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
$$
\lim_{h\to 0} \frac{h \| u - u_h\|^2_{L^2(\partial \kappa)} }{ \| u - u_h\|^2_{...
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1answer
32 views
Does convergence in $W^{1,1}([0,1])$ imply uniform convergence?
Let $I=[0,1]$. Let $f_n \in W^{1,1}(I)$ be continuous and suppose that $f_n$ converge in $W^{1,1}(I)$ to a smooth function $f$. Does there exist a uniformly convergent subsequence of $f_n$?
(There ...
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1answer
59 views
A question on equivalence of Sobolev norm
Define $\|.\|_{T^k(\Omega)}$ as $$ \|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)} $$
Is this norm ...
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36 views
Existance on a basic differential equation
I have a few questions about a basic differetial equation. Below, there are some of my considerations. I would be very greateful for your help.
Let $f\colon[0,T]\times\mathbb{R}\to \mathbb{R}$ be ...
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0answers
38 views
Polynomial approximation in Sobolev spaces
Let $U \subseteq \mathbb{R}^n$ be an open connected set. Let $p>1$, and let $f \in W^{1,p}(U)$. I have recently heard that $f$ can be locally approximated in $W^{1,p}$ by polynomials. That is, ...
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1answer
25 views
A question about weakly continuous functions from $\mathbb{R}_+ \rightarrow H^1 (\mathbb{R}^3)$
I wonder if the following is true
$$ 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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21 views
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
$$u_n(t)\to u_1(t)\qquad\text{weakly in} \qquad V, $$
$$u_n'(t)\to u_2(t)...
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1answer
14 views
Rellich–Kondrachov theorem - Continuity of the embedding
The Theorem says "$H^1(a,b)$ is compactly embedded in $L^2(a,b)$".
In the proof, it is written that the continuity follows directly from $$\Vert \cdot \Vert_{H^1}^2=\Vert \cdot \Vert_{L^2}^2+\vert \...
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0answers
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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0answers
16 views
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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0answers
16 views
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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0answers
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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0answers
19 views
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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26 views
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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13 views
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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35 views
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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0answers
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\...
3
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0answers
26 views
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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1answer
39 views
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 $u$:
$$
\Delta u =0\text{ on }\Omega,\...
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vote
1answer
47 views
Is the composition of a Sobolev function and a smooth function Sobolev?
Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain, and let $1<p<n$. Suppose that $f \in W^{1,p}(\Omega)$ is continuous*, and $g \in C^{\infty}(\mathbb{R})$.
Is it true that $g \...
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1answer
23 views
Two smooth functions glued together become a Sobolev function
Consider the square $S:=(0,1)\times (0,1)\subset \mathbb{R}^2$ and smooth functions $f,g:S\rightarrow (0,\infty)$, which can be extended up to the boundary by zero (i.e. $g(x),f(x):=0$ for all $x\in\...
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1answer
33 views
Prove that Laplace density (standard) lies in Sobolev space of order $\leq 3/2$
I'm trying to prove, that the function $f(x) = e^{-|x|}$ lies in $H^{s}(\mathbb{R}^2)$ for $s\leq \frac{3}{2}$.
Therefore I calculate the functions sobolev norm $$\|f\|_s^2 = \frac{1}{4\pi^2}\int_{\...
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0answers
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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0answers
49 views
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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23 views
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
64 views
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
22 views
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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1answer
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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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1answer
52 views
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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69 views
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
59 views
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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0answers
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}\}.$$
I would like to show that the closure of $V$ with ...
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0answers
20 views
Rescaling of homogeneous and inhomogeneous Sobolev spaces
We know that $\|u\|_{H^s} = \|(1+|k|)^s |\hat{u}(k)|\|_{L^2}$ and $\|u\|_{\dot H^s} = \|(|k|^{s}|\hat{u}(k)|\|_{L^2}$ are the inhomogeneous and homogeneous norms, respectively. If we define the ...
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0answers
58 views
Calculating Sobolev norm on boundary by extending the map on M as a harmonic map
It is known that the natural trace map $W^{1}(M) \ni \varphi \rightarrow \varphi|_{\partial M} \in W^{1/2}(\partial M)$ is continuous and onto. Since the Dirichlet problem $\Delta \varphi=0$, $\varphi|...
3
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1answer
30 views
Dirac Delta functions and Sobolev Embeddings
This is a question about the action of the Dirac delta on Sobolev spaces $H^s(\mathbb{R}^d) = W^{s,2}(\mathbb{R}^d)$.
We know that $\delta(\underline{x})\in H^s(\mathbb{R}^d)$ for $s<-d/2$.
In 2D,...
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1answer
40 views
For $v_1 \in H^1$ and $v_2 \in H_0^1$, then where's $v_1 + v_2$?
$H_0^1$ is functions of $H^1$ that have a property of vanishing (i.e. go to zero) on the boundary of the function's domain.
So $$H_0^1 \subset H^1$$
However, if one considers the sum:
$$v_1 + v_2$$
...
2
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0answers
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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vote
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: ...
1
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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 ...
1
vote
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 ...
