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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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Trace of Sobolev functions on a circle of radius R>0

I have a difficulty dealing with a seemingly elementary question. Let $f \in H^m(K)$, $m \geq 1$, where $K$ is a (possibly large) bounded domain in $\mathbb R^2$. Let $\Gamma$ be a smooth codimension ...
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25 views

A question on existence of a Sobolev Hilbert space, where convergence implies uniform convergence

Is there a Sobolev Hilbert space $H^k(\Omega)$($\Omega$ open subset of $\mathbb{R}^m$, with a smooth boundary), for some $k \in \mathbb{N}$, such that, any sequence in the space $C^0(\bar{\Omega})\cap ...
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22 views

inverse of a riem. metric in sobolev class still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth ...
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2answers
40 views

Sobolev space,notation

There are two notions of Sobolev space: $H_0^{k,p}(\Omega)$ and $H^{k,p}(\Omega)$ as a closure of $C_0^\infty(\Omega)$ and $C^\infty(\Omega)$, respectively,w.r.t. $\|\cdot\|_{W^{k,p}(\Omega)}$. In my ...
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34 views

Taking the limit of ratios of interpolation error

I don't know how to take the limit of the ratio of two functional error estimates, when both go to zero in particular $$ \lim_{h\to 0} \frac{h \| u - u_h\|^2_{L^2(\partial \kappa)} }{ \| u - u_h\|^2_{...
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1answer
32 views

Does convergence in $W^{1,1}([0,1])$ imply uniform convergence?

Let $I=[0,1]$. Let $f_n \in W^{1,1}(I)$ be continuous and suppose that $f_n$ converge in $W^{1,1}(I)$ to a smooth function $f$. Does there exist a uniformly convergent subsequence of $f_n$? (There ...
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59 views

A question on equivalence of Sobolev norm

Define $\|.\|_{T^k(\Omega)}$ as $$ \|f\|_{T^k(\Omega)} = \|f\|_{L^2(\Omega)} + \|(\sum\limits_{i=1}^d(\frac{\partial^{k}f}{\partial x_i^{k}})^2)^{\frac{1}{2}}\|_{L^2(\Omega)} $$ Is this norm ...
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36 views

Existance on a basic differential equation

I have a few questions about a basic differetial equation. Below, there are some of my considerations. I would be very greateful for your help. Let $f\colon[0,T]\times\mathbb{R}\to \mathbb{R}$ be ...
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38 views

Polynomial approximation in Sobolev spaces

Let $U \subseteq \mathbb{R}^n$ be an open connected set. Let $p>1$, and let $f \in W^{1,p}(U)$. I have recently heard that $f$ can be locally approximated in $W^{1,p}$ by polynomials. That is, ...
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1answer
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A question about weakly continuous functions from $\mathbb{R}_+ \rightarrow H^1 (\mathbb{R}^3)$

I wonder if the following is true $$ L_{\mathrm{loc}}^{\infty} (\mathbb{R}_+ ; H^1 (\mathbb{R}^3)) \cap C(\mathbb{R}_+ ; H^{-1} (\mathbb{R}^3)) \subset C_{\mathrm{w}} (\mathbb{R}_+ ; H^1 (\mathbb{R}^...
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Weak limit in Bochner-Sobolev space

Let "$V\subset H\subset V^*$" be an evolution triple. Let $(u_n)$ be a sequence of elements from $W^{1,2}(0,T;V,H)$ such that $$u_n(t)\to u_1(t)\qquad\text{weakly in} \qquad V, $$ $$u_n'(t)\to u_2(t)...
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1answer
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Rellich–Kondrachov theorem - Continuity of the embedding

The Theorem says "$H^1(a,b)$ is compactly embedded in $L^2(a,b)$". In the proof, it is written that the continuity follows directly from $$\Vert \cdot \Vert_{H^1}^2=\Vert \cdot \Vert_{L^2}^2+\vert \...
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29 views

Embeddings for negative Soboloev spaces

May one conclude from $u \in W^{-1,2}(\Omega)$ and $\phi \in L^\infty(\Omega)$ that also $u\phi \in W^{-1,2}(\Omega) $? Intuitively this seems clear and I tried to do this in the same way as the ...
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Palais-Smale Condition of a functional

Palais-Smale condition: Defined as in the strong formulation wiki link: https://en.wikipedia.org/wiki/Palais%E2%80%93Smale_compactness_condition I am unable to check how the following functional $J$ ...
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1answer
60 views

Does $W^{1,1}(\Omega)^*$ contain a countable separating set?

I am trying to show that a weakly precompact subset $S$ of $W^{1,1}(\Omega)$ is metrizable. If I am not mistaken, it is enough to show that $W^{1,1}(\Omega)^*$ contain a countable separating set. ...
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Unique Harmonic Extension in $H^1(\Omega)$

In lecture we had the following theorem: Let $\Omega \subseteq \subseteq \mathbb{R}^n$. Then $$H^1(\Omega) = H^1_0(\Omega) \oplus \{u \in H^1 : \Delta u = 0\}$$ where $\Delta u$ is understood in ...
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54 views

Can we stay invertible while approximating linear maps in Sobolev spaces?

Let $\Omega \subseteq \mathbb{R}^d$ be an open bounded domain. Fix an integer $1<k<d$. Let $f \in W^{1,k}(\Omega;\mathbb{R}^d)$ be a continuous map with $\det df > 0$ a.e. Consider the map ...
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Coercivity and PS condition implies global minimizer

I am unable to get any intution to prove the following statement. Any $f:X\to\mathbb{R}$ which is coercive and satisfies PS condition has a global minimizer, provided $X$ is a reflexive Banach space.
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Morrey embedding for Potential Spaces

I'm trying to prove that for $f\in H^{s,p}:=\{f/f=G_s*g,\,\,g\in L^p\}$ where $G_s$ is the bessel potential: $G_s(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}(1+|w|^2)^{-s/2}e^{iw\cdot x}dw$ the following ...
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Gagliardo–Nirenberg inequality for multi index derivatives.

I'm using a book that states the Gagliardo-Nirenberg inequality this way: Given a function $u:R^3 \rightarrow R$: $$ \|D^i u\|_{L^{2r/i}}\leq c_r \|u\|_{L^\infty}^{1-i/r}\|D^r u\|_{L^2}^{i/r} $$ For ...
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Particular case of Poincaré Inequality

I need to prove the following inequality: Let $\Omega\subset \mathbb{R}^n$ be a bounded open set. Let $\overline{u}=\frac{1}{|\Omega|}\int_{\Omega}u(x)dx$ be the average of a function $u$ defined in $...
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1answer
37 views

If $v_n \rightharpoonup v$ and $\phi \in C_c^{\infty}(\mathbb{R}^N)$, do we have $v_n \phi \rightharpoonup v \phi$?

I'm stuck trying to solve this question. If I have a sequence in the Sobolev space $D^{1,\vec{p}}(\mathbb{R}^N)$ (or $W_0^{1,p}(\mathbb{R}^N)$ for simplicity) which converges weakly to $v$ and $\phi \...
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34 views

Finite element approximation of weighted p-laplacian - error estimation?

i'm currently working on the following dirichlet problmen: \begin{cases} \text{div} (\sigma(x) |\nabla u|^{p-2} \nabla u) = f &\quad \text{in } \Omega\\ u = g &\quad \text{in } \partial\...
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Confused by a section in the introduction to Hormanders first PDE book?

I have a question on the introduction to Hormanders first PDE book. The introduction seems poorly (i.e. confusingly) written to me, hopefully the rest of the book is better. Anyway, he says classical ...
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35 views

Does weak convergence imply existence of an a.e. convergent subsequence? [duplicate]

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Dpr}{\mathcal{D}^{1,\vec{p}}(\R^N)}$ I'm reading this article by El Hamidi and Rakotoson. On page 745 (page 5 of the PDF), they construct a bounded sequence ...
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Does a continuous function in $W^{1,p}$ for $p<n$ lie in $W^{1,n}$?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain. Suppose that $f \in W^{1,p}(\Omega)$ for some specific $1<p<n$, and that $f$ is continuous. Is it true that $f \in W^{1,n}_{loc}(\...
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1answer
26 views

Continuous extension of an operator $\text{PI }: H^s(\partial\Omega)\to H^{s+1/2}(\Omega)$

I'm attempting to read a proof in Taylor's PDE, Vol I. Chapter 5. And I'm stuck on a detail in Proposition 1.7: Consider the following boundary problem for $u$: $$ \Delta u =0\text{ on }\Omega,\...
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1answer
47 views

Is the composition of a Sobolev function and a smooth function Sobolev?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain, and let $1<p<n$. Suppose that $f \in W^{1,p}(\Omega)$ is continuous*, and $g \in C^{\infty}(\mathbb{R})$. Is it true that $g \...
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1answer
23 views

Two smooth functions glued together become a Sobolev function

Consider the square $S:=(0,1)\times (0,1)\subset \mathbb{R}^2$ and smooth functions $f,g:S\rightarrow (0,\infty)$, which can be extended up to the boundary by zero (i.e. $g(x),f(x):=0$ for all $x\in\...
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1answer
33 views

Prove that Laplace density (standard) lies in Sobolev space of order $\leq 3/2$

I'm trying to prove, that the function $f(x) = e^{-|x|}$ lies in $H^{s}(\mathbb{R}^2)$ for $s\leq \frac{3}{2}$. Therefore I calculate the functions sobolev norm $$\|f\|_s^2 = \frac{1}{4\pi^2}\int_{\...
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32 views

Weak* convergence in $W^{1,\infty}(\Omega)$

Let $\Omega$ be a bounded domain in $\Bbb R^m$ with smooth enough boundary so that $W^{1,\infty}(\Omega)=\text{Lip}(\Omega)$. Let $(u_n)$ be a sequence in $W^{1,\infty}(\Omega)$. What does it mean ...
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In compact embedding Theorem, $u_0$ lies in $W^{1,p}(\Omega)$?

We know that $W^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$ if $\Omega\subset\mathbb R^N$ is a bounded open and $\partial \Omega$ is $C^1$, $1\leq p<N$ and $1\leq q<p^*:=\frac{Np}{N-p}$. Moreover, ...
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Understanding the proof of the Concentration-Compactness principle

$\newcommand{\R}{\mathbb{R}}$ I'm reading parts of the paper The Concentration-Compactness Principle in the Calculus of Variations. The Limit Case, Part 1 by P.L. Lions. I'm trying to understand the ...
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1answer
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Is it possible to extend the notion of $H$-convergence to the case of distributions?

The usual $H$-convergence is defined for operators of the following form (for the sake of simplicity, restrict ourselves with the one-dimensional case): $$ \frac{d}{d x}\left[A_\varepsilon(x) \frac{d ...
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1answer
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A subspace of $H^1(0,\infty)$

I am just wondering why a space like $H_0^1(0,\infty)=\{f\in H^1(0,\infty):f(0)=0\}$ is dense in $L^2(0,\infty)$ where $H^1$ is the Sobolev space? Thanks in advance. Math
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How to extend results from space of smooth functions to Sobolev spaces?

I regularly see proofs involving Sobolev spaces where the proof states it will show some result holds for, say, $u \in H_0^1(\Omega)$ where $\Omega$ is a smooth bounded domain. Then right away it ...
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Metrizability of a subset in the weak topology.

Let $X$ be a Banach space (not reflexive). It is well-known that $(X,w)$, which is $X$ with its weak topology, is not metrizable if $X$ is infinite dimensional. I want to know under which condition ...
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Showing a function is square integrable

I'm trying to show that the dirac delta function is in $H^{\frac{-n}{2}- \epsilon}(\mathbb{R}^{n}) \forall \epsilon > 0.$ Where $H^{s}(\mathbb{R}^{n})$ denotes Sobolev space of order $s$ on $\...
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1answer
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Sobolev Space basic calculation

I'm new to Sobolev Space I'm not quite understanding the following: Proposition: If $s$ and $s' \in \mathbb{R}, s < s'$ then we have $H^{s'}(\mathbb{R}^{n}) \subset H^{s}(\mathbb{R}^{n})$ where ...
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1answer
51 views

If $f\in H^1(0,1)$ is bounded below, then $1/f\in H^1(0,1)$

Let $f\in H^1(0,1)$ be such that $|f(x)|\ge a > 0$ for a.e. $x\in (0,1)$. I have problems showing that then $1/f\in H^1(0,1)$ by using the definition. Of course, we know that $f$ is absolutely ...
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37 views

Closure in $H^1((0,L)\times\mathbb{R})$

Let $\Omega=(0,L)\times\mathbb{R}$ and consider the space $$V:=\{f\in H^2(\Omega) \ \vert \ f'(0,y)=f'(L,y)=0 \ \forall \ y\in\mathbb{R}\}.$$ I would like to show that the closure of $V$ with ...
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Rescaling of homogeneous and inhomogeneous Sobolev spaces

We know that $\|u\|_{H^s} = \|(1+|k|)^s |\hat{u}(k)|\|_{L^2}$ and $\|u\|_{\dot H^s} = \|(|k|^{s}|\hat{u}(k)|\|_{L^2}$ are the inhomogeneous and homogeneous norms, respectively. If we define the ...
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Calculating Sobolev norm on boundary by extending the map on M as a harmonic map

It is known that the natural trace map $W^{1}(M) \ni \varphi \rightarrow \varphi|_{\partial M} \in W^{1/2}(\partial M)$ is continuous and onto. Since the Dirichlet problem $\Delta \varphi=0$, $\varphi|...
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1answer
30 views

Dirac Delta functions and Sobolev Embeddings

This is a question about the action of the Dirac delta on Sobolev spaces $H^s(\mathbb{R}^d) = W^{s,2}(\mathbb{R}^d)$. We know that $\delta(\underline{x})\in H^s(\mathbb{R}^d)$ for $s<-d/2$. In 2D,...
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1answer
40 views

For $v_1 \in H^1$ and $v_2 \in H_0^1$, then where's $v_1 + v_2$?

$H_0^1$ is functions of $H^1$ that have a property of vanishing (i.e. go to zero) on the boundary of the function's domain. So $$H_0^1 \subset H^1$$ However, if one considers the sum: $$v_1 + v_2$$ ...
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25 views

weak convergence of composition

Given $v_k \rightarrow v \ \ \ \text{weakly star in } L^2(0,T;W^{1,\infty}(\mathbb{R}^n)) $ $\eta _k(t,\cdot) \rightarrow \eta(t,\cdot) \ \ \ \text{in } C_\text{loc}(\mathbb{R}^n)\ $ uniformly in $t,...
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2answers
56 views

Characterization of precompactness/totally boundedness in Sobolev Spaces

within the paper about the Kolmogorov Riesz compactness theorem by Hanche-Olsen and Holden it is stated that for Sobolev Spaces any subset $\mathcal{F}$ of $W^{k,p}(\mathbb{R}^n)$ is totally bounded ...
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1answer
37 views

Why do I need partitions of unity to define Sobolev norms on compact manifolds?

I am considering the following possible definition of Sobolev spaces. Let $M$ be a compact manifold and $H^k(M)$ be the space of measurable functions $f\colon M\to\Bbb R$ with the following property: ...
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2answers
30 views

Completing proof of Sobolev embedding theorem

Up to this point I have proved that the approximation $||u_k||_{L^{p^*}}< C||\nabla u_k||_{L^{p}}$ for smooth compactly supported functions $u_k$. By density of $C^\infty_c$ in $W^{1,p}$, I would ...
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1answer
33 views

How can function be assumed smooth in proving Hardy's inequality in Evans book?

Evans proved the Hardy's inequality, Hardy's inequality. In the first step, it is said that the function may be assumed smooth. I wonder why we can assume this function is smooth. I guess it may ...