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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0answers
21 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 ...
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0answers
34 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 ...
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1answer
34 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.
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0answers
29 views

Composition of a compact support function with a increasing one

Let $u\in H^1(\mathbb{R}^n)$ have compact support and $c:\mathbb{R}\to\mathbb{R}$ is smooth, with $c(0)=0$ and $c'\geqslant 0.$ I am trying to prove $c(u(x))\in L^2(\mathbb{R}^n)$ or $c'(u(x))\in ...
3
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1answer
49 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
36 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
15 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 ...
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1answer
18 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 ...
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1answer
36 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 ...
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0answers
62 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 ...
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0answers
45 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 ...
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0answers
14 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 ...
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0answers
8 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
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0answers
28 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 ...
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0answers
14 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
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1answer
69 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 ...
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0answers
32 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 ...
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1answer
53 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
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1answer
81 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 ...
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0answers
20 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 ...
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1answer
34 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 ...
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0answers
30 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 ...
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2answers
66 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 ...
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0answers
13 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 ...
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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
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0answers
22 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 ...
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0answers
20 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
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1answer
28 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
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1answer
52 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 ...
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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
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1answer
30 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?
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1answer
41 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
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0answers
64 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
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0answers
44 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 ...
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3answers
46 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
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1answer
29 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
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2answers
49 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 ...
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2answers
33 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
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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
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1answer
32 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
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0answers
29 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 ...
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1answer
39 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
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1answer
65 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
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0answers
16 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
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1answer
60 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
55 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
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0answers
35 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
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0answers
57 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
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1answer
38 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 ...
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0answers
12 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 ...