Questions tagged [weak-derivatives]
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.
742 questions
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Step in Brezis' proof of chain rule for Sobolev spaces
I'm going through the proof of Corollary 8.11 in Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations which states:
Let $G \in C^1(\mathbb{R})$ be such that $G(0) = 0$, and ...
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Admissible test functions for weak formulation of Neumann problem
NOTICE TO ALL:
I do NOT want a solution to the below exercise. Please do not post one. I am working on that myself and don't want it spoiled.
I am working on the exercises from chapter 6 of Evans ...
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Chain rule for BV function
I have a small question about the chain rule for BV functions.
Let us consider $\varphi \in L^\infty(\mathbb{R}^d) \cap BV(\mathbb{R}^d)$ and $F \in C^1(\mathbb{R}, \mathbb{R})$. My question is: how ...
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$D(F\circ u)=(F'\circ u)Du$ (chain rule) in Sobolev spaces - still hold when $F'$ discontinuous or when $F$ only weakly diff'ble?
I just finished reviewing my solutions of the exercises in chapter 5 of Partial Differential Equations (2nd Ed) by L.C. Evans. We recall Problem 17:
Exercise 17. (Chain rule.) Let $F\in C^1(\Bbb R)$ ...
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Extension of weak solution to entire space
Assume that we have a weak solution to some elliptic pde, for simplicity say the inhomogeneous Laplace equation with right-hand side in $L^2$, i.e. we have $u\in W^{1,2}(E)$ with $E\subset\mathbb{R}^n$...
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Is $(\Delta-\lambda,H_0^2([0,1]^2))$ surjective in $L^2([0,1]^2)$?
I thought I had a solution for the problem below, but a friend remarked that $\sin(n\pi x)\sin(m\pi y)$ does not lie in $H_0^2([0,1]^2)$ because there is a Theorem in Brezis that states that $f\in H_0^...
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How to show that $\nabla u = 0$ almost everywhere on $\{u = 0\}$?
Let $u \in H^1_0(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is a bounded domain. I need to show that $\nabla u = 0$ almost everywhere on $\{u = 0\}$, but without using the weak derivative of $|u|, ...
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If $u \in H^1_0(\Omega)$, then $u^+ \in H^1(\Omega)$
Let $\Omega \subset \mathbb{R}^N$ be a bounded domain. I am trying to prove that, if $u \in H^1_0(\Omega)$, then $u^{+} = \max\{u,0\} \in H^1(\Omega)$. I would like a self-contained proof — that is, ...
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Explanation of weak derivative by Dorina Mitrea
In Distributions, Partial Differential Equations, and Harmonic Analysis (2nd Ed) by Dorina Mitrea, section 1.2 on p.4, there is something slightly confusing.
Here $\Omega$ refers to an open set of $\...
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Composition of $W^{1,p}$ function with Lipschitz function is still $W^{1,p}$
I am trying to prove that, given $ u \in W^{1,p}(\Omega) $, where $\Omega\subset \mathbb{R}^n$ is an open set, and $ f $ Lipschitz on $ \mathbb{R} $, under one of the following assumptions:
$ f(0) = ...
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what is the weak derivative of Singular function
I want to understand a bit about the weak derivative of singular function
Let u be singular. Using wiki to define the concept of singular functions I denote by $N⊂R$ a set such that $u$ has a ...
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Explicitly computing Riesz representation of elements in $H^{-1}(\Omega)$
I would like to know whether there is a way to explicitly construct the Riesz representation for the following example. For an open domain $\Omega \subset \mathbb{R}^n$ consider the embedding
$$
H_0^1(...
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Differentiability of norms and seminorms in $H_0^1(\Omega)$
I would like to know whether the following functions are Fréchet differentiable. Let $\Omega \subset \mathbb{R}$ be an open domain and consider the spaces $H_0^1(\Omega)$ and $L^2(\Omega)$. For $u \in ...
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Is a well-posed PDE with an additional "nice" term still well-posed?
PDE Problem.
Given functions $f,g,\mathbf{v}$, seek $u,p$ such that
$$\begin{align}
&\partial_t u + \mathbf{v} \cdot\nabla p \color{red}{-\epsilon\Delta u} = f\\
&\partial_t p + \mathbf{v} \...
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Interpolation inclusion of homogeneous Sobolev space
At the moment I am trying to digest the concept of homogenous Sobolev space and establish simple interpolation type inclusion between those spaces. Namely, I am wondering two things:
In the ...
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