# 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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### Is the image of gradient map from Sobolev space to Lebesgue space weakly closed?

Suppose f is a map defined between $W_0^{1,p}(\Omega)$ and $L^{p'}(\Omega)$ as follows - $u \mapsto |\nabla u|^{p-1}$. Is the range of this map weakly closed in $L^{p'}$?.
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### Distributional derivative of absolute value function

I'm tying to understand distributional derivatives. That's why I'm trying to calculate the distributional derivative of $|x|$, but I got a little confused. I know that a weak derivative would be ...
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### $L^2$ inner product split over sub-domains in $\mathbb{R}^3$

I have a bounded Lipschitz domains $\Omega, \Omega_1, \Omega_2 \subset \mathbb{R}^3$ such that $\overline{\Omega}=\overline{\Omega}_1 \cup \overline{\Omega}_2$ and $\Omega_1 \cap \Omega_2=\emptyset$. ...
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### Prove that each $W_0^{1,2}$-function is weakly differentiable

Let $\Omega\subseteq\mathbb R^n$ be open. $u\in\mathcal L^1_\text{loc}(\Omega)$ is called weakly differentiable $:\Leftrightarrow$ $\exists v\in\mathcal L^1_\text{loc}(\Omega;\mathbb R^n)$ with ...
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### Difference quotients and weak derivatives (Evans 5.8.2 theorem 3) [closed]

Could anyone give a proof on the remark below the theorem? Basically it is problem 11 I think the proof relates to example 19.19 (p.375) in enter link description here But I really do not ...
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### Computing weak derivatives on an open square

I am looking at the computation of weak derivative in the blog https://sunlimingbit.wordpress.com/2012/11/25/one-example-related-to-weak-derivative-2/ For equation (3), I have some confusions. Here ...
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### Does convergence in H1 imply pointwise convergence?

I'm trying to figure out if convergence in $H^1(a,b)$ implies pointwise convergence (by the way: what is the usual name of this space?). It is defined to be Hilbert space of absolutely continuous ...
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### Using a sequence of measures to create simple functions which approximate the Radon-Nikodym derivative of the limiting measure

I have a bunch of discrete probability measures with finite support: $\mu_1,\mu_2,\dots$, which strongly converge to an absolutely continuous probability measure $\mu$ in $\mathbf{R}^2$. That is, for ...
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### Does the existence of weak derivatives require the lower order derivatives also to exist?

I just want to confirm that for weak derivatives you don't require the lower order derivatives to exist in order for the higher order derivatives to exist?
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### $\nabla\phi_k\stackrel{L^2}{\to}\nabla\phi\Rightarrow\langle\nabla u,\nabla\phi_k\rangle\stackrel{L^2}{\to}\langle\nabla u,\nabla\phi\rangle$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $(\phi_k)_{k\in\mathbb{N}}\subseteq L^2(\Omega)$ with ...
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### How to solve a simple differential equation in the way of weak solutions?

I want to prove that the solution equation $y'=y$ is $y=Ce^x$, where $C$ is a constant. Here $y$ belongs to the space of linear operators on $C_0^\infty(\mathbb{R})$, and $y'$ is its weak ...
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### Interpolation between derivatives

I am trying to prove the following: Let $u \in W^{2,2}(\mathbb R)$. Then $\| u^\prime \|_{L^2}^2 \leq \| u \|_{L^2} \| u^{\prime \prime} \|_{L^2}$ holds (these are meant to be weak derivatives). ...
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### Definition of the Second Variational Derivative In terms of The first

I know that for functional $F$ the first variational derivative at $f$ with increment $h$ is defined as \begin{align*} \delta F[f,h]= \lim_{\alpha \to 0 } \frac{F[f+\alpha h]-F[f] ]}{\alpha }. ...
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### Compact support of derivatives of $u$ in weak sense

I would like to know if let $u:\mathbb{R}^n\to \mathbb{R}$ be a function in $W^{k,p}(\mathbb{R}^n)$ such that $u$ has compact support in $\mathbb{R}^n$ then, for each $|\alpha|\le k,$ $D^\alpha u$ ...
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### Weak derivative of $t^{\beta-1}$

Let $g(t)=t^{\beta-1}$, where $\beta\in (0,1)$. I am pondering, if there exists a weak derivative of $g$ in space $L^1(0,T)$, $T>0$. Firstly, we see that $g\in L^1(0,T)$. Now, we are looking for ...
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### Dirac Delta function to solve PDE in the sense of distribution

For each Borel set $A$ and $x\in\mathbb{R}^n$, denote Dirac measure centered at $x$ as \delta_x(A)=\begin{cases} 1 & \mbox{ if } x\in A \\ 0 & \mbox{ otherwise}. \end{cases} ...
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### Continuous function without a weak derivative

Let $f:\Omega\to\mathbb{R}$ be a continuous function. Is it necessarily true that $f$ has a derivative in the weak sense? That is, is there some $v:\Omega\to\mathbb{R}$ such that for every test ...
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### Show that this function is weakly differentiable

I need to show that the function $$u(x,y) = 1-x_1^2 \quad x_1>0 \\ u(x,y)=1+x_1^2 \quad x_1 \leq 0$$ is weakly differentiable on the unit ball. It is clear what the ...
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### Questions about weak derivatives

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
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### Second derivative Dirac delta distribution times $(x-a)^2$, intepretation

I'm not sure if this calculation is correct and if I interpret it correctly (from old exam). Show that $(x-a)^2 \delta ''_a = 2 \delta _a$. We have for distributions $f$ and test functions ...
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### Energy functional for a differential equation

Is there a variational formulation for the following differential equation: $\frac{\partial}{\partial x}(D(u,x)\frac{\partial u}{\partial x})=0$ $x$ varies over $[0,1]$, $D$ is bounded, is ...
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### Weak derivative (Sobolev spaces)

I'm reading "Functional Analysis" - Michel Willem and I can't understand the definition of weak derivative from chapter 6, namely the definition of $$\partial^\alpha f.$$ Can you give me a concrete ...
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### A step of change of variables I don't understand when show derivative of a function in $L^n$?

How (16) can goes to (17) and how to show $loglog(1+\frac{1}{|x|})\in L^n$ for $n>1$? My attempt: Let $|x|=r$, then $(x^2)^{\frac{1}{2}}=r$. Differentiate on both sides, I get ...
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### Proof-verification: $C^1$-functions are weak differentiable

I wanted to prove that the strong derivative coincides with the weak derivative if the function is regular enough, i.e.: Let $U \subset \mathbb{R}^n$ be an open subset and $f \in C^1(U)$, then for ...
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### divergence form of the determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
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### Question about this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

In this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$ I don't understand this part: $$\int_U \partial_i u^\epsilon\,v\,dx\to 0\tag w$$ for all $v\in L^2(U)$, but now we only ...
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### How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?

How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$? Since $|u|\in L_p$, I only need to show weak derivative of $u$ exists and $Du \in L_p$. Can anyone give me some hint? Thanks!
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### How to prove the chain rule with respect to weak derivatives?

The following theorem is on the textbook "weak differentialble functions". I found it confusing from the absolutely continuous part. I am writing to ask is this the only way to prove it? Can anyone ...
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### Difference quotient bounded in $L^1$ does not imply the weak partial derivative exists and is in $L^1$

Let $U\subset \mathbb{R}^{n}$ be open and $V \subset\subset U$ ($V$ is open and compactly contained in $U$). $C$ is a fixed constant. Given $u\in L^1(U)$. Show by example that if we have ...
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### Proof of the Poincare inequality for $W_0^{1,2}((a,b))$.

I have a question about an exercise for which I already have the solution, which I do not unterstand completely. Let $a, b \in \mathbb R$ with $0 < a < b$. Then we have \begin{align*} ...
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### first order weak derivative of function $f(x) = |x|$

let $f(x)= |x|$ how can I calculate the first order weak derivative of this function in $x=0$? Does anyone have an idea on how to calculate this?