For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

learn more… | top users | synonyms

0
votes
0answers
13 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 ...
1
vote
0answers
12 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 ...
1
vote
0answers
10 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
10 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
20 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
46 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
0answers
28 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
11 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
3 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 ...
0
votes
0answers
21 views

Proving a Sobolev-type ineqauality

Given $I=(0,1)$ and $u\in W^{2,p}(0,1)$ for $p>1$. I am trying to prove that for any $\epsilon>0$, the following hold: $$ \|u\|_{L^\infty(I)}+\|u'\|_{L^\infty(I)}\leq ...
0
votes
0answers
19 views

Examples on Sobolev Spaces and weak derivatives [closed]

We know that if ${W^{1,1}((0,1)^{2})}\equiv H^{1}((0,1)^{2})$ and if the weak derivative of $u$ $\epsilon$ ${W^{1,1}((0,1)^{2})}$ satisfies $D_{j}$$u=0$ a.e. in $(0,1)^{2}$, then how to show that ...
-2
votes
0answers
6 views

Examples on Weak derivatives and Sobolev Spaces

I know that $u(x)=\log\log(1+|x|^{-1})$ is unbounded function.But how to show that $u$ $\epsilon$ ${W^{1,n}(B(0,1))}$ for $n\geq1.$ I know that the weak derivatives $D_{j}u$ for ...
0
votes
0answers
12 views

Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...
3
votes
1answer
43 views

The question regrading to density argument in analysis.

I know the density argument in $L^p$ space, in Sobolev spaces, and even in BV are really sweet in many many cases. However, some times author just work on nice functions and comment by "the rest can ...
0
votes
0answers
25 views

Another way to show $\nabla u =0$ a.e. on $u=0$ for $u\in H^1(\Omega)$

This is an exercise on Evans PDE book, Ch5. It provides another way to prove $\nabla u =0$ a.e. on $u=0$ for $u\in H^1(\Omega)$ then the one in Evans & Griapy's book. The statement is as ...
0
votes
1answer
33 views

Prove a Poincare-Like Inequality

Here is the question. Given any $\alpha>0$ and $u\in H^1(\Omega)$, $\Omega=B(0,1)$ in $n$ dimensions. Then we have $$\int_\Omega |u|^2 dx\leq C(\alpha)\int_\Omega |\nabla u|^2dx $$ provided that ...
0
votes
1answer
21 views

The Relation between Holder continuous, absolutely continuous, $W^{1,1}$, and $BV$ functions

I am trying to find out the relation between those spaces. Take $I\subset R$ on the real line. $I$ can be unbounded. Then I have: We first assume $I$ is bounded. If $u\in C^{0,\alpha}(I)$, for ...
0
votes
0answers
13 views

Embedding of fractional Sobolev spaces

I have a question regarding fractional Sobolev spaces. Given an open bounded set $\Omega\subset \mathbb{R}^{N}$ (Lipschitz, for instance), $s\in (0,1)$ and $1\leq p<q<\infty$, does the following ...
1
vote
1answer
21 views

How to get a smoothing operator from a rapid decreasing function?

From John Roe: Elliptic Operators, topology and asymptotic methods, page 82-83. Let $\mathcal{D}$ be a Dirac operator on the spin bundle $S$, then any section $s\in L^{2}(S)$ has a "Fourier ...
0
votes
0answers
26 views

dense subspace of $C(0,T)$

I want to prove that the space H of functions which are continuous in [0,T] with weak derivative in $L^2[0,T]$ and their value in 0 is 0, is dense in the space of continuous functions in [0,T] with ...
0
votes
2answers
51 views

Sobolev space on union of two open sets

Let $\Omega_1,\Omega_2 \subset \mathbb{R}^n$ be open sets. Let $p \in [1,\infty]$. Let $u: \Omega_1 \cup \Omega_2 \to \mathbb{R}$ be a function such that $u|_{\Omega_1} \in W^{1,p}(\Omega_1)$ and ...
0
votes
0answers
9 views

reference request about sobolev space and BV space

I am studying Sobolev Space and BV space by using Leoni's and Evans & Gariepy's book. I was wondering that where can I find some explicit example and some computational question of those space. A ...
0
votes
0answers
21 views

Question about Sobolev spaces. Controlling Divergence.

I have a question about Sobolev spaces, I think I just need a reference. For $\Omega$ an open and bounded subset of $\mathbb{R}^d$, and $\vec{\Phi}\colon \Omega \to \mathbb{R}^d$ a vector valued ...
0
votes
0answers
21 views

Find an operator $Z$ in $H^1(0, \infty)$ with $\langle u,Zv\rangle = \int \bar{u}v dx$

I'm working with operators associated to bilinear forms. What I need to find is a continous, linear operator $T$ defined on $H^1((0, \infty))$ [note that $H^1 = W^{1,2}$ is the Sobolev space] such ...
0
votes
0answers
18 views

A function in $W^{1,p}(U)$, $U=B^0(0,1)$

Let $\{r_k\}_{k=1}^{\infty}$ be a countable, dense subset of $U=B^0(0,1)$ and a given function by $$u(x) = \sum_{k=1}^{\infty}\frac{1}{2^k}|x - r_k|^{-\alpha},\,\, x \in U$$ For which values of ...
1
vote
1answer
23 views

Counterexample for existence of a minimiser in a variational problem

I'm trying to find an example of a minimisation problem of the form $$ \inf \{ J(u) := \int_{\Omega} f(x)|u(x)| + |\nabla u(x)|^2:\, u \in H^1, \, \int u = 1\}$$ with $\Omega$ an open and bounded ...
2
votes
1answer
41 views

Uniformly bounded sequence in Hilbert-Sobolev space

Let $\Omega \subset \mathbb{R}$ be a bounded open set with $C%1$ boundary and $H^1(\Omega) = W^{1,2}(\Omega)$ be the Hilbert-Sobolev space. Let ${u_n}$ be a sequence of functions which are uniformly ...
0
votes
0answers
14 views

finding the solution space of helmholtz equation with mixed boundary?

Let $\Omega\subset \mathbb{R}^2$ be a bounded set and $\Omega' \subset \Omega$ such that $\partial\Omega \cap \partial\Omega' = \emptyset$ $$ \left\{ \begin{align} k^2 u + \Delta u &= 0 \quad ...
1
vote
1answer
29 views

Is the embedding $L^2(0,T;H^1) \subset L^2(0,T;L^2)$ compact?

Is the embedding $L^2(0,T;H^1(\Omega)) \subset L^2(0,T;L^2(\Omega))$ compact?
0
votes
1answer
31 views

The approximation of BV functions

From Evans & Gariepy 's book, I learned that generally for any $u\in BV(R^n)$, we can find $u_n\in BV(R^n)\cap C^\infty (R^n)$ such that $$ \lim_{n\to\infty} \|u_n-u\|_{L^1(R^n)} = 0$$ and ...
3
votes
0answers
63 views

Equivalence of two definitions of weak solution (from a book, I don't understand something!!!!)

Consider $$y_t - \Delta y = f$$ $$y(0) = y_0$$ with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions: We have $y \in ...
2
votes
0answers
43 views

Proving a PDE has a particular weak form (check my proof please!)

Let $u_t - \Delta u = f$ hold in $L^2(0,T;H^{-1})$ for a solution $u \in L^2(0,T;H^1_0)$ with $u_t \in L^2(0,T;H^{-1})$. This means $$\int_0^T \left(\langle u_t(t), v(t)\rangle + \int_\Omega \nabla ...
-1
votes
0answers
26 views

Inequality in $H^2$

I have tried to prove this result, but it seems too hard. Need Help. Let $U\subseteq\mathbb{R}^n$ a bounded set with smooth boundary, and the differential operator: ...
1
vote
3answers
44 views

Closed Operator on a Sobolev space

I am wondering if the following differential operator $A:D(A)( \subset {\bf{H}}) \to {\bf{H}}$ defined on the sobolev space $\mathbf{H}=H_{0}^{k}(0,1)\times {{L}^{2}}(0,1)\text{ }$ is a closed ...
1
vote
1answer
27 views

Density of $C\infty([0,T];V)$ in $W(0,T;V,V)$.

Let $W=\{ u \in L^2(0,T;V) : u_t \in L^2(0,T;V)\}$ where $V$ is a Hilbert space in the Gelfand triple $V \subset H \subset V^*$ and $u_t$ is the weak time derivative. Is it true that ...
3
votes
2answers
46 views

Fractional Sobolev space $H^{1/2}(-\pi,\pi)$

Let $H^{1/2}(-\pi,\pi)$ be the space of $L^2$ functions whose Fourier series coefficients $\{c_n\}_n$ satisfy the summability constraint $\sum_n |n| |c_n|^2 < \infty$. Are functions in ...
0
votes
2answers
32 views

$G(f) \le \|f\|_{H^s(\mathbb R)},\; s>2 \Rightarrow G(f) \le \|f\|_{H^2(\mathbb R)}$?

If a quantity of a function $f$, call $G(f)$ satisfies $$ G(f) \le \|f\|_{H^s(\mathbb R)} $$ for all $s>2$, then can I conclude that this holds for the limiting case $s\to 2$: $$ G(f) \le ...
5
votes
0answers
38 views

$H^{1/2}$ function but not better

I am looking for an example of a function $f: [0, 1] \longrightarrow \mathbb{R}$ that is in the Sobolev space of order $1/2$, $H^{1/2}([0, 1])$, but not in the Sobolev space of order $1/2 + ...
1
vote
1answer
31 views

Estimating the rate of convergence of $|S_Nf-f|$ given that $\|f\|_{H^s}\leq 1$

Given that the Soloblev space norm $$\|f\|_{H^s}^2=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ and the inequality $$\|f(\cdot +\theta)-f\|_{L^2}\leq 2\pi ...
0
votes
0answers
28 views

bounding supremum with Sobolev embedding theorem

I consider $d \geq 2$ and the domain $ D = [0,1]^d$. I consider a function $f : D \to \mathbb{R}$ that is $k$ times continuously differentiable. [$k$ can be specified as large as possible.] I am ...
1
vote
1answer
37 views

Estimating the modulus of continuity of translation in $L^2$ by a Sobolev norm of the function

For any $s\in \mathbb{R}$ define the Hilbert space $H^s(\mathbb{T})$ by means of norm $$\|f\|^2_{H^s}=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ Show that for any $0\leq ...
0
votes
1answer
63 views

The proof of a Sobolev embedding inequality by a compactness argument

I want to prove the following If $N\ge 3$ there exists a constant $c_0=c_0(\Omega)$ such that for all $\alpha\ge 1$ and $z\in H^1(\Omega)$ \begin{align} ...
1
vote
0answers
14 views

$\left\|f\right\|_{L^{\infty}(\mathbb R^d)} \leq K \left\|f\right\|_{H^{s}(\mathbb R^d)}$

The Swchartz, $\mathcal S(\mathbb R^d)=\left\{f\in C^{\infty}(\mathbb r^d): \sup_{x\in \mathbb R^d}(1+|x|^{2})^{\frac{k}{2}}\sum_{|\alpha\leq l|}|D^{\alpha}f(x)|< \infty\right\}$, for all $k, l \in ...
2
votes
1answer
41 views

weak solution of poisson equation

Consider the equation , with $u\in H^1$ $$\begin{cases}\Delta u = f & \text{in }\Omega\\ \displaystyle \frac{\partial u}{\partial \nu} = 0 & \text{in } \partial \Omega\end{cases}$$ where $\nu$ ...
3
votes
1answer
53 views

Closure of $C_0^{\infty}$ in $W^{k,p}(\Omega)$

Why is it that in the definition of $W_0^{k,p}(\Omega)$ for $\Omega$ with boundary smooth enough, we only have $D^{\alpha}u$ for all $0\leq|\alpha|\leq k-1$ vanishing at the boundary and not ...
0
votes
0answers
29 views

Traces of Sobolev functions in an unbounded domain

I have a doubt concerning the trace of Sobolev functions. Let $C=\Omega\times(0,\infty)$ an infinite cylinder of basis a smooth domain $\Omega$ of $\mathbb{R}^{N}$ and consider the classical Sobolev ...
1
vote
0answers
35 views

Discrete Poincare Inequality

For the sake of this question, let $\Omega \subset \mathbb{R^2}$ be a regular domain. In variational problems involving the Sobolev space $W^{1,1}(\Omega)$ (or $BV(\Omega)$) one often uses the Sobolev ...
1
vote
1answer
34 views

Two theorems about approximation by smooth functions

Let $U$ be an open subset of $\mathbb{R}^{n}$. The following are two theorems taken from the chapter about Sobolev Spaces of the Evans' book. Theorem 1 Assume $u\in W^{k,p}(U)$ for some $1\le ...
0
votes
0answers
11 views

Sobolev spaces and Lipsschitz continuity [duplicate]

How to show that u $\epsilon$ ${W^{1,\infty}(\Omega)}$ if and only if u is Lipschitz continuous. But I suggested to use the fact that u is Lipshtz means that there is a constant $L>0$ such that ...
1
vote
1answer
38 views

Is the sum of two Sobolev spaces defined on two different sets the Sobolev space defined on the union of these two sets?

Is it true that $H^1(\Omega_1 \cup \Omega_2 )=H^1(\Omega_1)+H^1(\Omega_2)$? Below, we already have a counterexample. Let me ask further. If I impose $\Omega_1 \cap \Omega_2=\emptyset$, is there still ...