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
48 views

$\|u\| = \Big(\int |\nabla u|^p \Big)^\frac{1}{p}$ as a norm on $W_0^{1,p}(\Omega)$ where $\Omega$ is bounded

Let $\Omega$ be a bounded open set of $\mathbb{R}^N$, define $\|u\|_1 = \Big(\int |\nabla u|^p \Big)^\frac{1}{p}$ as a norm on $W_0^{1,p}(\Omega)$ for $2\leq p<\infty$, where $$\int |\nabla u|^p ...
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42 views

Lebesgue's points Sobolev functions

Given $u\in W^{1,p}_{loc}(U)$, define $$u_{x,r}:=\frac{1}{|B(x,r)|}\int_{B(x,r)}u(y)dy. $$ I proved that $$ \frac{d}{dr}u_{x_0,r}\le Cr^{\frac{\varepsilon}{p}-1} $$ for $r\in ...
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56 views

I want to proof that the lebesgue measure of the set below is positiva. Help me!

Let $\Omega$ be a domain limited with smooth boundary in $\mathbb{R}^{n}$ and consider the Sobolev space $H^{1}_{0}(\Omega)$ equiped with the norm $||u||=\int_{\Omega}|\nabla u|^{2}dx$. Let ...
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1answer
47 views

$H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$

When $\Omega$ is a bounded open set of $\mathbb{R}^N$ with the help of Poincare inequality, we know that $H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$ is a Hilbert space. ...
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165 views

Density of smooth compactly supported functions in Sobolev space over unbounded domain.

Prove that $C^{\infty}_ {c}(\mathbb{R}^n)$ is dense in $W^{k,p}(U)$ for any open $U\subset \mathbb{R}^n$ with $\partial U\in C^1.$ In which $p\in [1,\infty)$ Note: In Lawrence Evans's PDE text, the ...
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2answers
49 views

Which Sobolev-Space to use to formulate weak biharmonic equation, $H^2_0$ or $H_0^1\cap H^2$?

For the weak formulation of the biharmonic equation on a smooth domain $\Omega$ $$ \Delta^2u=0\;\text{in}\;\Omega\\ u=0, \nabla u\cdot \nu=0\; \text{on}\; \partial\Omega $$ why does one take ...
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1answer
26 views

Integral over the unit ball in $\mathbb{R}^n$

Let $f(x)=|x|^r$ on $B_1(0)$ real valued function.Where $B_1(0)$ is the unit ball in $\mathbb{R}^n$. I am trying to show that if $r>1-n$ f has a weak derivative. ATTEMPT: I know from the ...
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1answer
98 views

Semilinear equation (PDE)

I've found this hard exercise on chapter 6 of Evans' book. I have no idea on how to proceed. Assume $u\in H^1(\mathbb{R}^n)$ has compact support, and assume that it is a weak solution of the semi ...
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1answer
62 views

Poincare type inequality on compact manifold

I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact manifold with or without boundary. The inequality I am looking for is the equivalent of $ ...
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1answer
53 views

for a compact manifold $M$, is the dual space of $H^1(M)$ equal to $H^{-1}(M)$?

Let $M$ be a compact Riemannian manifold. Is it true that $$(H^1(M))^* = H^{-1}(M)?$$ is there some intuitive explanation why? Or some reference? Thanks Here $H^1$ is the usual Sobolev space of $u ...
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3answers
50 views

Strongly convergent to zero in $L^2$ but $H^1$ norm not vanishing

Let $\Omega$ be some open, bounded, smooth subset of $\mathbb{R}^n$. I'm wondering whether it is possible for a sequence of functions $f_n:\Omega \rightarrow \mathbb{R} $ to be strongly convergent to ...
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1answer
39 views

Equivalence of dual spaces of Sobolev Spaces

I have a quick question: Is the following equivalence true for Sobolev Spaces $(W^{1,p}(\Omega))^{*} = W^{-1,p}(\Omega) = (W^{1,p}_{0}(\Omega))^{*}$ where $W^{1,p}_{0}(\Omega)$ is the closure of ...
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0answers
65 views

Function with divergence, curl and normal trace on boundary equals zero is zero

Let $u\in H^1(\Omega)$ with $\nabla\times u=0$ in $\Omega\subset\mathbb{R}^3$ (open bounded domain), $u\times n=0$ on $\partial\Omega$ (where $n$ is a a normal vector to $\partial\Omega$), ...
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1answer
40 views

Choosing a clever “test function” in Sobolev spaces.

Given $\mathbf{f}$ with $f_1,...,f_N\in L^2(\Omega)$ $$\int_\Omega \mathbf{f} \cdot \nabla v = 0 \quad\forall v \in H_0^1(\Omega)$$ we have $\mathbf{f} = \mathbf{0}$ a.e. since $\mathbf{f} \in ...
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56 views

Neumann eigenvalue problem for the Laplacian

Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$. Consider the Neumann eigenvalue problem $$ \Delta u =\lambda u \, \text{at} \, \Omega \\ \frac{\partial u}{\partial \vec{\nu}}=0 ...
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2answers
102 views

Interpolation inequality in sobolev space

Let $U$ be a bounded, connected open subset of $\mathbb R^n$ with $C^1$ boundary $\partial U$. Asume $|\beta| \leq k-1$ and $k$ is a integer. Show that for each $\epsilon >0$ there exists a ...
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1answer
37 views

Bounded variation of function $(f-M)^+$ and the measure of the set where it is concentrated

I read this statement in the book by Evans & Gariepy, page 215, last two lines. Here $f\in BV(R^n)$ and for fixed $\epsilon>0$ and $N>0$, we define $$A_\epsilon^N :=\{x\in \mathbb ...
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1answer
50 views

Convergence in $L^p$, Cauchy in $L^\infty$

If $u_n$ is a convergent sequence in $L^p$ with $u_n \to u$, and $u_n$ is convergent is $L^\infty$, is it true that the limit in $L^\infty$ must be $u$? Is it true if $u_n$ are all test functions, ...
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1answer
37 views

Question about difference quotient in Sobolev space

Let $u\in W^{1,p}(R)$ be given, $1\leq p<\infty$. We define $$ \tau_h(u)(x):=\frac{u(x+h)-u(x)}{h} $$ be the difference quotient. We all know that up to a subsequence $\tau_h(u)\to u'$ in the ...
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1answer
30 views

Trace on $H^1(\Omega_1\cup\Omega_2)$ (one little question in the conclusion of my proof)

Let $\Sigma$ a smooth surface that separates $\Omega_1$ and $\Omega_2$ (open and bounded sets) and let $(q^n_1)_{n\in\mathbb{N}}$ and $(q^n_2)_{n\in\mathbb{N}}$ sequences in $H^1(\Omega_1)$ and ...
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0answers
14 views

Prove that the subespace is closed [duplicate]

We consider $\Omega\subset\mathbb{R}^3$ (open bounded with smooth surface) and a surface $\Sigma\subset\Omega$. $\Sigma$ divides $\Omega$ in $2$ open bounded subsets: $\Omega_+$ and $\Omega_-$. ...
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0answers
97 views

Weak subsolution and composition with convex smooth function

I'm trying to solve a problem in Partial Differential Equations by Lawrence C. Evans. It goes like this Assume $u\in H^1(U)$ is a bounded weak solution of \begin{equation}-\sum_{i,j=1}^n ...
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1answer
32 views

For which $s$ is the function $(||x||^{s-2}x_i)^2$ integrable on the unit ball of $\mathbb R^n$?

Initial task is to find out, for which $s$ stands $u=||x||^s \in H^1(\Omega)$, where $\Omega = B(1,0)\subset\mathbb{R}^n$ and $H^1(\Omega)$ is a Sobolev space $W^{1,2}(\Omega)$. As to prove this, we ...
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73 views

Proving continuity on Sobolev space with weak topology

Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to ...
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1answer
174 views

Show that $u(x)=\ln\left(\ln\left(1+\frac{1}{|x|}\right)\right)$ is in $W^{1,n}(U)$, where $U=B(0,1)\subset\mathbb{R}^n$.

The entire problem statement is: Let $n>1$ and let $U=B(0,1)\subset\mathbb{R}^n$. Show that $u:U\to\mathbb{R}$ given by $$u(x)=\ln\left(\ln\left(1+\frac{1}{|x|}\right)\right)$$ is in ...
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1answer
39 views

Poincare Inequality for 1-Dimensional Problem.

I am referring to the book Introduction to Functional Analysis to Boundary Value Problems and Finite Element by Daya Reddy (page ...
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0answers
27 views

A $L^1$-bounded sequence from a $H^m$-bounded sequence

I am trying to show the following: for any $m > 0$ and $\alpha \in \mathbb{N}^n$, assume $(f_j)$ is a sequence of functions which is bounded in $H^m(\mathbb{R}^n).$ Assume moreover that all the ...
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1answer
66 views

Holder continuity using Sobolev imbedding

We assume for any $V\subset \subset U$ and $1<p<\infty$ $||u||_{W^{2,p}(V)}\le C(||\Delta u||_{L^p(U)}+||u||_{L^1(U)})$ for some $C=C(V,U,p)$. Given, $B=\{x∈R^3,|x|<1/2\}$ and we suppose ...
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1answer
111 views

The derivative of total variation

Define function $u$ on $[0,1]$ such that $$ u(x)= \begin{cases} x^2\cos\frac{1}{x} &0<x\leq 1\\ 0 & x=0 \end{cases} $$ and by $V(x):= \text{Var}_{[0,x]}u$, i.e., the total variation of ...
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1answer
35 views

Sobolev space $W^{1,2}((0,1))$ and boundary ODE - how does integration by parts goes?

As a part of a question about $W^{1,2}((0,1))$, I want to get a boundary ODE on $g$ and don't quite know how to integrate (?) in order to get the equation. let $g\in C^2[0,1]$ be our variable, $f\in ...
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1answer
50 views

Energy functional in Sobolev Space

Let $E[u]:=\int_W |\nabla u|^2 +V(x)u^2 dx$ be the energy functional on functions $u\in H^1_0(W)$ and $V\in L^\infty(W)$. Can someone help me show that $E$ is bounded below for all $u\in H^1_0(W)$ ...
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31 views

Existence of strong solutions to parabolic p-Laplace equation

Can I find a reference to where the existence of strong solution $u \in L^2(0,T;W^{1,p})$ with $u_t \in L^2(0,T;L^2)$ is proved to the equation $$u_t - \Delta_p u = f$$ $$u(0) = u_0$$ ...
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30 views

How do we define fractional Sobolev spaces on manifolds?

Fix $s > 1/2.$ The trace operator is surjective from $H^{s}(\mathbb{R}^{n+1})$ to $H^{s-\frac{1}{2}}(\mathbb{R}^{n}).$ If $\Omega$ is a bounded open set of $\mathbb{R}^{n}$ with smooth boundary, ...
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1answer
39 views

For $u\in H^s(\mathbb{R})$ with $s>n/2$, show that $\lim_{x\to\infty}u(x)=0$

For $u\in H^s(\mathbb{R}^m)$ with $s>m/2$, show that $\lim_{x\to\infty}u(x)=0$ By the Sobolev embedding theorem $H^s(\mathbb{R}^m)\hookrightarrow C_b(\mathbb{R}^m)$ and this should pretty ...
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1answer
44 views

divergence form of the determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
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1answer
42 views

Is it a closed set of $H^1(\Omega)$?

Let $\Omega\subset\mathbb{R}^3$ an open bounded domain (without holes) with boundary $\partial\Omega$ and let $\Omega_1\subset\Omega$. We consider the domain $\Omega\setminus\Omega_1$, that is, a ...
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1answer
95 views

Discrete Sobolev Poincare inequality proof in Evans book

In page 275 of Evans book, the Poincare's inequality has been proven via contradiction. I was wondering how can one extend this prove to prove Sobolev-Poincare inequality: $||u-u_\Omega||_{L^{p*}}\le ...
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41 views

Counterexample for Trace operator in BV space.

Suppose $\Omega\subset R^N$ is open bounded with Lipschitz boundary. Then for $u\in W^{1,p}(\Omega)$, there exists a linear operator $T$: $W^{1,p}(\Omega)\to L^{p}(\partial\Omega)$ such that the ...
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23 views

Something like a trace inequality of $H^1(\Omega)$

I have the following question: Let $\Sigma$ an a surface inside of an open domain $\Omega\subseteq\mathbb{R}^3$, where $\Sigma$ divides $\Omega$ in 2 open domains (for example: $\Sigma$ could be a ...
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1answer
58 views

question regarding to study Sobolev space by Fourier transform

I am reading Sobolev space by using Fourier transform approach. Here I have some questions that treated to be "obvious" by textbook but I can not understand it. We define operator ...
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1answer
128 views

Almost everywhere convergence of a bounded sequence in $H_0^1(\Omega)$

Let $\Omega \subset \mathbb{R}^N$ be a smooth domain with bounded complement (e.g. $\Omega = \mathbb{R}^N \setminus \overline{B(0;1)}$), and $(u_n)$ be a bounded sequence in $H^1_0(\Omega) = ...
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64 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
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0answers
22 views

Inverse continuous

According Dehman, Gérard and Lebeau in "Stabilization and control for the nonlinear Schrödinger equation on a compact surface" (Math. Z. (2006) 254:729–749, DOI 10.1007/s00209-006-0005-3) is claimed ...
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24 views

Auxiliary Backward problems

I am reading a paper that deals with a certain finite element method. We have the weighted Sobolev space $\mathcal{W}_{\mu}$ that consists of functions for which the norm $\Vert u \Vert_{\mu}^{2} ...
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90 views

Why is Existence and Uniqueness for Navier-Stokes Easier in 2-D than in 3-D?

I know that existence and uniqueness for incompressible viscous flow in the 2-D case has already been established$^1$, and that doing the same for the 3-D case has yet to be shown. Not only that, but ...
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1answer
40 views

Schrödinger Equation in Sobolev Space

Hi can anyone help me with this question? Let $u\in C^2(B(0,1))$, where $B(0,1)$ is the unit ball, such that $u$ vanishes at the boundary and let $u$ solve the Schrödinger equation $\Delta ...
3
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1answer
32 views

I would like to know if this particular orthonormal set is an orthonormal basis.

It is straightworward to see that $$\Biggl\{\frac {\sin (mx)}{1+m^2}\Biggr\}$$ is an orthonormal set with respect to the norm $$\|u\|^2=\int_0^{2\pi}|u|^2+|\nabla u|^2$$ (i.e. the norm in the Sobolev ...
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1answer
48 views

Estimate for functions in Sobolev space $H^s$

Can anyone help me with this question Use the Fourier transform to prove that if $u\in H^s(\mathbb{R}^n)$ for an integer $s$ such that $s>n/2$ then $u\in L^\infty (\mathbb{R}^n)$.
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0answers
19 views

differential quotient in Sobolev space by Fourier transform

I am studying Michiel's PDE book. Here I have a question about one theorem in chapter about Sobolev space. At the beginning of this chapter, he define $\tau_y(u)(x)=u(x+y)$ and by assuming $u\in ...
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1answer
46 views

Reference request for Sobolev embedding for the case $p>N$

Here I am trying to prove the COMPACT embedding for the case $p>N$. The exercise shows that for $p>N$, then $W^{1,p}(\Omega)$ is compact embedded in $C^{0,\alpha}(\bar{\Omega})$ where ...