2 u_r)\f{u_r}{|u_r|}+D_u g(r^2 u_r)r^{2}|u_r| \right).
\eeq
Substituting \eqref{ELu_r} into the above identity, we obtain
\begin{align*}
&\f{d}{dr}W(u,x_0,r)-\int_{\pa B_1}r|\f{d}{dr}u_r|^2\,\mathcal{H}^{n-1}\\
=& \int_{B_1}\bigg\{ -\left( g(r^2 u_r) \f{u_r}{|u_r|}+D_u g(r^2 u_r)r^{2}|u_r| \right)\f{d}{dr}u_r\\
&\qquad +D_u g(r^2 u_r)\cdot \f{d}{dr}(r^2 u_r)|u_r|+  g(r^2 u_r)\f{u_r}{|u_r|}\cdot \f{d}{dr}u_r\bigg\}\,dx\\
=&2 r \int_{B_1}D_u g(r^2 u_r)\cdot u_r|u_r|\,dx
\end{align*}
Hence we have proved \eqref{Weiss}.
\end{proof}

\begin{rmk}
For other $\al\in[0,2)$, the analogue result still holds for
\beqo
W(u,x_0,r)=\f{1}{r^{n+2\ka-2}}\int_{B_r(x_0)}\f12|\na u|^2+W(u)\,dx-\f{\ka}{2r^{n+2\ka-1}}\int_{\pa B_r(x_0)}|u|^2 \,d\mathcal{H}^{n-1}.
\eeqo
where $\kappa:=\f{2}{2-\al}$ and $W(u)=g(u)|u|^\al$. The derivative of $W(u,x_0,r)$ is given by
\beqo
\f{d}{dr} W(u,x_0,r) =r\int_{\pa B_1}|\f{du_r}{dr}|^2\,d\mathcal{H}^{n-1}+\ka r^{\ka-1}\int_{B_1}D_ug\cdot u_r|u_r|^\al
\eeqo
For our purpose we only need the statement for $\al=1$. The proof for general $\al\in [0,2)$ is identical and we omit it here.
\end{rmk}

\begin{proposition}\label{prop growth}
Let $\al=1$ and $u$ be a bounded entire minimizer and let $\G(u)$ be as defined in \eqref{notations}. There exist constants $r_0$ and $C$, which only depends on $\|u\|_{L^\infty}$ and the potential function $W(u)$, such that
\beq\label{growth estimate}
|u(x)|\leq C \dist(x,\G(u))^2,\quad |\na u(x)|\leq C\dist(x,\G(u))
\eeq
whenever $\dist(x,\G(u))\leq r_0$.
\end{proposition}

\begin{proof}
This proposition and the proof are almost identical to \cite[Theorem 2]{asuw} (except now for a more general potential function). We present the whole argument here for completeness.

The statement of the proposition is equivalent to
\beqo
\sup\limits_{x\in B_r(x_0)}|u(x)|\leq Cr^2,\quad \sup\limits_{x\in B_r(x_0)}|\na u(x)|\leq Cr,
\eeqo
whenever $x_0\in\G(u)$, $r\leq r_0$. By \eqref{ELeq} and the standard theory of elliptic regularity, it suffices to show
\beq\label{growth est int form}
\f{1}{r^n}\int_{B_r(x_0)}|u|\,dx\leq Cr^2,\quad \forall x_0\in \G(u),\; r\leq r_0
\eeq
where $C$ and $r_0$ only depend on $\|u\|_{L^\infty}$ and the potential function $W$.

Note that since $|u|$ is uniformly bounded, we have $u\in C^{1,\g}$, which further implies $|\na u|$ is uniformly bounded. As a result, there is a constant  $r_0$ such that $\dist(x,\G(u))\leq 2r_0$ implies $|u(x)|\leq \theta$, where $\th$ is the constant in Lemma \ref{nondeg}. Also, $W(u(x))$ has the form $g(u(x))|u(x)|$ for some smooth function $g(u)\geq C>0$ when $\dist(x,\G(u))\leq 2r_0$.

Since $|\na u|$ is bounded and $r_0$ is a constant, we have that $W(u,x_0,r_0)$ is uniformly bounded by some constant $C_1$ independent of $u$ and $x_0$. Here $W(u,x_0,r_0)$ is the quantity defined in \eqref{def weiss}. Using Lemma \ref{weiss lemma}, we compute for $r<r_0$
\beq\label{compute integral growth}
\begin{split}
\f{1}{r^{n+2}}\int_{B_r(x_0)}g(u)|u|\,dx& =W(u,x_0,r)-\f{1}{r^{n+2}}\int_{B_r(x_0)}\f12|\na u|^2\,dx\\
&\qquad +\f{1}{r^{n+3}}\int_{\pa B_r(x_0)}|u|^2\,d\ch^{n-1}\\
&=W(u,x_0,r)-\f{1}{r^{n+2}}\int_{B_r(x_0)}\f12|\na (u-p(x-x_0))|^2\,dx\\
&\qquad +\f{1}{r^{n+3}}\int_{\pa B_r(x_0)}|u-p(x-x_0)|^2\,d\ch^{n-1}\\
&\leq W(u,x_0,r_0)+\int_{r}^{r_0}2s\int_{B_1} |D_ug||\f{u(x_0+sx)}{s^2}|^2\,dx\,ds\\
&\qquad +\f{1}{r^{n+3}}\int_{\pa B_r(x_0)}|u-p(x-x_0)|^2\,d\ch^{n-1},
\end{split}
\eeq
for every $p(x)\in\mathcal{H}$, where $\ch$ is defined by
\beqo
\begin{split}
\ch:=&\{p(x): \; p(x)=(p_1(x),...p_m(x)), \text{ each }p_i(x) \text{ is a }\\
 &\quad  \text{homogeneous
harmonic polynomial of second order.} \}
\end{split}
\eeqo
We would like to point out that the homogeneity and harmonicity of $p(x)$ is used in the second equality of \eqref{compute integral growth}.

We already know that the first term in the last step of \eqref{compute integral growth} is bounded by a constant $C_1$ independent of $u$ and $x_0$. For the second term, since $u(x_0+x)\leq C|x|^{\f53}$ when $|x|\leq r_0$ by the $C^{1,\f23}$ regularity (c.f. Remark \ref{c1,alpha remark} and observation below \eqref{notations}), we have
\beqo
\int_{r}^{r_0}2s\int_{B_1} |D_ug||\f{u(x_0+sx)}{s^2}|^2\,dx\,ds\leq C\int_r^{r_0}s^{-3}\int_{B_1} |sx|^{\f{10}{3}}\,dx\,ds\leq C_2
\eeqo
for some constant $C_2$. Because $g(u)\geq C>0$ in $B_r(x_0)$, in order to prove \eqref{growth est int form}, it suffices to show that there is a constant $C_3$, independent of $u$ and $x_0$, such that for any $x_0\in\G(u)$ and $r\leq r_0$,
\beq\label{last term bound}
\min\limits_{p\in\ch} \f{1}{r^{n+3}}\int_{\pa B_r(x_0)}|u-p(x-x_0)|^2\,d\ch^{n-1}\leq C_3.
\eeq

Let $p_{x_0,r}$ be the minimizer of the integral $\int_{\pa B_r(x_0)}|u-p(x-x_0)|^2\,d\ch^{n-1}$ among $p\in\ch$. Then $p_{x_0,r}$ satisfies
\beq\label{ortho}
\int_{\pa_{B_r}(x_0)}(u(x)-p_{x_0,r}(x-x_0))\cdot q(x-x_0)\,d\ch=0\quad \forall q\in\ch.
\eeq

Suppose by contradiction that \eqref{last term bound} is not true, then there is a sequence of entire minimizers $\{u_k\}$ (uniformly bounded), a sequence of points $x_k\in \G(u_k)$ as well as a sequence of radii $r_k\ri 0$ such that
\beqo
M_k:=\f{1}{r_k^{n+3}}\int_{\pa B_{r_k}(x_k)}|u_k-p_{x_k,r_k}(x-x_k)|^2\,d\ch^{n-1}\ri\infty.
\eeqo

Define
\beqo
v_k(x):=\f{u_k(x_k+r_kx)}{r_k^2},\qquad w_k:=\f{v_k-p_{x_k,r_k}}{\sqrt{M_k}}.
\eeqo
Then we immediately get
\beqo
\int_{\pa B_1(0^n)}|w_k|^2\,d\ch^{n-1}=1
\eeqo
and we have
\beq\label{compute w_k}
\begin{split}
&\int_{B_1(0^n)} \f12|\na w_k|^2\,dx-\int_{\pa B_1(0^n)}|w_k|^2\,d\ch^{n-1}\\
=&M_k^{-1}\left( \int_{B_1(0^n)} \f12|\na (v_k-p_{x_k,r_k})|^2\,dx-\int_{\pa B_1(0^n)}|v_k-p_{x_k,r_k}|^2\,d\ch^{n-1} \right)\\
=&M_k^{-1}\left( \int_{B_1(0^n)} \f12|\na v_k|^2\,dx-\int_{\pa B_1(0^n)}|v_k|^2\,d\ch^{n-1} \right)\\
\leq &M_k^{-1} W(u_k,x_k,r_k)\\
\leq & M_k^{-1}\left(  W(u_k,x_k,r_0)+\int_{r_k}^{r_0} 2s \int_{B_1} |D_u g||\f{u_k(x_k+sx)}{s^2}|^2 \,dx\,ds\right)\\
\ri & 0\quad \text{as }k\ri\infty.
\end{split}
\eeq

So $w_k$ is uniformly bounded in $W^{1,2}(B_1)$. Also we note that by \eqref{eleq for al=1} each $w_k$ satisfies the equation
\beqo
\D w_k=\f{1}{\sqrt{M_k}} \left( \f{v_k}{|v_k|}g(u_k)+D_ug(u_k) \right)\chi_{\{|u_k|>0\}},
\eeqo
which implies
\beqo
|\D w_k|\leq \f{C}{\sqrt{M_k}}\ri 0,\quad \text{as }k\ri\infty.
\eeqo
By Schauder estimates, $w_k$ is uniformly bounded in $C_{loc}^{1,\g}(B_1)$ for any $\g< 1$. Therefore we can extract a subsequence, still denoted by $w_k$, that converges to $w_0$ with the following properties
\begin{enumerate}
\itemsep0.5em
\item $w_k\ri w_0$ weakly in $H^1(B_1)$, strongly in $L^2(\pa B_1)$, $\int_{\pa B_1} |w_0|^2\,d\ch^{n-1}=1$.
\item $w_k\ri w_0$ in $C^{1,\g}_{loc}(B_1)$ for any $\g<1$;
\item $\D w_0=0$;
\item $|w_0(0^n)|=|\na w_0(0^n)|=0$;
\item $\int_{\pa B_1} w_0\cdot q\,d\ch^{n-1}=0$ for any $q\in\ch$. This property follows from \eqref{ortho}.
\end{enumerate}

By \cite[Lemma 4.1]{Weiss3}, we know that for any $w_0$ satisfying (3) and (4),
\beqo
\int_{B_1}|\na w_0|^2\,dx\geq 2\int_{\pa B_1}|\na w_0|^2\,d\ch^{n-1}.
\eeqo
On the other hand, from \eqref{compute w_k} we know
\beqo
\int_{B_1}|\na w_0|^2\,dx\leq  2\int_{\pa B_1}|\na w_0|^2\,d\ch^{n-1}.
\eeqo
Therefore we have $\int_{B_1}|\na w_0|^2\,dx= 2\int_{\pa B_1}|\na w_0|^2\,d\ch^{n-1}$ which implies (again by \cite[Lemma 4.1]{Weiss3}) that each component of $w_0$ is a homogeneous harmonic polynomial of second order, i.e. $w_0\in \ch$. This is in contradiction with properties (1) and (5). The proof is complete.
\end{proof}

\section{$(n-1)$-Hausdorff measure of the free boundary for $\al=1$} \label{section 5}

In this section, we continue working with the potential function $W(u)$ satisfying H1 and H2 with $\underline{\al=1}$. Assume $u$ is a bounded entire minimizer of the energy $J$. We would like to study the $(n-1)$-Hausdorff measure of $\pa^* I_0$ and prove the second part of Theorem \ref{main}, i.e. the inequality \eqref{free bdy lower bdd}, \eqref{est free bdy}.

Firstly we focus on the local estimate of $\pa^* \{u=a_i\}$ and we use the same notations and assumptions as in Section \ref{growth est}. Take $a_1=0^m$ and $W(u)=g(u)|u|$ for some $g\in C^2(B_\theta)$, $\th$ as in Lemma \ref{nondeg}. $u$ satisfies the Euler-Lagrange equation \eqref{ELeq}.

Thanks to the growth estimate \eqref{growth est} and the non-degeneracy Lemma \ref{nondeg}, we have for every $x_0\in\G(u)$ (recall that $\G(u)$ is defined in \eqref{notations}) and small $r$,
\begin{align}
\label{control of u}c_1r^2& \leq \sup\limits_{x\in B_r(x_0)}|u(x)|\leq c_2r^2,\\
\label{control of nablau} c_1r &\leq \sup\limits_{x\in B_r(x_0)}|\na u(x)|\leq c_2r.
\end{align}

\begin{thm}\label{local estimate}(Local estimate of $\G$)
There are constants $r_0$ and $C_0$ such that
\beq\label{bdy_meas:local}
\mathcal{H}^{n-1}(\G(u)\cap B_{r_0}(z))\leq C_0\quad  \text{for every }\;z\in \G(u).
\eeq
\end{thm}

\begin{proof}

Take the constant $r_0$ such that for any $x$ that satisfies $\mathrm{dist}(x,\G(u))\leq 2r_0$, $|u(x)|\leq \th$. We will fix a ball $B_{2r_0}(z)$ for some $z\in \G(u)$ in the rest of the proof.

We define
\beqo
v_i:=\pa_{x_i} u,\;\; i=1,2,...,n,\qquad \S_\e(u):=\{x\in B_{r_0}(z)\cap\{|u|>0\}: |\na u|<\e\}.
\eeqo
By differentiating the Euler-Lagrange equation \eqref{ELeq}, formally we have
\begin{equation}\label{equation_for_vi}
\begin{split}
\D v_i=&|u|^{-1}g(u)v_i+ |u|^{-1}(D_ug\cdot v_i)u )-|u|^{-3}g(u)(v_i\cdot u)u\\
&+|u|^{-1}(v_i\cdot u)D_ug+|u|(D^2_u g\cdot v_i).
\end{split}
\end{equation}
Take the function $\psi_\e(x):\BR^+\ri [0,1]$ defined by
\beqo
\psi_\e(x)=\begin{cases}
1, & x\geq \e,\\
\f{x}{\e}, & x\in [0,\e).
\end{cases}
\eeqo
We also choose a smooth cut-off function $\phi \in C_c^{\infty }(B_{2r_0}(z),\BR)$ such that
\beqo
\phi\equiv 1 \text{ in }B_{r_0}(z),\quad |\na \phi|\leq \f{C}{r_0}
\eeqo

Let
\beqo
\ta:=B_{2r_0}(z)\cap \{|u|>0\}.
\eeqo
The key of the proof is to estimate the following integral
\beq\label{finite-perimeter-integral}
I:= \int_{\ta}\na v_i\cdot \na\left[ \psi_\e(|v_i|)\f{v_i}{|v_i|} \phi \right]\,dx,
\eeq
from which estimate \eqref{meas est:Sigma_e} below follows.

\textbf{Claim. }There exists a constant $C(g,r_0)$, which is independent of $\e,\,z$, such that
\beq\label{claim est I}
I\leq C(g,r_0).
\eeq
\begin{proof}[Proof of the Claim]
Define
\beqo
\eta:=\psi_\e(|v_i|)\f{v_i}{|v_i|}\phi
\eeqo
We first show that $\eta\in W_0^{1,2}(B_{2r_0}(z),\BR^m)$. Indeed, by direct computation we have
\begin{align*}
\pa_j \eta&= \psi_\e'(|v_i|)\pa_j|v_i|\f{v_i}{|v_i|}\phi+\psi_\e(|v_i|)\f{\pa_j v_i}{|v_i|}\phi\\
&\quad -\psi_\e(|v_i|)v_i\f{\pa_j|v_i|}{|v_i|^2}\phi+\psi_\e(|v_i|)\f{v_i}{|v_i|}\pa_j\phi
\end{align*}
By definitions of $\psi_\e$, $\phi$ and the $W^{2,2}$ estimate of $u$, the right-hand side is $L^2$-integrable. Combining with the fact that $\phi\in C_0^{\infty}(B_{2r_0}(z))$, we get $\eta\in W_0^{1,2}(B_{2r_0}(z),\BR^m)$.

We notice that since $\D v_i$ is very singular when $|u|\ri 0$, so we can not directly perform integration by parts by moving all the derivatives on $v_i$ in domain $\ta$. Instead, we will switch $\pa_i$ and $\nabla$.

For any $f\in C_0^\infty(B_{2r_0}(z),\BR^m)$, by integration by parts we have
\beqo
\int_{B_{2r_0}(z)}\na v_i\cdot\na f\,dx=\int_{B_{2r_0}(z)} \Delta u\cdot \pa_if\,dx
\eeqo
This can be generalized to the vector-valued function $\eta$ in $W_0^{1,2}(B_{2r_0}(z),\BR^m)$, so we get
\beq\label{ibp}
\int_{\ta}\na v_i\cdot\na \eta\,dx=\int_{B_{2r_0}(z)} \na v_i\cdot\na \eta\,dx=\int_{B_{2r_0}(z)} \D u\cdot \pa_i\eta\,dx=\int_{\ta} \D u\cdot \pa_i\eta\,dx
\eeq
Above we have exploited the fact that $D^2 u$ and $\na \eta$ vanish almost everywhere on $\{|u|=0\}$. So it suffices to prove
\beq\label{claim after ibp}
\int_{\ta} \D u\cdot \pa_i\eta \,dx\leq C(g,r_0).
\eeq

We define the set $\ta_\delta:=B_{2r_0}(z)\cap \{|u|>\d\}$, it is obvious that $\ta_\d\subset \ta$ for any $\d>0$ and $\ta=\lim_{\d\ri 0 } \ta_\d$. Then we have
\begin{align}
\nonumber &\int_{\ta} \D u\cdot \pa_i\eta\,dx\\
\nonumber=&\lim\limits_{\d\ri 0} \int_{\ta_\d} \D u\cdot\pa_i \eta\,dx\\
\label{two parts}=&\lim\limits_{\d\ri 0} \int_{\ta_\d} -\D v_i\cdot \eta\,dx+\lim\limits_{\d\ri 0} \int_{\pa \ta_\d} \Delta u\cdot \eta \g_i d\s
\end{align}

For the first term in \eqref{two parts}, we further compute
\begin{align}
\nonumber &-\int_{\ta_\d} \D v_i\cdot [\psi_\e(|v_i|)\f{v_i}{|v_i|}\phi]\,dx\\
\label{estimate:integral}=&-\int_{\ta_\d} \psi_{\e}(|v_i|)|v_i|^{-1}\phi \bigg( |u|^{-1}g(u)|v_i|^2-|u|^{-3}g(u)(v_i\cdot u)^2\\
\nonumber &\qquad\qquad\qquad \qquad  +2|u|^{-1}(D_ug\cdot v_i)(u\cdot v_i)+|u|(v_i\cdot D^2_ug\cdot v_i) \bigg)\,dx
\end{align}

By the Cauchy-Schwartz inequality,
\beqo
|u|^{-1}g(u)|v_i|^2-|u|^{-3}g(u)(v_i\cdot u)^2\geq 0.
\eeqo
Substituting this into \eqref{estimate:integral} gives
\beq
\begin{split}
\label{First part:bounded by C}&\int_{\ta_\d}-\D v_i\cdot \eta\,dx \\
\leq &\bigg| \int_{\ta_\d} \psi_{\e}(|v_i|)|v_i|^{-1}\phi \bigg( 2|u|^{-1}(D_ug\cdot v_i)(u\cdot v_i)+|u|(v_i\cdot D^2_ug\cdot v_i) \bigg)\,dx\bigg|\\ \leq &C(g, r_0).
\end{split}
\eeq
The integral is bounded by a constant $C(g,r_0)$ (doesn't depend on the choice of $z, \,\d,\, \e$) because $|v_i|$, $D_u(g)$, $D_u^2g$, $u$ are all uniformly bounded by a constant in $B_{2r_0}(z)$.

For the second part in \eqref{two parts}, we apply \eqref{ELeq} to obtain
\beq
\begin{split}\label{second part}
&\int_{\pa \ta_\d} \D u\cdot \eta\gamma_i d\s\\
=&\int_{\pa\{|u|>\d\}\cap B_{2r_0}(z)}\D u\cdot \eta\g_i d\s\\
=&\int_{\pa\{|u|>\d\}\cap B_{2r_0}(z)}\left(g(u)\f{u}{|u|}+|u|D_ug(u)\right)\left( \psi_\e(|v_i|)\f{v_i}{|v_i|}\phi) \right) \g_i d\s\\
=&\int_{\pa\{|u|>\d\}\cap B_{2r_0}(z)} g(u)\pa_i|u|\f{\psi_\e(|v_i|)}{|v_i|} \phi \g_i d\s+ \int_{\pa\{|u|>\d\}\cap B_{2r_0}(z)}|u|\pa_i g(u)\f{\psi_\e(|v_i|)}{|v_i|} \phi \g_i d\s\\
=&:\mathrm{I}+\mathrm{II}
\end{split}
\eeq

We notice that on $\pa\{|u|>\delta\}$, if $\left|\nabla |u|\right|\neq 0$, then the outer normal vector can be written as $\g=\f{-\na |u|}{\left| \na |u|\right|}$, so we obtain that $\mathrm{I}\leq 0$.

For the term $\mathrm{II}$, we perform integration by parts again to get
\beq
\begin{split}\label{estimate:bdy int}
\lim\limits_{\d\ri 0}\mathrm{II}\leq &\lim\limits_{\d\ri 0}\d
\left|\int_{\pa\{|u|>  \delta\}\cap B_{2r_0}(z)} \pa_ig(u)\f{\psi_\e(|v_i|)}{|v_i|}\phi\g_i\,d\sigma\right|\\
\leq &\lim\limits_{\d\ri 0}\delta \int_{\{|u|>\delta|\cap B_{2r_0}(z)\}}\left|\pa_i(\pa_ig(u)\f{\psi_\e(|v_i|)}{|v_i|}\phi)\right|\,dx=0
\end{split}
\eeq
We note that in the last step of \eqref{estimate:bdy int}, the limit is zero since it is the multiplication of $\d$ (goes to zero) and a bounded integral (the bound depends on $\e$, but doesn't depend on $\d$).

Combining \eqref{estimate:integral}, \eqref{First part:bounded by C}, \eqref{second part} and \eqref{estimate:bdy int} will conclude the proof of the Claim.

\end{proof}

On the other hand, we compute
\begin{align*}
I&=\int_{\ta}\big(\na v_i \na\psi_\e(|v_i|)\f{v_i}{|v_i|}\phi\big)+\big( \na v_i\cdot \na(\f{v_i}{|v_i|})\psi_\e(|v_i|)\phi \big)\\
&\qquad\qquad\qquad  +\big( \na v_i \f{v_i}{|v_i|}\na\phi \psi_\e(|v_i|) \big)\,dx\\
&=\f{C}{\e}\int_{\ta\cap\{0<|v_i|<\e\}} |\na |v_i||^2\phi\,dx\\
&\qquad + \int_{\ta\cap \{|v_i|>0\}} \left( |v_i|^{-1}|\na v_i|^2- |v_i|^{-1}|\na|v_i||^2\right)\psi_\e(|v_i|)\phi \,dx\\
&\qquad +\int_{\ta}\left( \na|v_i|\na\phi\psi_\e(|v_i|) \right)\,dx
\end{align*}
Note that we have
\beqo
\begin{split}
&\qquad\qquad\qquad |v_i|^{-1}|\na v_i|^2- |v_i|^{-1}|\na|v_i||^2\geq 0,\\
&\int_{\ta}\left( \na|v_i|\na\phi\psi_\e(|v_i|) \right)\,dx\leq (\int_{\ta}|\na |v_i||^2)^{\f12}(\int_{\ta} |\na \phi\psi_\e(|v_i|)|^2)^{\f12}\leq C(r_0).
\end{split}
\eeqo
Combining with \eqref{claim est I}, we conclude that
\beq\label{meas est:Sigma_e}
\int_{\S_\e}|\na|v_i||^2\,dx\leq C(g,r_0)\e, \quad \forall \e<<1.
\eeq

We need the following lemma.
\begin{lemma}\label{vi energy}
There are constants $\e_0$ and $C$ such that for every $\e\leq \e_0$,
\beqo
\sum\limits_{i=1}^n\int_{B_\e(z)\cap\Om(u)}|\na|v_i||^2\,dx\geq C\cl^n(B_{\e}(z)),\quad \forall z\in\G(u).
\eeqo
\end{lemma}
\begin{proof}
Recall that $\Om(u)=\{|u|>0\}$. If the statement is false, there exists a sequence of uniformly bounded entire minimizers $\{u_j\}_{j=1}^\infty$, $\{z_j\in\G(u_j)\}$,  $\{\e_j\}$ as well as $\{C_j\}$ such that
\begin{align}
&\nonumber\qquad\qquad \lim\limits_{j\ri\infty}\e_j=0,\quad \lim\limits_{j\ri\infty} C_j=0,\\
&\label{contradict1}\sum\limits_{i=1}^n\int_{B_{\e_j}(z_j)\cap\Om(u)}|\na|\pa_iu_j||^2\,dx< C_j\cl^n(B_{\e_j}(z_j)).
\end{align}
Define $f_j(x): B_1(0^n)\ri \BR^m$ as
\beqo
f_j(x):=\f{u_j(z_j+\e_jx)}{\e_j^2}.
\eeqo
Then by Proposition \ref{prop growth}, \eqref{control of u} and \eqref{contradict1}, we have
\begin{enumerate}
\itemsep0.5em
\item $|f_j(0^n)|=|\na f_j(0^n)|=0\quad \text{for every }j$,
\item $\|f_j\|_{C^{1,\g}(B_1(0^n))} (\g<1)$ is uniformly bounded.
\item $\sum\limits_i^n \int_{B_1}|\na |\pa_i f_j||^2\,dx\leq C_j\om_n,$
\item $\sup_{B_{\f12}(0^n)} |f_j(x)|\geq C>0$ for some constant $C$.
\end{enumerate}
Using all these properties, we can get the following convergence results up to some subsequence,
\begin{align}
&\nonumber \qquad\qquad\quad f_j\ri f \text{ in }C^{1}(B_{1}(0^n)),\\
&\label{convergence1}|f(0^n)|=|\na f(0^n)|=0,\quad \sum\limits_i^n \int_{B_1}|\na |\pa_i f||^2\,dx=0. \\
& \label{convergence2}\quad \qquad\qquad \sup_{B_{\f12}(0^n)} |f(x)|\geq C>0.
\end{align}
Note that \eqref{convergence1} implies that $f\equiv 0$ in $B_1(0^n)$, which yields a contradiction with \eqref{convergence2}. The proof of Lemma \ref{vi energy} is complete.
\end{proof}

According to Besicovitch covering lemma, we can find a covering of $\G(u)\cap B_{r_0}(z)$ by a finite family of balls $\{B_j\}_{j\in J}$, such that each ball is of radius $\e$ and centered on $\G(u)\cap B_{r_0}$, and no more than $N_n$ balls from this family overlap, where $N_n$ is independent of $\e$ and of the set $\G(u)\cap B_{r_0}$. By the estimate \eqref{control of nablau}, we have $B_{\e}(z)\cap\Om(u)\subset \S_{C\e}$ for some constant $C$. Consequently, we obtain
\begin{align*}
\sum\limits_{j\in J}\cl^n(B_j)&\leq C \sum\limits_{j\in J}\sum\limits_{i=1}^n\int_{B_j\cap\{|u|>0\}}|\na(|v_i|)|^2\,dx\quad \text{(by Lemma \ref{vi energy})}\\
&\leq C\sum\limits_{i=1}^n\int_{\S_{C\e}}|\na(|v_i|)|^2\,dx\leq C(g,r_0)\e \quad \text{(by \eqref{meas est:Sigma_e}).}
\end{align*}
This implies
\beqo
\sum\limits_{j\in J}\e^{n-1}\leq C(g,r_0),
\eeqo
for some constant $C(n,r_0)$ independent of the choice of $z$. Finally letting $\e\ri 0$, we get
\beqo
\mathcal{H}^{n-1}(\G(u)\cap B_{r_0}(z))\leq C(g,r_0).
\eeqo
The proof of Theorem \ref{local estimate} is complete.
\end{proof}

\begin{rmk}\label{local_rmk}
Using Theorem \ref{local estimate}, we can prove that for any $R$, there exists $C(R)$ such that
\beqo
\ch^{n-1}(B_R(x)\cap \G(u))\leq C(R)
\eeqo
To prove this, one simply covers the set $B_R(x)\cap \G(u)$ by identical small balls $\{B_{r_0}(z_i)\}$ such that $z_i\in \G(u)$ for every $i$. We omit the details.
\end{rmk}

Now we use the local estimate Remark \ref{local_rmk} and Theorem \ref{first part main theorem} to prove the global estimate of the $\mathcal{H}^{n-1}$ measure of the free boundary $\pa^* I_0$.

\begin{thm}[Second part of Theorem \ref{main}]\label{global estimate}
Let $\al\in (0,2)$, $x_0\in \BR^n$. There are constants $c_1,r_0$ such that for any $r>r_0$,
\beqo
\mathcal{H}^{n-1}(\pa^* I_0 \cap B_r(x_0))\geq c_1r^{n-1}.
\eeqo
And when $\al=1$, there are constants $c_2, r_0$ such that for $r\geq r_0$,
\beqo
\mathcal{H}^{n-1}(\pa^* I_0 \cap B_r(x_0))\leq c_2r^{n-1}.
\eeqo
\end{thm}

\begin{rmk}
Unlike the local estimate, here all the constants depend on $u$.
\end{rmk}

\begin{proof}
Without loss of generality we take $x_0=0^n$. According to Theorem \ref{two phase existence}, for sufficiently large $r$, there are two phases $a_1,a_2\in A$, which depend on $r$,  such that
\beqo
\cl^n(B_r\cap \{u=a_j\})\geq cr^n, \quad j=1,2.
\eeqo
Using the relative isoperimetric inequality we obtain that
\beqo
\begin{split}
&\ch^{n-1}(\pa^*\{|u-a_1|>0\}\cap B_r)\\
\geq &C\left(\min\{\cl^n(B_r\cap \{u=a_1\}), \cl^n(B_r\backslash \{u=a_1\} )\} \right)^{\f{n-1}{n}}\geq c_1r^{n-1},
\end{split}
\eeqo
which gives the lower bound. Note that this estimate is valid for any $0<\al<2$.

For the upper bound, we fix $\al=1$ and examine more closely the proof of Theorem \ref{first part main theorem}. Again we consider the domain $\tilde{S}_{kL}$ and classify all the sub-cubes $\{S_i\}_1^{(2k)^n}$ into five classes $T_1$--$T_5$. If $S_i\in T_4$, then $u(x)\equiv a_{j_0}$ for all $x\in \overline{S_i}$, which implies $\ch^{n-1}(S_i\cap \pa^* I_0)=0$. Moreover, for any $x_0\in\pa S_i$, by the definition of $T_4$ it holds that
\beqo
\max\limits_{x\in B_L(x_0)} |u(x)-a_{j_0}|\leq \theta.
\eeqo

By the proof of Lemma \ref{value of L}, we obtain that $B_{L/4}(x_0)\subset \{u=a_{j_0}\}$ and consequently $x_0\not\in \pa^* I_0$. As a result, we have
\beqo
\ch^{n-1}(\overline{S_i}\cap \pa^* I_0)=0.
\eeqo

Using estimates in Theorem \ref{first part main theorem} and Remark \ref{local_rmk}, we have for large enough $k$
\beq\label{upper bd free bdy}
\begin{split}
&\ch^{n-1}(\pa^*I_0\cap \tilde{S}_{kL})\\
\leq & \sum\limits_{S_i\in T_1\cup T_2\cup T_3\cup T_5} \ch^{n-1}(\pa^*I_0\cap \overline{S_i}) \\
\leq &(|T_1|+|T_2|+|T_3|+|T_5|) C(L)\\
\leq & c_2(W,u) k^{n-1}.
\end{split}
\eeq
The upper bound follows immediately from \eqref{upper bd free bdy} and the proof is complete.

\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
What did BlackBerry's co-founders just tell us?
Dan Primack
FORTUNE — BlackBerry co-founders Mike Lazaridis and Douglas Fregin today said in an SEC filing that they are considering a buyout offer for the struggling smartphone maker, in which they currently hold a combined 8% stake. There is no mention of potential partners, although the pair is known to have previously reached out (unsuccessfully) to private equity firms.
What's interesting here isn't that the pair has interest in BlackBerry (BBRY). For example, there were reports last month that Lazaridis had reached out to private equity firms like The Blackstone Group (BX) and The Carlyle Group (CG). Instead, it's that they aren't aligned with Canadian conglomerate Fairfax Financial, which recently "agreed" to buy BlackBerry for $4.7 billion.
I put "agreed" in quotation marks because the Fairfax deal had more holes than the Bushwood Country Club golf course. For starters, Fairfax had not yet conducted due diligence, and had the right to terminate the deal without paying any penalty. More importantly, Fairfax said that it was leading an investor group without identifying any of the other members (for its part, Fairfax only planned to roll over its 10% stake).
There had been some widespread assumptions that Fairfax planned to fill the equity hole by partnering with other existing shareholders like Lazaridis and Fregin. But the new filing would suggest the two are not working in concert. Or at least not yet. It also suggests that their talks with private equity firms have not yet borne fruit.
So today's development really just creates more questions than answers. Which really sums up the entire BlackBerry situation to date.
Sign up for Dan Primack's daily email newsletter on deals and deal-makers: GetTermSheet.com

redpajama_set_name: RedPajamaCommonCrawl


Bullies are a time-honored personality type. (To be honest, we are all bullies to some degree, or at least, capable of being bullies.) Bullies are instinctively drawn to the weak and defenseless; mentally ill folk always make the list. Left unchecked; bullies morph into monsters, I know. Philly, my home town, is among the nation's deadliest cities, thug violence is commonplace. Indeed, I was once beaten unconscious with lead pipes and left for dead in a snowbank.
Back when I was cab driving, a hard-bitten veteran told me, "There is only one way to deal with a gang of "punks" coming for you. You don't run, you don't talk, and you don't make deals. You figure out which one is the leader and you stick a knife in his face." My own mother, a reasonable and patient individual, once tried to run my father over with a 1956 Pontiac Chieftan (a very large car) simply because she could not endure being bullied any longer.
The confrontation approach may win short-term but always fails long-term for the simple reason that it plays to the bully's area of strength; violent brutality. To defeat the bully you must understand, and eliminate, your fear of him. When he realizes you will accept a beat-down if you must, the power he holds over you slips in between his fingers. When he looks into your eyes and sees you looking back, the mean, sadistic thrill he craves is gone. At that point he will seek out a more timid individual.
That is the joyful power and freedom that come from going toe-to-toe, and not flinching. If that is not enough for you, if you are full of hate and resentment, if you dream of reducing this wretched excuse for a human being to a quivering, pathetic blob of sopping flotsam; then it is time to remove the ruthless sword of humor from its sheath.
When you make it obvious you find the bully pathetic and laughable, he is vanquished. And it's a very reasonable assessment because bullies are the very antithesis of what they appear to be. Coming across mean and rough is merely their way of masking cowardice and self-loathing.
Nothing ever just gets better; what happens is; if you're lucky, you get better.


Nifty futures, in reality, are index futures where the underlying is the S&P CNX Nifty index.
However, in India, index futures trading commenced in 2000 on the National Stock Exchange (NSE).
Moreover, for the Nifty futures contracts, the permitted lot size is 50, and in multiples of 50.
Striker Stock Research is the Best Stock Advisory Company in India to provide Nifty Future Tips.
In fact, like other futures contracts, Nifty futures contracts also have a three-month trading cycle — the near-month, the next month and the far-month. Nifty Future Tips.
So, after the expiry of the near-month contract, a new contract of a three-month duration would be introduced on the next trading day. Best Digital Marketing Course in Indore.
In reality, investors can trade in Nifty futures by having a margin amount in their account.
However, this margin is a percentage of the contract value. Nifty Future Tips.
Usually, it is about 10-12 percent. Nifty Future Tips.
So, by trading in index futures, an investor is buying and selling the basket of stocks comprising the index, in their respective weights.
In fact, stock index futures are traded in terms of a number of contracts. Nifty Future Tips.
However, each contract would be to either buy or sell a fixed value of the index.
Moreover, the value of the contract will be the multiplication of lot size by the index value.
There are times when you sell the stock, but there is an upside in the market, thus resulting in lost potential profits. Nifty Future Tips.
Index futures help you mitigate this risk.
By buying index futures when you are short on the stock, you can minimize the number of potential profits lost.
Also, you may have face the times when you own a portfolio and are uncomfortable about market conditions.
You can hedge this risk by selling index futures.
The concept rests on the fact that every portfolio has index exposure.
Further, the counting of risks is as per the fluctuations in the index.


This question raised multiple threads of inquiries, like – What exactly is a river? What is a living river? What is a dying river? Do we ignore or abuse rivers because we haven't understood them?
Unlike the word 'river,' in reality, it is one of most ecologically and sociologically complex entity. To address the complexity of the river, a river can defined as – "a channel carrying freshwater, but a hydrological, geomorphic, ecological, biodiversity-rich, landscape level system that serves as a key part of the freshwater cycle, balancing dynamic equilibrium between snowfall, rainfall, surface water and groundwater, and provides a large number of social and economic services to the people and ecosystems all through its watershed"1.
But then the question arises – what is a living river? Or what are the characteristics of a healthy river? Ramaswamy R Iyer, who was one of the leading water experts of our country, outlined three essential features of the river – Aviralta or uninterrupted flow, Flow variation, and Nirmalta or absence of pollution – and without these, a river is no longer a river or is dead.
"Aviralta or uninterrupted flow: Flow connectivity of a river cannot be compromised. By leaving empty long stretches of river bed without the river, as in the case of hydroelectric projects, river identity is lost. Unconnected water bodies simply are no longer a river anymore.
Flow variation: Violent and destructive fluctuations in the water flow, where a river remains dry for 20-24 hours and encounters a 8 m water wall in the remaining 4 hrs, as in the case of run-of-the-river projects, can no longer be called a river.
If we analyze the status of the Godavari in Nashik, as per the above mentioned outline, we will encounter a frightening reality that the Godavari is on the verge of dying or dead in some stretches.
Nonetheless, there is still hope left, only if, the citizens of Nashik begin worshiping and respecting Godavari, not only intangibly through rituals but also in materiality through their actions.


I promised last week to post samples of what we made at my open house, so here are the pictures.
This first card is made with the Friends Never Fade stamp set (pg. 99 IB&C) and the Perfect Punches stamp set (pg. 131 IB&C). The colors were taken from the Green House Gala DSP (pg. 173 IB&C). I used Real Red, Chocolate Chip, Baja Breeze, and Whisper White cardstocks, Real Red, Daffodil Delight and Always Artichoke markers, and Chocolate Chip and Baja Breeze ink pads (Baja Breeze was used to sponge around the edges of the white images). The Chocolate Chip CS was run through the Big Shot with the new Vintage Wallpaper Textured Impressions Embossing Folder, then a layer of the DSP was added to the left side of the Chocolate Chip CS with a piece of Real Red 5/8" Satin Ribbon brought down the center to "cover up" the seam. The greeting was stamped with Chocolate Chip on Whisper White and then punched out with the 2"x 1-3/8" Extra-Large Oval Punch (pg. 202 IB&C) and attached with Dimensionals. This card idea came from one of the groups that I'm on, but I forget who the original creator was. If it's you, please let me know and I'll be happy to give you credit!
That's one of the great things about stamping - you don't have to come up with ideas on your own. The Idea Book & Catalog has LOTS of ideas to help you start stamping, and then there are a lot of other groups that you can join or places you can go to look at samples created by other people. I would guess that I saw this card on the Late Night Stampers Yahoo group that I'm a part of. They have two groups - one that anyone can join and the other is exclusively for Stampin' Up! demonstrators. I love the creativity and sharing that goes on in that group! Actually, most demonstrators are more than willing to share their creative ideas and are flattered when you CASE one of their creations! I know that I am!
The second card is a monochromatic card made with one of the new Level 1 Hostess sets called Because I Care. Demonstrators were able to get this set for FREE when we ordered a certain item in our pre-order from the new catalog -- another benefit to being a demonstrator! Stampin' Up! often has a special just for demonstrators to show their appreciation for us!
This card is made with Bashful Blue and Marina Mist cardstock and inks. The paper on the left side of the card is our new Presto Patterns Specialty DSP (pg. 172 IB&C). The picture in the catalog doesn't do this paper justice! It's sooo cool! The paper is white with white glossy designs on it. To get it to look like this on the card, we just brayered Marine Mist ink over the top and then lightly buffed it with a tissue to bring out the white pattern. Presto!--really cool looking paper in any color you want! The main image has a piece of the new 1/8" Taffeta Ribbon in Basic Black across the upper left hand edge. The saying is stamped and punched out with the Large Oval Punch and layered onto Basic Black cardstock punched with the Extra-Large Scallop Oval punch.
The last project was a sampler of the new In Colors for this year. I only had the cardstock here when I had my open house, so everyone made this little sampler of cardstock so they could have each of the new In Colors for their own at home. The new In Colors are shown on page 165 of the new IB&C are are luscious!
Here are a few other pictures of my open house before everyone got here. Once people started coming, I got too busy and forgot to take any pictures! But everyone who came said they had a great time. I also offered a special this year - anyone who ordered at least $35 in merchandise from the new Idea Book & Catalog got to take their catalog home for FREE! I usually like to have some special offer at my open house and it changes from year-to-year. After all, I don't want to get in a rut!
I made up a board to show our colors in their new color families. I LOVE the new color families! They are all so rich looking!
I also gave everyone who came a cardstock sampler. It has each of the new colors in the core collections such as Daffodil Delight, Tangerine Tango, Early Espresso, etc. The cardstock was punched out with the new 2-3/8" Scallop Circle Extra-Large punch (pg. 201 IB&C). Love it!
All-in-all, I had a great day and I think everyone who came really enjoyed themselves (at least that's what they told me!). If you couldn't make it to my open house but still want to get a new IB&C, please contact me and let me know!