Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

3,525 questions
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Multiplication operator from Sobolev space to $L^p$ space is compact?

Let $M_q: W^{1,\tilde{p}}(\mathbb{R}^2) \rightarrow L^p(\mathbb{R}^2)$, where $\tilde{p}$ is the Sobolev conjugate of p, $\left(\frac{1}{\tilde{p}}=\frac{1}{p}-\frac{1}{2}\right)$. What we know is ...
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Dual of Sobolev space $H^{s}(\mathbb{R}^n)$ Taylor Michael.

Why the dual of $H^{s}(\mathbb{R}^n)$ is $H^{-s}(\mathbb{R})$? I know that dual of $H^{s}(\mathbb{R}^n)$ is $\left\{T:H^{s}(\mathbb{R}^n)\to \mathbb{C}:T \text{ bounded and linear functional} \right\}$...
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$W_{loc}^{1,2}$ regularity of nonnegative subharmonic functions

I'm trying to solve the following excercise: Let $v \in C(D)$ be a nonnegative subharmonic function in an open set $D$. Prove that $v \in W_{loc}^{1,2}(D)$ The problem has the following hint: ...
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Showing that $u\in H^k(\mathbb{R}^n)\Leftrightarrow <\xi>^k\hat{u}\in L^2(\mathbb{R}^n)$ Taylor Michael.

Note: $<\xi>=(1+|\xi|^2)^{1/2}$ Why $u\in H^k(\mathbb{R}^n)\Leftrightarrow <\xi>^k\hat{u}\in L^2(\mathbb{R}^n)$? To think that to use the density of the space of Schwartz in the space of ...
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PDE's as minimisation problems

I am reading through Chapter 8 of Brezis. Particularly Section 8.4, which deals with some examples of boundary value problems on an interval. For the most part I understand the steps in the proofs ...
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How are Sobolev spaces on compact Riemannian manifolds defined?

For an open subset $\Omega\subset \mathbb R^n$ one can define the Sobolev space $$H^1(\Omega):=W^{1,2}(\Omega)=\{u \in L^2(\Omega) \, \vert \, \partial u \in L^2(\Omega)\}.$$ Is there a "simple" way ...
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Why are Sobolev spaces useful?

Why are Sobolev spaces useful, and what problems were they developed to overcome? I'm particularly interested in their relation to PDEs, as they are often described as the 'natural space in which to ...
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Is the Sobolev space $W^{1,p}(\Omega)$ always a Hilbert space if $\Omega \subset \mathbb{R}^2$ is bounded and has a continuous boundary?

Well... The title says it all. I know that for $p = 2$ the space $W^{1,p}(\Omega)$ is a Hilbert space, because $L^2(\Omega)$ is. But what if $1 < p < \infty$? Is there some theorem or statement,...
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Convergence of Mollifier in $W_0^{l,p}(\Omega)$

I want to prove that for $u \in W_0^{l,p}(\Omega)$ we have convergence of mollifiers $u_\rho \rightarrow u$. I appreciate that $u_\rho \rightarrow u$ in $L^p(\Omega)$ so just need to show convergence ...
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$H^1(\Omega)$ in Euclidean space vs $H^1(\Sigma)$ on a compact surface

For an open subset $\Omega \subset \mathbb R^n$ the Sobolev space $H^1(\Omega)=W^{1,2}(\Omega)$ is defined as \begin{equation} H^1(\Omega)=\{ u \in L^2(\Omega) \, \vert \, \partial^{\alpha}u \in L^2(\...
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Estimation in Sobolev norm

I try to understand the last part of Interior Regularity theorem. The theorem says: if $u\in W^{1,2}(\Omega)$ , where $\Omega$ is in $R^n$ and bounded, and $u$ is the weak solution of $\Delta u = f$ ...
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Dense in Weighted Sobolev Space [on hold]

We know that the $C^\infty$ function space on a bounded interval $I$ of $\Bbb R$ is dense in the Sobolev space on the same domain. My question is: Is the $C^\infty$ function space on a bounded ...
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Sequentially continuity

Given a map $E:L^\infty\rightarrow C^1(\overline{\Omega}), g_n\rightarrow g$ in $L^\infty(\Omega)$. $u_n=E(g_n), n\in \mathbb{N}$ and u=E(g) and show that every time we take a subsection we can find ...
Let $\Omega$ be a bounded open set with lipschitz boundary, How can we show that the functional defined by $f:W^{1,p}\rightarrow\mathbb{R}$ $f(u)=\int_{\Omega}|Du|_{\mathbb{R}^N}^p$ is sequentialy ...