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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2
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1answer
151 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 ...
1
vote
1answer
33 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 ...
2
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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 ...
0
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1answer
59 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 ...
2
votes
1answer
99 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 ...
0
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1answer
33 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 ...
1
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1answer
46 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 $u\in H^1_0(W)$ ...
0
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0answers
28 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 $$u_t - \Delta_p u = f$$ $$u(0) = u_0$$ ...
0
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0answers
21 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, ...
0
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1answer
38 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 ...
1
vote
1answer
39 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 ...
0
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1answer
40 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 ...
1
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1answer
78 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 ...
1
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0answers
32 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 ...
0
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0answers
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 ...
1
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1answer
52 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 ...
1
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1answer
114 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) = ...
2
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0answers
62 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 ...
3
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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 ...
1
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0answers
22 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} ...
7
votes
2answers
85 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 ...
1
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1answer
37 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
votes
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 ...
1
vote
1answer
46 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 $u\in L^\infty (\mathbb{R}^n)$.
1
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0answers
15 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 ...
0
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1answer
35 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 ...
1
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1answer
68 views

Poincare inequality on arbitrary finite domain for the case that $\|u-u_\Omega\|_{L^q}\leq \|\nabla u\|_{L^q}$ where $1<q<p$

This is exercise 12.25 from Leoni's book. Update: @Lukas Geyer has provided a counterexample for the case that $\Omega$ has a very bad boundary, which suggests that original exercise might be wrong. ...
0
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0answers
28 views

Equivalence discrete H^2 Sobolev norms

My aim is showing the equivalence of two discrete Sobolev norms. On $\mathbb{Z}^d$, $d\ge 2$, one defines the discrete derivative in the direction of the coordinate vector $\vec e_j$ as $$ ...
1
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0answers
27 views

Evans PDE mappings into better spaces

Evans PDE chapter 5.9 theorem 4 (mappings into better space), Evans wrote in the proof: In addition, $\bar{u}^{'}\in L^2(0,T;L^2(V))$ , with the estimate: \begin{equation} ...
1
vote
1answer
134 views

Weak derivative as an $L^2$ limit of the difference quotient

Let $u \in H^1(\mathbb{R})$. Show that $$\left\| \frac{u(x+h)-u(x)}{h} - u' \right\|_2 \to 0\quad \text{ as } h \to 0, $$ where $u' \in L^2(\mathbb{R})$ is the weak derivative of $u$. In other words, ...
2
votes
1answer
56 views

Sobolev multiplication theorem

I would like to know whether multiplication defines a bounded map $$H^{1/2} \otimes H^{1/2} \to H^{-1/2}$$ dimension of the domain is $3$. I have checked two different sources and one said that it ...
7
votes
1answer
169 views

Use $C^\infty$ function to approximate $W^{1,\infty}$ function in finite domain

This is exercise 10.21 from Leoni's book. The exercise asks me to prove that for any $u\in W^{1,\infty}(\Omega)$ where $\Omega$ is open FINITE, there exists a sequence $(u_n)\subset C^\infty(\Omega)$ ...
0
votes
1answer
23 views

Derivative of a function in the context of Sobolev spaces

Consider $B_1$ the unitary ball in $R^n$ centered in the origin and $2 \leq p < \infty$. Let $ \psi \in W^{1,p}(B_1)$. Let $h \in W^{1,p}(B_1) $ with $h - \psi \in W^{1,p}_{0}(B_1)$ with ...
-1
votes
1answer
31 views

How can I complete my proof: Sobolev space W^(1,p) is complete? Using Convergence theorem

I'm trying to prove that W^(1,k) (R) is complete. The steps i Had so far: let {fn} be a cauchy sequence in W^(1,k). therefore {fn} and {dfn} are cauchy sequences in L^p(R), and therefore converge ...
0
votes
1answer
30 views

Local Estimates for higher order homogeneous elliptic operators

For $u\in W^{2k}_2(\mathbb{R^n})$, $k\geq 1$, it is well known (see, for example, Exercise 12.9.4 in Krylov, N. "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces") that the following ...
2
votes
2answers
39 views

Estimate in Sobolev Spaces

Let $u\in H^0(U)\cap H^1_0(U)$ and $v_k\in C^\infty_c$(U) such that $v_k\rightarrow u$ in $H^1_0(U)$ and $w_k\in C^\infty (U)$ such that $w_k \rightarrow u $ in $H^2(U)$. I want to show that $ \int_U ...
0
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1answer
23 views

Prove the function $u(x):=1-|x|^{2-N}$ is in $W^{2,p}$ on $\{x\in \mathbb R^N;\,\,|x|>1\}$

This is exercise 10.11 from Leoni's book. Take $\Omega:=\{x\in \mathbb R^N;\,\,|x|>1\}$ and let $$u(x):=1-|x|^{2-N}$$ for $N\geq 3$. I am trying to prove that $\frac{\partial^2u}{\partial ...
1
vote
1answer
47 views

Extension theorems in Sobolev spaces: Solving for constants

I saw this problem in PDE book and tried searching for the idea behind solving it which I have not been able to find yet. If we have $n\ge2$, $B=\{x\in\mathbb R^n:|x|<1\}$ and ...
1
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0answers
23 views

Show that it is a element of $(H^1(\Omega))'$

Let $\Omega$ a open regular subset of $\mathbb{R}^n$, $n\ge 1$. I want to prove that if $u\in L^2(\Omega)$, the distribution $\partial_{x_i} u $ is an element of $(H^1(\Omega))'$ (for $1\le i \le ...
0
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0answers
37 views

Gelfand triples for Product Spaces

For $V = H^1(\Omega)$ and $H=L^2(\Omega)$. If we identify H with it's dual space $H^*$, then we have the following relation: \begin{equation} V \subset H \subset V^* \end{equation} Does this also ...
0
votes
1answer
51 views

Trace of $L^p$ function

For $U$ a bounded domain in $\mathbb{R}^n$, why is it that, in general, an $L^p$, $1\leq p<+\infty$, function does not have a trace on the boundary of $U$? Thanks in advance.
3
votes
1answer
54 views

Proving a Sobolev-Type inequality (also it is related to variational problem)

This is question 8.23 part $4$ from H. Brezis Functional analysis I already have that for any $f\in L^p(I)$, $p>1$ and $I=(0,1)$ there exists a unique $u\in H_0^1(I)$ satisfying ...
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0answers
42 views

Compositions and products on Sobolev spaces

Does anybody have a good textbook reference for someone who wants to begin studying products and compositions in Sobolev spaces, where the underlying domain is either $\mathbb{R}^n$ or an open subset ...
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0answers
35 views

How to interpret dual space convergence in norm $L^2(0,T;V^*)$ (sobolev--bochner spaces)

Let $V \subset H \subset V^*$ be a Gelfand triple. Define $W = \{ u \in L^2(0,T;V) \mid u' \in L^2(0,T;V^*)$. A result is that $C^\infty([0,T];V)$ is dense in $W$. So given $w \in W$, there exists ...
0
votes
1answer
56 views

Convergence with respect to the graph norm

Let $T:A\subset E\to F$ be a linear closed operator with dense domain. (E,F are Banach spaces) and $x_n$ a sequence in A such that $||x_n-x||_E\to 0$. Let $||x||_1:=||x||_E+||T(x)||_F$ be the graph ...
0
votes
1answer
145 views

$L^p$ norm of a gradient

Suppose $f:\mathbb{R}^n\to \mathbb{R}$ and let $Df=(f_{x_1},f_{x_2},..., f_{x_n})$, the gradient of $f$. A special case of the Gagliardo-Nirenberg inequality says that $$||f||_{p^*}\leq ...
1
vote
0answers
67 views

Derivatives of Differential operators

Let $V=H^1(\Omega)$ and $U = L^{\infty}(\Omega)^2$. Take for example, a differential operator, $L:U\times V \rightarrow V^* $ such that: \begin{equation} L(u,y) = \Delta y + \vec{u}(x)\nabla y ...
1
vote
1answer
70 views

Question regarding the dual space of $H_0^1(\Omega)$

Given $\Omega\in R^N$ open bounded with smooth boundary. We define $H^{-1}$ to be the dual space of $H_0^1(\Omega)$ and from Evan's PDE book, chapter $5$, we know that for any $f\in H^{-1}$, there ...
1
vote
0answers
20 views

Does this gradient map have a closed range?

Let $\mathbb{T}^n$ be $n$-dimensional torus. Let $H^1(\mathbb{T}^n)$ be the Sobolev space of functions in $L^2(\mathbb{T}^n)$ whose weak derivative is in $L^2(\mathbb{T}^n)$. Then the gradient map ...
0
votes
0answers
23 views

orthonormal basis or Parseval frame for Sobolev spaces

Consider the uni-variate Sobolev space of order $m$: $$ \mathcal{W}_2^m=\{ f:[0,1] \rightarrow \mathbb{R}|f,f^{(1)},\dots ,f^{(m-1)}\, \text{are absolutely continuous and}\, f^{(m)}\in L^2\}.$$ It is ...