# Tagged Questions

For question about weak derivatives, a notion which extends the classical notion of derivative and allows us to consider derivatives of distributions rather than functions.

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2answers
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### Laplace transform of the derivative of the Dirac delta function

$$\int_{0}^{\infty} \delta'(t) e^{-st} \ dt = \delta(t)e^{-st} \Big|_{0}^{\infty} + s \int_{0}^{\infty} \delta(t) e^{-st} \ dt$$ = 0 - \lim_{t \to 0} \delta(t)e^{-st} + s e^{-st}\Big|_{t=0} = - ...
1answer
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### Finding the weak derivative of order $3$ of $f(x)=\operatorname{sgn} \sin(x)$ where $\operatorname{sgn}$ is the sign function

Let $$f(x)=\operatorname{sgn} \sin(x)$$ where $\operatorname{sgn}$ is sign function. I need to find the weak derivative of order 3 for $f(x)$?
1answer
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### Use difference quotient with not uniform bound to appoximate weak derivative

Suppose U is an open set,not necessarily bounded or has Lipschitz boundary, $f\in L^p(U)$ ,define the difference as usual: $$D^h_i f=\frac{f(x+he_i)-f(x)}{h},\ \ \forall x\in U'\subset\subset U$$ ...
1answer
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### Why is the weak limit of the derivatives the derivative of the weak limit here?

In [1, chapter 8.2.1.b, p.466] the author uses the following argument: Let $U \subset \mathbb{R}^N$ be an open, bounded domain with smooth boundary. Given a bounded sequence \$(u_k)_{k \in ...
2answers
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### Weakly differentiable but classically nowhere differentiable

Is there any example of a function which is weakly differentiable but none of its versions are classically differentiable (or differentiable only on a set of measure 0) ? Thanks
1answer
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### Weak Differentiability of Holder functions

Is it true that every Holder function is weakly differentiable? If not please give counterexample. Thanks
2answers
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### about weak derivative ( Sobolev Spaces )

the following afirmation is true ? Consider $\Omega$ a bounded and smooth domain . Let $u \in W^{1,p} ( \Omega)$ ( p>1). Supose that $u \geq 0$. let $\alpha >1$ . Then \$\nabla u ^{\alpha} = ...
0answers
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### A weak chainrule [Urbano, Intrinsic Scaling]

Hey I'm reading the book on intrinsic scalign by Urbano an there is a certain issue i have problems with. Essentially the problem is the following. Let $\Omega\subset \mathbb{R}^n$ be a bounded ...
1answer
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### Derivative in the sense of distributions

I have a question regarding calculating the derivative in the distribution sense of the following function: $$f(x) = \frac{d^2 }{d x^2}|\cos|x||$$ Maybe someone can point me in the right ...
2answers
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### Weakly convergence in $W^{1,q}, 1<q<\infty$

Let $(x_n)$ be weakly convergent against $x$ in the Sobolev space $W^{1,q},1<q<\infty$. Now I have to show, that $(\dot{x_n})$ converges weakly against $\dot{x}$ in $L^q$. (With the point I ...
1answer
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### Prove that $f$ does not have a weak derivative

Consider a function $f:\mathbb{R} \rightarrow [0,1 ]$ defined by: \$\begin{equation*} f(x)=\left\{ \begin{array}{rl}0 & \text{if } x\leq 0,\\ 1 & \text{if } x\geq 1, \\ 1/2 & \text{if } ...
1answer
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### Distributional/weak time derivative basic question

Suppose we have $u \in L^2(0,T;H^1(\Omega))$, and $v \in L^2(0,T;H^{-1}(\Omega))$ is the weak time derivative of $u$, so by definition it satisfies $$\int_0^T u(t)\phi'(t) = -\int_0^T v(t)\phi(t)$$ ...
0answers
99 views

### When the weak derivative just is the strong (or classical) derivative?

When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).
1answer
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### Weak derivative of $\operatorname{sgn}(x_1)$

Let $x\in \mathbb{R}^{n}, x = (x_1,\ldots,x_n)$, and $f(x) = \operatorname{sgn}(x_{1})$. Is $f$ weakly differentiable on $U = B(0,1)$, i.e. unit ball in $\mathbb{R}^{n}$, and what is the weak ...
1answer
222 views

### Quick question on definition of derivative in the sense of distribution

Consider a function f such that on $(-\infty,0)$ and $(0,\infty)$, f is differentiable. At 0, there is a point of discontinuity. e.g. $f(x) = 0$ for $x\leq 0$ and $f(x)=x$ for $x>0$ Then if we ...
1answer
168 views

### Easy question on derivative in the sense of distribution

I would like help proving this elementary result: Let $f\in L^{1}_{loc}(a,b)$. Let $x_0 \in (a,b)$ Let $F(x)=\int^{x}_{x_0} f$. Then $F'=f$ in the sense of distributions. i.e How do I show ...
1answer
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### Consistency of derivative definitions in Sobolev spaces

Just for the sake of completeness, I begin defining the Sobolev space $H^m(\mathbb{R}^n), \; m \in \mathbb{N}$, as the following set: \$H^m(\mathbb{R}^n) = \{u \in L^2 : P^{\alpha} F u \in L^2,\; ...
1answer
126 views

### Weak derivatives

How can I prove that $|\nabla u|=|\nabla|u||$ when $u$ is regular enough for example Lipschitz or $W^{1,1}_{loc}$. Other question is about the pointwise derivative when $f:[0,1]\to R$ is BV is that ...
0answers
229 views

### Weak derivative

Let $u \in C(\Omega)$ be a function with weak derivative $Du \in C(\Omega)^n$. How does one prove that $Du$ coincides with the classical derivative? Is the mean value theorem for integration ...
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