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2 reversed inclusion for $H^1$ and $L^4$

I am not going to try to find the most general conditions under which lower semi-continuity holds but for that I suggest the standard reference for all of this is "Direct methods in the Calculus of Variations" by Bernard Dacorogna which covers all of this in full detail. I will give a brief outline of the answer however.

In order for $\int f(x,u,du)dx$ to be lower semi-continuous with respect to weak convergence one does not in general need any sort of convexity in the $u$ variable and what is more important generally is

1) Coercivity (see below) of the functional $f$ in the $du$ variable.

2) The right embedding along with continuity of $f$ in the $u$ variable with respect to the topology of strong convergence in this space.

Example: A good example is $E(u) = \int_0^1 |\nabla u |^2 + (1-u^2)^2$ with $u = 0$ on $\partial \Omega$ in $\mathbb{R}^d$. Observe that when $d \leq 4$ one has $L^4 H^1(\Omega) \subset \subset H^1(\Omega)$ L^4(\Omega)$and consequently the non-convex term depending on$u$is lower order. Therefore one can pass to the limit in any minimizing sequence even though$(1-u^2)^2$is very non convex (but it is continuous with respect to strong convergence). Notice however that for the energy$\bar E(u) = \int_0^1 |\nabla u|^2 + u^4$in$\mathbb{R}^5$with$u$prescribed on$\partial \Omega\$, the second term is not lower order but here we may use convexity to conclude lower semi-continuity of the second term.

The function $p \mapsto f(x,z,p)$ is coercive if there exists some constant $C > 0$ so that $f(x,z,p) \geq C|p|^q$ for some range of $q$. It then depends on what spaces one is working in but the goal is to use an embedding theorem such as $L^q \subset\subset W^{1,p}$ for $1 \leq q < p^*$ and to conclude that for a minimizing sequence $u_n$, there is in fact a strongly convergent subsequence in $L^q$ for some $q$. Then one will generally expect some sort of continuity of $f$ in the $u$ variable.

There are many more interesting examples but in almost all cases in practice the goal is to show that one has strong convergence of the $u_n$s in your minimizing sequence in some $L^p$ space. The book by Braides focuses mostly on asymptotics of functionals which depend on some large (or small) parameter and I'm not sure how much he talks about the assumptions needed in the direct method.

1

I am not going to try to find the most general conditions under which lower semi-continuity holds but for that I suggest the standard reference for all of this is "Direct methods in the Calculus of Variations" by Bernard Dacorogna which covers all of this in full detail. I will give a brief outline of the answer however.

In order for $\int f(x,u,du)dx$ to be lower semi-continuous with respect to weak convergence one does not in general need any sort of convexity in the $u$ variable and what is more important generally is

1) Coercivity (see below) of the functional $f$ in the $du$ variable.

2) The right embedding along with continuity of $f$ in the $u$ variable with respect to the topology of strong convergence in this space.

Example: A good example is $E(u) = \int_0^1 |\nabla u |^2 + (1-u^2)^2$ with $u = 0$ on $\partial \Omega$ in $\mathbb{R}^d$. Observe that when $d \leq 4$ one has $L^4 \subset \subset H^1(\Omega)$ and consequently the non-convex term depending on $u$ is lower order. Therefore one can pass to the limit in any minimizing sequence even though $(1-u^2)^2$ is very non convex (but it is continuous with respect to strong convergence). Notice however that for the energy $\bar E(u) = \int_0^1 |\nabla u|^2 + u^4$ in $\mathbb{R}^5$ with $u$ prescribed on $\partial \Omega$, the second term is not lower order but here we may use convexity to conclude lower semi-continuity of the second term.

The function $p \mapsto f(x,z,p)$ is coercive if there exists some constant $C > 0$ so that $f(x,z,p) \geq C|p|^q$ for some range of $q$. It then depends on what spaces one is working in but the goal is to use an embedding theorem such as $L^q \subset\subset W^{1,p}$ for $1 \leq q < p^*$ and to conclude that for a minimizing sequence $u_n$, there is in fact a strongly convergent subsequence in $L^q$ for some $q$. Then one will generally expect some sort of continuity of $f$ in the $u$ variable.

There are many more interesting examples but in almost all cases in practice the goal is to show that one has strong convergence of the $u_n$s in your minimizing sequence in some $L^p$ space. The book by Braides focuses mostly on asymptotics of functionals which depend on some large (or small) parameter and I'm not sure how much he talks about the assumptions needed in the direct method.
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