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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.

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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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1answer
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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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31 views

$W_{loc}^{1,2}$ 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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1answer
25 views

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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approximating functions in $W^{1,2}$ with Lipschitz functions with image contained in fixed ball

Fix $r>0$. For $h>0$ let $u_h\in W^{1,2}(B_h,B_r)$, where $B_h$ and $B_r$ are the ball of radius $h$ and $r$ centered at the origin in $\mathbb R^n$, respectively. I want to approximate the $...
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28 views

Composition of Sobolev map and smooth function.

Let $\Omega = S^1 \times [a, b] $ and $f\in C^\infty (\Omega )$. When is $\;H^s(\Omega, \Omega) \rightarrow H^s(\Omega, \Omega), u \mapsto f \circ u$ well defined and smooth? I've only found results ...
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1answer
21 views

Properties of Functions in $W^{1,p}(\Omega)$ and Their Weak Derivatives

Let $\Omega \subset \mathbb{R}^{N}$ and $W^{1,p}(\Omega)$ be Sobolev Space. Then we let $u\in W^{1,p}(\Omega)$ and define (i) $u^{+} := \max\{0,u\}$ (ii) $u^{-} := \max\{0,-u\}$ Then, we claim that $u^...
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1answer
33 views

Definition de $L^1((0,T);W^{1,\infty}(\mathbb R))$.

I have some trouble about the definition of the space $L^1((0,T);W^{1,\infty}(\mathbb R)$, on one side the formal definition given by Bochner consists in the set of measurable functions $u$ from $(0,T)...
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1answer
44 views

Why chose $H^{-1}(\Omega)$ instead of $L^2(\Omega)$?

For given $f \in L^2(\Omega)$ Poisson's equation reads $$- \Delta u=f \quad \text{on }\Omega.$$ So the variational problem becomes: For given $f \in H^{-1}(\Omega)$ find $u \in H_0^1(\Omega)$ such ...
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1answer
17 views

Strict inequality of the H1 seminorm of a curve emerging from a convex set and its convex projection

Let $\gamma(t) \in W^{1,2}([0,1]; \mathbb{R}^d)$ such that $\gamma(0) \in C$, with $C \subset \mathbb{R}^d$ some closed convex set, and $ \gamma(t) \not \in C, \ \forall t\in (0,1]$. Furthermore we ...
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1answer
72 views

Extension of a Sobolev Function

Let $\Omega _1$ and $\Omega_2$ are smooth open sets and $A= \partial \Omega_1 \cap \partial \Omega_2.$ Let $\Omega ^\prime = \Omega_1 \cup \Omega_2 \cup A$ be an open set. Let $f$ is defined on $\...
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29 views

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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22 views

weak solution to non-homogeneous initial-boundary value problem

Let $\Omega\subset \mathbb{R}^n$ be a bounded Lipschitz domain, and consider a general linear parabolic equation of the divergence form: $$ u_t-\partial_i(a^{ij}\partial_ju)+b^i\partial_i u+cu=f, \...
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18 views

Linear functional on Sobolev space [closed]

Suppose $\Omega$ is a smoothly bounded domain in $\mathbb{R}^n$, $n\ge 3$, and let $\{ u_m\}$ be a bounded sequence in $H_0^1 (\Omega)$ which converges to $u$ in the weak topology. Define $T_m v=\int_{...
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2answers
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If compact embedding $H^1_0(\Omega) \hookrightarrow H^{-1}(\Omega)$ is not surjective, how can $H^{-1}(\Omega)$ be the dual of $H^1_0(\Omega)$?

Let $\Omega$ be bounded with Lipschitz boundary. By Rellich compactness, $\iota: H^1_0(\Omega) \hookrightarrow L^2(\Omega)$ is compact embedding. It is also dense. By Riesz representation, $L^2(\...
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A caratheodory function [closed]

Let $\Omega$ be a bounded set with lipschitzien boundary. $1 <q <p <r$ and $ u\in W^{1,p}(\Omega)$ i can't understand why is it a carathéodory function because ir $u (z)$ is negatif the ...
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23 views

Does anyone know any books provide complete picture of mathematical spaces, hierarchical view, relations, natural progression among them

I was wondering if anyone is aware of any books that provide a comprehensible and complete picture of mathematical spaces? I've looked into many real analysis as well as complex for a complete list of ...
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27 views

Is the definition of the Sobolev space H^1(M) on a compact manifold that simple?

In https://hebey.u-cergy.fr/NotesSharpSP.pdf right at the beginning Hebey says Given $(M,g)$ a smooth compact $n$-dimensional Riemannian manifold, one easily defines the Sobolev spaces $H^p_k(M)$,...
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0answers
35 views

Looking for a convergent sequence in $W^{1,2}[0,\infty)$ with some properties.

Define $$W^{1,2}[0,\infty) := \{ f \in L^2[0,\infty): f' \in L^2[0,\infty)\}.$$ It is a Hilbert space with the inner product $$ \langle f,g \rangle = \langle f,g \rangle_{L^2} + \langle f',g' \rangle_{...
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1answer
47 views

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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1answer
36 views

Confusion about Sobolev spaces

Reading through Chapter 8 of Brezis. We see that $C_{c}^{\infty}(\mathbb{R})$ is dense in $W^{1,\,p}(I)$. Later on in the chapter we define $W^{1,\,p}_{0}(I)$ to be the closure of $C_{c}^{1}(I)$ in $W^...
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1answer
24 views

Extending Poisson equation from half unit disc

Let $B_+$ and $B$ denote the upper half and full unit disc in $\mathbb{R}^2$ respectively. Suppose $f \in L^2(B_+)$ and that $u \in H_0^1(B_+)$ is a weak solution to the Dirichlet BVP $-\Delta u =f$ ...
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2answers
34 views

How does $\| \nabla u \|_{L^2}^2 \leq C \|\nabla u \|_{L^2}$ imply $\| u\|_{W^{1,2}}^2 \leq \| u\|_{W^{1,2}}$?

So I'm working on some notes and I found this inequality that I really can't make sense of it. This is what is going on: Let $u \in W_0^{1,2}$ and let $f \in L^2$ both on some $\Omega \subseteq \...
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1answer
67 views

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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0answers
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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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Does a function $f \in W^{n,p}(U)$ which vanishes on $U-\overline{V}$ automatically lives in $W^{n,p}_0 (V)$?

Suppose $U,V$ are open sets in $\mathbb{R}^n$ and $\overline{V} \subseteq U$. Suppose $f \in W^{n,p} (U)$ is $0$ on $U-\overline{V}$. Can we say that the restriction of $f$ to $V$ (denoted again by $...
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1answer
20 views

Intersection of all Sobolev spaces with negative order

Call $H^{s}$ the usual $L^2$-based Sobolev spaces on, say, a closed manifold, for $s \in \mathbb R$. The intersection $\bigcap _{s<0} H^s $ contains $L^2$. Is this intersection equal to $L^2$? ...
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56 views

Which space is bigger? (Sobolev Space)

First, let us define the following space : \begin{align*} H^{1}(\Omega) :=\{u :\Omega\to\mathbb{R}\,|\,\forall\alpha\text{ multiindex },0\leq|\alpha|\leq1,||D^{\alpha}u||_{L^{2}(\Omega)}<\infty\} \\...
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1answer
31 views

Sobolev Inequality in 1D

Reading through Brezis and I am having trouble understand the proof of the following theorem: There exists a constant $C$ such that $\|u\|_{\infty}\leq C\|u\|_{W^{1,\,p}}$ $\forall u\in W^{1,\,p}$, $...
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1answer
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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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0answers
29 views

Sobolev space extension operator

Let $1\leq p\leq\infty$. There exists a bounded linear operator $P:W^{1,\,p}(I)\rightarrow W^{1,\,p}(\mathbb{R})$, called an extension operator, satisfying the following properties: (i) $Pu|_{I}=u\...
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0answers
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Contraction semigroup of nonlinear heat equations

Let $(S(t))_{t\geq 0}$ be a semigroup generated by $\Delta$ operator in $L^{2}(\Omega)$ for a bounded smooth domain $\Omega \subset \mathbb{R}^{N}$. Here we assume \begin{align} \begin{cases} D(B) = \{...
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2answers
102 views

Sobolev Embedding into $L^{\infty}$

I tried to find the question here, but I couldn't. I'm a bit puzzled by Sobolev embeddings at the moment. In my lecture notes, I found the statement "If $\Omega \subseteq \mathbb{R}^d$ is ...
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0answers
14 views

Density of $C^\infty(\overline{\Omega})$ in $W^{k,2}(\Omega)$

First, $\Omega\subseteq \mathbb{R}^n$ and $$\Omega=\prod_{i=1}^n\left(a_i,b_i\right). $$ I know that the space of test functions $C^\infty_c(\Omega)$ is indeed dense in $W^{k,2}(\Omega)$. Is there ...
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0answers
46 views

Differentiability under the integral sign

Do we have some kind of Lebesgue theorem in the case of an integral defined over a boundary? $\frac{d}{dt}\int_{\partial\Omega}f(t,x)d\sigma$
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26 views

Differentiability of functional

Let $\Omega$ be a bounded domain with Lipschitz border. We define $G:W^{1,p} \rightarrow\mathbb{R}$ by $$G(u)=\int_{\partial\Omega}|u|^pd\sigma.$$ How can I show that $(G'(u),h)=\int_{\partial\Omega}|...
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0answers
48 views

Inequality of a p-laplacian

Let $\Omega$ be a bounded domain with lipschitzian boundary.$2\leq p$ Why do we have $$\int_{\Omega}\langle|\nabla u|^{p-2}\nabla u- |\nabla v |^{p-2}\nabla v,\nabla v\rangle\leq \int_{\Omega}(|\nabla ...
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0answers
19 views

A boundary equality and dual of the trace's image [duplicate]

Let $\Omega$ be a bounded domain with lipschitzian boundary. How can I show that $\left\langle\frac{\partial u}{\partial n},h\right\rangle_{W^{\frac{1}{p' },p}}=0, \ \forall h\in W^{1,p}\implies \...
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Maximal monotone functional

Given a bounded domain $\Omega$ $h:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ defined by $$ h(u)=\int_{\Omega}|Du|^{p-2}(Du,Dh)_{\mathbb{R}^N}dz$$ why is it continuous and maximal monotone ?
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How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
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1answer
38 views

Is $C^k$ function space is dense in Sobolev space? [on hold]

It is known that $C^\infty$ function space dense in Sobolev space. But I want to know that is this statement true also for $C^k$ function space?
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23 views

Inequality involving periodic functions and Sobolev space.

Set $n\in\mathbb{N}$. For $\psi=u+iv$ in $H^1_T(\mathbb{R}^n)=\lbrace \psi\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)$, I wonder if $$ \int_Q |\nabla u|^2\geq C\int_Q (u-1)^3 ...
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1answer
30 views

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 ...
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1answer
29 views

Sequentially lower semicontinuity

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 ...
1
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1answer
692 views

Equivalent norms on Sobolev space $H^1$ for bounded Lipschitz domains

While reviewing some lecture notes, I stumbled upon the following proposition. $\newcommand{\vertiii}[1]{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert #1 \right\vert\kern-0.25ex\right\vert\...
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0answers
37 views

Evans pde proof of global approximation theorem

I have a question regarding step 3 of the proof (shown in picture below). I am wondering if there is a need for us to consider the subset $V\Subset U$ instead of directly using $U$ when showing the ...
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1answer
49 views

Reference request: Laplace-Beltrami eigenfunction bases for Sobolev spaces

I'm working on a smooth $(d-1)$-dimensional surface $M\subset \mathbb{R}^d$. Let $(\phi_k)_{k\in\mathbb{N}}$ be an orthonormal basis of $L^2(M)$ consisting of the eigenfunctions of the Laplace-...