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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0
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1answer
23 views

Intuitive question about the boundary values of a Sobolev function

Let $B_R$ a ball of radius $R$ in $R^n.$ Let $u \in H^1(B_R)$ and let $u^{1} \in H^1(B_R)$ with $u^1 - u^{+} \in H^1_{0}(B_R)$. ($u^{+}$ denotes the positive part of the function $u$). Let $v(y) : = ...
0
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2answers
36 views

Reference for denseness of testfunctions in sobolevspace

for my thesis I need a reference for a proof that $C_0^\infty(\mathbb R)$ is dense in $W^{2,2}(\mathbb R)$ in respect to the Sobolev-$\| \cdot \|_{W^{2,2}}$-Norm. I have tried Google but I can't find ...
0
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1answer
16 views

How can I conclude this “gluing property” for these Sobolev functions?

Let $u \in W^{1,2}(\Omega) \cap C(\Omega)$, where $\Omega$ is an open bounded domain in $R^n$ with smooth boundary.Let $B(x,R) \subset \overline{B(x,R)} \subset \Omega $ a ball. Consider $u^{\star} ...
4
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2answers
87 views

Significance of Sobolev spaces for numerical analysis & PDEs?

I never had an option to take a Functional Analysis module. I am tied up with other work for the next two months so I won't get a chance to self-study it until September. So one thing I was wondering ...
0
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0answers
25 views

Simple question about Sobolev Spaces

I'm studying Sobolev Spaces and my lecturer told us that $H^1_0(U)=H^1(U)$. Is this true for all $U\subseteq \mathbb{R}^n$? Even $U=\mathbb{R}^n$? And why exactly is this true?
3
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1answer
86 views

Multivariable calculus chain rule for weak derivatives

Let $g:(0,1) \rightarrow \mathbb R^n$ be absolutely continuous, $F \in W^{1,2} (\mathbb R^n).$ Is it true that a.e. it holds $$ \dfrac{dF(g(t))}{dt} = \nabla F(g(t)) \cdot g'(t) \quad ? $$ What I ...
3
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1answer
99 views

Counterexample to Rellich-Kondrachov Compactness Theorem, case $q=p^*$

I was searching some counterexample for I was searching some counterexample for Rellich-Kondrachov Compactness Theorem (You can see: PDE, Evans, chapter 5), for the case $q=p^*$ and I found this ...
0
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0answers
60 views

The Dirichlet problem for the Laplace equation: classical solutions versus weak solution

Let $B_R$ a ball in $\mathbb{R}^n$. Consider $u^{\star} \in H^{1}(B_R) $ and $f \in H^{1}(B_R) \cap C(\overline{B_R})$. Suppose that $u^{\star}$ minimizes $$\int_{B_R} |\nabla u|^2, u \in \{ v \in ...
0
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0answers
24 views

Functions equal in Sobolev spaces

Consider the Sobolev space $H^k(\mathbb{R}^n)$, where $k, n \geq 1$ are integers. Also, consider $u, v \in H^k(\mathbb{R}^n)$ such that $u$ and $v$ are equal almost everywhere in the Lebesgue measure ...
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0answers
39 views

About the compact embedding $W^{1,p}(U)\subset\subset L^p(U)$

How can I get this compact embedding $W^{1,p}(U)\subset\subset L^p(U)$ for a open, bounded and $C^1$ subset $U$ of $\mathbb R^n$? Basicly This is the Evans' Remark on pg. 289 2nd edition. But I ...
1
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1answer
37 views

Can I conclude that this Sobolev function is Lipschitz?

Let $u \in H^{1}(\Omega)$ ($\Omega \subset R^n$ a bounded domain with smooth boundary). Suppose that there is a constant $C>0$ such that $$ |u(x) - u(y)| \leq C |x-y|,$$ for every Lebesgue point ...
0
votes
1answer
65 views

Interpolation between derivatives

I am trying to prove the following: Let $u \in W^{2,2}(\mathbb R)$. Then $\| u^\prime \|_{L^2}^2 \leq \| u \|_{L^2} \| u^{\prime \prime} \|_{L^2}$ holds (these are meant to be weak derivatives). ...
1
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0answers
32 views

Elliptic regularity of second order pseudos

Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that ...
1
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1answer
36 views

Showing that there is a unique solution for the following equation

Let $I = (0, 1)$, $a : H_0^2(I) \times H_0^2(I) \to \mathbb{R}$ a continuous bilinear form defined by $$a(u, v) = \int\limits_I u'' (x) v'' (x) dx.$$ Show that for every $f \in L^2(I)$ there is a ...
5
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0answers
82 views

Sobolev type embedding

Consider a compact manifold $M$ and a point $q \in M$. Let us say that that the following inequality holds: $$ \Vert \varphi u\Vert_{L^p} \leq C\Vert \varphi u\Vert_{H^1},$$ where $\varphi \in ...
0
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1answer
43 views

Norm in homogeneous Sobolev space

I'm having doubts on the definition of the norm in the space $H_0^2$ defined over the (rectangular) domain $\Omega \subset \mathbb{R}^2$ as follows $$ H_0^2(\Omega) = \left\{ v \in L^2(\Omega): v = ...
2
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1answer
103 views

Definition of Hölder Spaces of maps between manifolds (also Sobolev Spaces of these maps)

Let $M$ be a Riemannian manifold, $N$ a manifold, $k\in \mathbb{N}$, $\alpha\in(0,1)$ Question: How exactly is the Hölder space $C^{k,\alpha}(M,N)$ and the sobolev space $W^k_p(M,N)$ defined? Are ...
2
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1answer
60 views

functions in $H_0^1(\Omega) \cap C(\overline{\Omega})$ are zero on the boundary

I want to solve the following problem Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set with $C^\infty$-smooth boundary. Show that for any $u \in H_0^1(\Omega) \cap C(\overline{\Omega})$ ...
0
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1answer
96 views

Weak derivative of logarithm function

everybody! I am trying to calculate the weak derivative of the function $$u(x)=\log\log\left(1+\frac{1}{|x|}\right)$$ where $x \in B(0,1)$. I know that $$\nabla ...
1
vote
1answer
633 views

Question about Fourier transforms of gradient, curl and divergence

Consider a vector field $v:\mathbb{R}^3\rightarrow\mathbb{R}^3$. Denote by $F_u$ the Fourier transform of a scalar or vector field $u$. Can one finds an equality relation between ...
1
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1answer
47 views

Compact support of derivatives of $u$ in weak sense

I would like to know if let $u:\mathbb{R}^n\to \mathbb{R}$ be a function in $W^{k,p}(\mathbb{R}^n)$ such that $u$ has compact support in $\mathbb{R}^n$ then, for each $|\alpha|\le k,$ $D^\alpha u$ ...
1
vote
1answer
162 views

Integration by parts formula on unbounded manifold

Let $M$ be a closed Riemannian manifold and set $X = M \times [0,\infty)$ with the trivial product metric induced. If $u$ and $v$ are functions defined on $X$, how do I know that the formula $$\int_X ...
5
votes
1answer
69 views

Schrödinger operator with delta (zero range) interaction.

I am reading the book of Albeverio named Solvable models in quantum mechanics. In the first chapter it is explained how to realize the operator $"-\Delta+\delta_0"$ as a self adjoint operator on ...
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2answers
54 views

Characterization of $H^k$ by Fourier transform

Let $u\in H^k(\mathbb{R}^n)$ a function with complex-valued. Question 1: If $u\in H^k(\mathbb{R}^n)$ and $D^\alpha u\in L_2(\mathbb{R}^n)$, by definition for each $|\alpha|\le k,$ then we have ...
1
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1answer
115 views

Self-adjoint extension of the Laplacian

Let $M$ be a complete Riemannian manifold and $-\Delta$ denote the Laplace-Beltrami operator on $M$. We can prove that $(-\Delta f, g) = (\nabla f, \nabla g) = (f, -\Delta g)$, when $f, g \in ...
0
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0answers
20 views

Step of proof Hardy's Inequality [duplicate]

I trying to prove the Hardy's Inequality, by Evans book. I need of a little help in the step: If $u\in L^1(B(0,r))$ satisfying $$(2-n)\int_{B(0,r)}\frac{u^2}{|x|^2}dx=2\int_{B(0,r)}u ...
0
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1answer
48 views

The Sobolev type embedding for negative Sobolev space

Given $\Omega\in \mathbb R^N$ open bounded smooth boundary, assume $u_n$, $u\in L^q(\Omega,\mathbb R^d)$ for some $d\in\mathbb N$ and $1<q<\frac{N}{N-1}$. We also assume that $u_n\to u$ weakly ...
1
vote
1answer
56 views

Density of $C_c^\infty$ in $W_0^{1,2}$

Let $\Omega \subset \mathbb{R}^N$ be a bounded open set and let $(f_n) \subset L^2(\Omega)$. Suppose there exists $f \in L^2(\Omega)$ such that $$\int_{\Omega} f_n \varphi \rightarrow \int_{\Omega} f ...
1
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1answer
52 views

About the definition of Sobolev Spaces

I'm studying Sobolev Space and I have a question about the definition: Def.: The Sobolev Space $W^{k,p}(U)$ consists of all locally summable functions $u:U\to \mathbb{R}$ such that for each ...
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0answers
39 views

A question in partial differential equation.

Suppose a.b is bounded in $L^2(0,T;H^{-1}(\Omega))$; $a\geq \alpha>0$ almost everywhere in $\Omega\times(0,T)$ and $a\in L^\infty(\Omega\times(0,T))$. Is $b\in L^2(0,T;H^{-1}(\Omega))$?
1
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1answer
46 views

Integral of $au^2$ where $a$ is continuous and $u \in W_0^{1,2}(\Omega)$

Let $\Omega \subset \mathbb{R}^N$ be a smooth domain with bounded complement. Let $a \in C(\Omega)$ and let $u \in W_0^{1,2}(\Omega)$. Suppose that $a > 0$ in $\Omega$ and $\displaystyle ...
1
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0answers
90 views

How to prove $H_0^1(\Omega)=H_0(div;\Omega)\cap H_0(curl;\Omega)$

It was shown in the book "Finite Elements Methods for Navier-Stokes Equations" by Girault and Raviart that $$H_0^1(\Omega)=H_0(\operatorname{div};\Omega)\cap H_0(\operatorname{curl};\Omega).$$ The ...
0
votes
1answer
38 views

Question about a bounded sequence in $W^{1,1}$ admitting a weakly convergent subsequence

Let $B$ be the unit ball in $\mathbb{R}^{12}$ and $f_n\in L^7(B)$ such that $\lVert f_n \rVert_{W^{1,4}(B)}$ is bounded. Is it true that there exists a subsequence weakly convergent in $W^{1,1}(B)$? ...
0
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1answer
112 views

Remark below the proof of Rellich-Kondrachov compactness theorem

The following if a remark left at the end of the proof of the RK compactness theorem p.274 Evans. Here I think I got through everything more or less, but have trouble proving the final claim $$ ...
6
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0answers
66 views

Trace theorems for arbitrary differentiability $k$, with embedding constants under control as $k\to\infty$

The usual trace theorem (with non-optimal exponents, but I don't care for those at the moment) says that $$ W^{1,p}(\Omega)\hookrightarrow L^p(\partial\Omega) $$ for Lipschitz domains. When ...
2
votes
2answers
393 views

Laplace operator defined on a Sobolev space

Consider the Laplace operator $$A:W^{2,2}(\mathbb{R})\to L^2(\mathbb{R})\;\;\\A u = -u^{\prime \prime}$$ I want to know why this operator is closed (I'm using the closed graph theorem): Let ...
1
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2answers
29 views

Does $a_n \in H^1(\mathbb R^n)$ and $b_n \rightharpoonup 0$ in $H^1(\mathbb R^n)$ imply $\langle a_n, b_n \rangle \to 0$?

I have a question mainly in functional analysis. Suppose that $H^1(\mathbb R^n)$ is the standard Sobolev space that we all know. My question is as follows: Does $a_n \in H^1(\mathbb R^n)$, $|a_n| ...
1
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2answers
87 views

Counter example in Sobolev spaces: Is there a standard example for showing that $H^1(\mathbb{R}^2)\not\subset L^\infty(\mathbb R^2)$?

One way to define the Sobolev space $H^s(\mathbb{R}^n)$ is as the completion of the $C^\infty$ functions on $\mathbb{R}^n$ under the norm $$\lVert f \rVert_{H^s(\mathbb{R}^n)}:= \left( ...
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0answers
46 views

$L^2$-Sobolev space

I am looking at the proof of the following lemma and I don't understand the conclusion. Lemma: For $k\geq 0$ integer, let $f=\sum_{n\in \mathbb{Z^d}}{c_n\chi_n}\in L^2(\mathbb{T}^d)$. Then $$f\in ...
0
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1answer
49 views

If $u \in H^1(0,\infty)$, does $u_x(\infty) = 0$?

Let $u \in H^1(0,\infty)$. Then is it true that $u'(x) \to 0$ as $x \to \infty$? I am wondering by Green's formula/IBP in this setting; do I get something like $$\int_0^\infty u_{xx}v = ...
0
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1answer
69 views

Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
2
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1answer
48 views

Can we conclude that $v_{n}\rightarrow v$ in $L^{\infty}\left(\Omega\right)$ if $p>N$

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, $v_{n}\rightharpoonup v$ in $W_{0}^{1,p}\left(\Omega\right)$ , $\left\Vert v_{n}\right\Vert _{W_{0}^{1,p}}=1$ $\forall n$ . So we ...
0
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2answers
65 views

is a convex continuous function absolutely continuous

Does a continuous convex function $\mathbb{R} \to \mathbb{R}$ belong to $W^{1,1}_{loc}$ ? thank you.
1
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1answer
224 views

Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?

Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded? If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ ...
2
votes
1answer
156 views

Related question to : If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

This problem was asked in here. I need to ask something. Provide details for the following alternative proof that if $u \in H^1(U)$, then $$Du = 0 \text{ a.e. } \, \text{ on the set } \{u=0\}. $$ ...
9
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0answers
223 views

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
2
votes
1answer
81 views

Showing Sobolev space $W^{1,2}$ is a Hilbert space

I have the Sobolev space $W^{1,2}$ consisting of all continuous functions $f \in L^2(\mathbb{R})$ such that there exists an $f'$ with $f(b) - f(a) = \int_a ^b f'(t) dt$. $W^{1,2}$ has inner product ...
1
vote
1answer
120 views

Extension of bounded operators between norm spaces

Let $X$ and $Y$ be two Banach Spaces and $X_1$ be a subspace of $X$. If $T$ is a bounded linear operator from $X_1\to Y$, then, this is an extension of $T$ from $X\to Y$ such that $\|T\|_{X}\le ...
5
votes
1answer
121 views

Regularity theorem for PDE: $-\Delta u \in C(\overline{\Omega})$ implies $u\in C^1(\overline{\Omega})$?

I stumbled over this question in the context of PDE theory: Let $U$ be connected,open and bounded in $\mathbb{R}^n$ and $u \in C^0(\overline{U}) \cap C^2(U)$ and $\Delta u \in C^0(\overline{U})$ with ...
1
vote
2answers
112 views

Example 3, Sobolev space Evans

In the following example p.260 Evans. I think I understand everything except for one calculus fact in the second last equation: $$ \int_{\partial ...