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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About the gradient of a function in $H^{1}(\Omega)$

let $\Omega \in \mathbb{R}^{n}$ a bounded domain and $u \in H^{1}(\Omega)$ a real function. In the Leoni's Book - A First Course in Sobolev Spaces, the author define $\nabla u = (D_{1} u,\dots, ...
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24 views

If $u \in W^{1,p}(I) \bigcap C_c(I)$ then $u \in W_{0}^{1,p}(I) $ where $ W^{1,p}(I)$ is the Sobolev Space

I want to show the following statement: If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, i.e. the space consisting of the functions that are ...
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1answer
31 views

Showing that $(G' \circ u)u' \in L^p(I)$ where $G \in C^1(\mathbb{R})$ and $u \in W^{1,p}(I)$

I am trying to prove the rule of differentiation of a composition for weak derivatives in Sobolev spaces following the proof given in Corolary 8.11 in Functional Analysis, Sobolev Spaces and Partial ...
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19 views

Convergence of the series $\sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha}$

Suppose $S=\{y_k\}_{k=1}^{\infty}$ is a countable dense subset of the open unit ball $B(0,1)$ in $\mathbb{R}^n$. We write \begin{equation} u(x) = \sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha} ...
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1answer
31 views

Does the weak divergence exist for each $\mathcal L^2(\Omega;\mathbb R^d)$-function?

Let $\Omega\subseteq\mathbb R^d$ be open. $v:\Omega\to\mathbb R$ is called weak divergence of $u:\Omega\to\mathbb R^d$ $:\Leftrightarrow$ $$\int_\Omega v\varphi\;{\rm d}\lambda=-\int_\Omega\langle ...
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1answer
22 views

is $u\in L_{loc}^1(\mathbb{R}^2)$ and 2. $u\in W_{loc}^{1,1}(\mathbb{R}^2)$?

Let $U\in C^1((0,\infty))$, $u:\mathbb{R}^2\to \mathbb{R}$ defined by $$u(x,y)=U(\sqrt{x^2+y^2}).$$ Under which conditions on $U$ is 1.$u\in L_{loc}^1(\mathbb{R}^2)$ and 2. $u\in ...
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1answer
21 views

Doubts on distributional derivative: null derivative and non negative derivative

suppose we are in $\mathbb R^n$ and fix $E\subset \mathbb R^n$ and let $\chi_E$ be the indicator function of the set $E$. I'm reading an article that says that a vector field $X$ on $\mathbb R^n$, ...
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28 views

Distributional derivative $Xu=0$ then $u$ is constant over the flow lines of $X$

Suppose we have a Lie group on $\mathbb R^n$, something like $(\mathbb R^n, \cdot)$, where with $\cdot$ we denote the group law on $\mathbb R^n$. Call $\mathfrak g$ the Lie algebra of $(\mathbb R^n, ...
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1answer
29 views

Functions of the form $u(x,y) = f(x-y)$ are weak solutions of $u_x + u_y = 0$

This is a problem out of Logan's Applied Math book. Section 6.7, problem 2. Show that for any locally integrable function f on $\mathbb{R}$ the function $u(x,y) = f(x-y)$ is a weak solution to ...
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42 views

an iff proof on the existence of weak derivative

I have trouble understanding the following proposition. Proposition $f,g\in L_{\text{loc}}^1(\Omega)$. Then $g=D^{\alpha}f$ iff. there exists $f_m\in C^{\infty}(\Omega)$ such that $f_m\to f$ in ...
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1answer
14 views

Are there $H^1_0(\Omega)$-functions in the plane that are discontinuous over curves?

Consider a bounded domain $\Omega \subset \mathbb{R}^2$ with Lipschitz-boundary and a curve $\gamma : [0,1] \rightarrow \Omega$. Is it possible to construct a function $f \in H^1_0(\Omega)$ which is ...
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1answer
20 views

$u\in W^{1,1}(U) \Rightarrow u^+\in W^{1,1}(U)$ , where $U\subset\mathbb{R}^n$ open

Im trying to show that for an open set $U\subset\mathbb{R}^n$ and a function $u\in W^{1,1}(U)$, also the positive part $u^+$ is in $W^{1,1}(U)$. My idea is the following: Let $E\subseteq U$ defined ...
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1answer
53 views

necessary and sufficient conditions for the existence of solution in the space $W^{k,p}$

I am learning about weak derivatives and sobolev space. In particular I need help to learn the proving strategy/technique. I have trouble proving on how to show a solution belongs to some sobolev ...
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33 views

$-u''+u=\delta_y$ where $u(0)=u(1)=0$

Let $0<y<1$ be arbitrary. What is the weak solution of the differential equation $-u''+u=\delta_y$ where $u(0)=u(1)=0$ then? The weak form of the equation above is given by ...
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1answer
20 views

Calculate the weak formulation of $\int_\Omega \left(\alpha v(x) + \beta\Delta\Delta v(x)\right)w(x)dx=\int_\Omega \gamma w(x)dx$

I would like to calculate the weak formulation of the following equation: $$ \alpha v(x)+\beta\Delta^2 v(x)=\gamma $$ Which brings me to this formula: $$ \int_\Omega \left(\alpha v(x) + \beta ...
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0answers
52 views

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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1answer
83 views

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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30 views

Weak chain rule the other way round.

If $f: \mathbb{R} \to \mathbb{R}$ is weakly differentiable and $u: \Omega \to \mathbb{R}$ is smooth, where $\Omega \subset \mathbb{R}^n$, can we show that for any test function $\varphi \in ...
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24 views

Why are these functions Locally Integrable?

Apparently the following are locally integrable: (a) $x^r\in L_1(0,\infty)$, $\forall r\in\mathbb{R}$. But this couldn't be right because $\frac{1}{x}\notin L_\text{loc}(\mathbb{R})$ (since ...
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1answer
21 views

How to handle a diffusion equation with a variable diffusion coefficient with finite elements?

Based on the notes here I created a finite elements solver for the stationary heat equation (Poission's equation) $$-u''(x) = f(x)$$ However I would like to solve the stationary heat equation that ...
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2answers
148 views

Exercice 4 chapter 7 Evans book

I want to solve the following problem: Suppose $H$ is a Hilbert space, if $u_{n}\rightharpoonup u$ in $L^{2}(0,T;H)$ and $u'_{n}\rightharpoonup v$ in $L^{2}(0,T;H^{'})$. I want to proof the $v=u'$. ...
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34 views

Weak differentiability and diffeomorphisms

Let $U,V\subset\mathbb{R}^n$ be open sets and assume the existence of a $\mathcal{C}^1$-diffeomorphism $\phi:U\rightarrow V$. Let $u\in W^{1,p}(U)$, $1\leq p\leq\infty$, and define ...
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0answers
16 views

Derivative is a Radon measure if time increment bounded

Here, $u$ belongs to some $L^p$ space. Can someone tell me what result the author uses to say that $u_t$ is a Radon measure?
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1answer
54 views

Weak convergence and weak convergence of time derivatives

I am working in $H^1(S^1)$, the space of absolutely continuous $2\pi$-periodic functions $\mathbb R\to\mathbb R^{2n}$ wih square integrable derivwtives. I have a sequence $z_j$ (for the record, it ...
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1answer
62 views

Weak derivative of modulus

I am stuck on an introductory problem on Sobolev spaces. Any help would be appreciated! Suppose $\Omega\subseteq\mathbb{R}^n$ is bounded and open and $u:\Omega\rightarrow\mathbb{R}$ has an ...
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1answer
38 views

Distributional derivative of $2$-variable indicator function

Let us say we have the indicator function $\chi_{\{|x|\leq 1\}}$ in $\mathbb{R}^2$. How can I write out the weak derivative of this indicator function? Is it $\delta_{|x|=1}$? Or it should be vector ...
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43 views

How do I show that a distribution is a function?

While reading Grigor'yan's book on the heat kernel I have encountered the following definition of a Sobolev space on a Riemannian manifold $M$: $W^2 (M) = \{ u \in W^1 (M) : \Delta u \in L^2 (M) ...
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2answers
61 views

Mistake in reasoning about Sobolev spaces

I am new to Sobolev spaces and, while trying to construct a proof, I make some subtle mistake that I cannot detect. The setting: let $C \subset \Bbb R^n$ be a closed, measure-$0$ set. Let $U = \Bbb R ...
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1answer
41 views

Heaviside function and weak solutions

If a solution to a PDE has a solution f(x) in which f(x) is a Heaviside function, independent on its argument, can I say the solution is unstable, therefore being weak?
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44 views

A question on vanishing viscosity in Evans PDE

In Evans book on PDE (2nd edition) he uses the vanishing viscosity method to prove existence of a weak solution to the following linear hyperbolic system ...
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1answer
55 views

Is the regularity of $u$ necessary to deduce this result? (Evans PDE)

One of the exercises in Evans book on PDEs (at the end of chapter 7) is given as follows: Assume $$u_k\rightharpoonup u\quad\mbox{in}\quad L^2(0,T;H^1_0(U)),$$ $$u_k'\rightharpoonup ...
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1answer
36 views

If $\nabla u\in L^2(\mathbb{R}^n)$, then $u\in L^2_{loc}(\mathbb{R}^n)$

Suppose $u\in D'(\mathbb{R}^n)$ (i.e., $u$ is a distribution) can be identified with a weakly differentiable function such that $\nabla u\in L^2(\mathbb{R}^n)$. Show that $u\in ...
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1answer
36 views

Continuously differentiable functions are weakly differentiable

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $u\in C^1(\Omega)$. I want to show, that $u$ is weakly differentiable, i.e. $$\int_\Omega\psi\frac{\partial u}{\partial ...
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1answer
49 views

Weak convergence in the Sobolev space and compact embeddedness

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sopolev space $u\in C^0\left(\overline\Omega\times [0,\infty)\right)\cap C^{2,1}\left(\Omega\times ...
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1answer
48 views

$L^2$ 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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1answer
37 views

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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2answers
61 views

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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1answer
38 views

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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2answers
127 views

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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0answers
18 views

$\nabla\phi_k\stackrel{L^2}{\to}\nabla\phi\Rightarrow\langle\nabla u,\nabla\phi_k\rangle\stackrel{L^2}{\to}\langle\nabla u,\nabla\phi\rangle$

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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1answer
26 views

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 ...
3
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202 views

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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1answer
62 views

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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19 views

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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1answer
32 views

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 $|\alpha|\le k,$ $D^\alpha u$ ...
0
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1answer
34 views

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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1answer
33 views

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 \begin{equation} \delta_x(A)=\begin{cases} 1 & \mbox{ if } x\in A \\ 0 & \mbox{ otherwise}. \end{cases} ...
2
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2answers
136 views

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 ...
0
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1answer
43 views

Show that this function is weakly differentiable

I need to show that the function \begin{equation} u(x,y) = 1-x_1^2 \quad x_1>0 \\ u(x,y)=1+x_1^2 \quad x_1 \leq 0 \end{equation} is weakly differentiable on the unit ball. It is clear what the ...
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0answers
33 views

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 ...