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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1answer
107 views

Is $T$ a compact mapping from $W_{0}^{1,2}\left(\Omega\right)$ into itself? [closed]

Let $\Omega$ be an open bounded subset in $\mathbb{R}^{6}$ and $f$ be in $L^{8}\left(\Omega\right)$. For any $w$ in $W_{0}^{1,2}\left(\Omega\right)$, define $T\left(w\right)$ be in ...
2
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0answers
48 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
0
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0answers
29 views

If $\partial\Omega\in C^{2+\alpha}$ and $-\Delta\Theta=f\text{ in }\Omega$ with $f\in C_0^\infty(\Omega)$, then $\Theta\in C^{2+\alpha}$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with $\partial\Omega\in C^{2+\alpha}$ for some $\alpha>0$ $f\in C_0^\infty(\Omega)$ $\Theta\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be the ...
0
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2answers
49 views

Example of an $H^{-1}$ function that isn't $L^2$

I'm going back over some PDE and Sobolev space theory, and the following is puzzling to me. Consider a nice domain $\Omega$ and the space $H^1_0(\Omega)$ of functions with $L^2$ first derivatives, ...
2
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1answer
47 views

Stuck in the proof of Extension theorem

I am reading the extension theorem in sobolev spaces in the book '' Partial Differential Equation'' by Evan and I get stuck at one point. Let $U\subset\mathbb{R}^n$ is open and bounded, and $\partial ...
0
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1answer
27 views

Absolute Continuity defined by Necas

I read a definition of the absolute continuity in Necas' book "Direct Methods in the Theory of Elliptic Equations": Let $\Omega$ be a domain in $\mathbb{R}^n$ , $P$ a line verifying ...
2
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1answer
46 views

Conditions under which a function vanishing on the boundary belongs to $H_0^1$

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set ,and $u \in C(\overline{\Omega}) \cap C^1(\Omega) \cap H^1(\Omega) $ be a function such that $u \big|_{\partial \Omega}=0 $.Prove that $ u ...
2
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0answers
69 views

How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary. My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that ...
4
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1answer
94 views

Prove that a series converges in $L^2(\Omega)$

Let $\Omega$ be a bounded smooth domain and let $\varphi_k$ be eigenfunctions of the Neumann Laplacian with eigenvalues $\lambda_k$. Let $u \in H^1(\Omega)$. I want to show that for all $y \in ...
3
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1answer
40 views

If the weak derivative $\nabla u$ of $u\in L^2(\Omega)$ exists, then $\int_\Omega|\nabla u|^2=\int_\Omega|\nabla u^+|^2+\int_\Omega|\nabla u^-|^2$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded $u\in \mathcal{L}^2(\Omega)$ be weakly differentiable, i.e. $$\int_\Omega u\nabla\psi\;d\lambda^n=-\int\psi\nabla u\;d\lambda^n\;\;\;\text{for all ...
2
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1answer
55 views

Reference about Sobolev spaces

I'm looking some book from which I could learn more about Sobolev spaces. I'm interested rather in abstract theory: some topics which I would like understand in detail include: general construction ...
4
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0answers
55 views

Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$

Assume $\Omega$ is open bounded domain in $\mathbb R^n$ Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$ with inner product ...
0
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1answer
18 views

About Chebyshev inequality for integrals

Let $u \in H^1(\Omega) \cap C(\Omega)$ ($\Omega \subset R^n$ a smooth and bounded domain) a nonnegative function. Let $B(x,R) \subset \overline{ B(x,R) } \subset \Omega $ a ball. Suppose that ...
2
votes
1answer
38 views

A classical solution of Poisson's equation is also a weak solution

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be a solution of $$\left\{\begin{matrix}-\Delta u&=&f&&\text{in }\Omega\\ ...
1
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1answer
29 views

About equi-integrability

Suppose $\Omega\subset \mathbb R^N$ is bounded and lipschitz boundary. Suppose $u_n, u\in H^1(\Omega)$ such that $u_n\to u$ weakly in $H^1$. Then can I conclude that $$ ...
0
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1answer
40 views

Dual space of Sobolev Space

I'm studying pde's and I'm on the topic of Sobolev Spaces and I have a small question regarding the dual of a Sobolev Space. I'm trying to understand a proof about the characterization of the dual of ...
0
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1answer
27 views

Is function $u$ nice when all $\Delta^k u$ are nice?

Let $\Omega \subset\mathbb{R}^d$ has smooth boundary and $$\Delta^k u \in W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega) \qquad k\geq0$$ Show that $u\in W^{n,2}(\Omega)$ for all $n\in \mathbb{N}$. This ...
0
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1answer
17 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
30 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
13 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} ...
3
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2answers
51 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
21 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
61 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 ...
2
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1answer
44 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 ...
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0answers
46 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 ...
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0answers
23 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 ...
0
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0answers
28 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
vote
1answer
32 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
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1answer
52 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). ...
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0answers
28 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
24 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
79 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
26 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
49 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 ...
1
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1answer
38 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
votes
1answer
45 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
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1answer
51 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
vote
1answer
18 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
91 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
57 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 ...
1
vote
2answers
42 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 ...
0
votes
1answer
67 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 ...
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0answers
16 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
votes
1answer
28 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
46 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
35 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
36 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
vote
1answer
42 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
vote
0answers
67 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
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1answer
24 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 $W^{1,1}(B)$? ...