Questions tagged [sobolev-spaces]

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

27
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2answers
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Sobolev space $H^s(\mathbb{R}^n)$ is an algebra with $2s>n$

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have $\lVert ...
9
votes
1answer
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Problem 9 - Chapter 5 - Evans' PDE (First Edition)

In the $1$st edition, this was question $5.9$. The question is: Integrate by parts to prove: $$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\...
6
votes
1answer
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Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
5
votes
1answer
3k views

relation between $W^{1,\infty}$ and $C^{0,1}$

I know that $f \in C^{0,1}_{loc}(U)\Leftrightarrow f \in W^{1,\infty}_{loc}(U)$ and I have a reference for this. I would like a reference or a explanation for $C^{0,1} = W^{1,\infty}$ on domain convex....
12
votes
1answer
3k views

Sobolev embedding for $W^{1,\infty}$?

From the Evans'PDE book I learned $W^{1,\infty}(U)$ coincide with Lipschitz continuous $C^{0,1}(U)$ with $U\in\Bbb{R}^n(n\geqslant1)$ is bounded and $\partial U\in C^1$. I wonder the counterpart ...
15
votes
1answer
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How is the acting of $H^{-1}$ on $H^1_0$ defined?

I have a question about the Sobolev Space $H^1_0(U)$, where $U$ is a open subset of $\mathbb{R}^n$. Let us denote with $H^{-1}(U)$ the dual space of $H^1_0$. How is the acting of $H^{-1}$ and $...
17
votes
1answer
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How to prove the spectrum of the Laplace operator?

How can I prove that the spectrum of the Laplace operator $$\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$$ is $\sigma(\Delta)=[-\infty,0]$?
13
votes
1answer
6k views

The Sobolev Space $H^{1/2}$

This is a very stupid question. In my course on linear PDEs, the professor used $H^{1/2}$ without defining it, and I have been looking on google trying to find a definition, but the only related thing ...
8
votes
1answer
2k views

Sobolev Embedding (Case: p=N)

Let $\Omega\subset\mathbb{R}^N$ be a regular bounded domain. Suppose $p=N$, then by Sobolev theorem, we have that for fixed $q\in [1,\infty)$ $$\|u\|_q\leq C\|u\|_{1,N}\ ,\forall\ u\in W^{1,N}(\...
5
votes
4answers
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Evans PDE Problem 5.15: Poincaré inequality for functions with large zero set

I have trouble proving the following problem (Evans PDE textbook 5.10. #15). Could anyone kindly help me solving the problem? I know that I should somehow use Poincaré inequality but I still cannot ...
6
votes
1answer
597 views

Sobolov Space $W^{2,2}\cap W^{1,2}_0$ norm equivalence

I would like to know why on $W^{2,2}\cap W^{1,2}_0$ the norms $$ ||u|| _{W^{2,2}}=\sum_{|\alpha|\leq 2}||D^\alpha u||_{L^2}$$ and $$||\Delta u||_{L^2}$$ are equivalent.
5
votes
2answers
346 views

Sobolev spaces - about smooth aproximation

Consider $\Omega $ a open and bounded set of $\mathbb R^n$ . Let $u \in H^{1}(\Omega)$ a bounded function. I know that there exists a sequence $u_m \in C^{\infty} (\Omega) $ where $u_m \rightarrow u$...
10
votes
2answers
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Chain rule in the Sobolev space $W^{1,p}$

(Chain rule) Assume $F : \mathbb{R} \to \mathbb{R}$ is $C^1$, with $F'$ bounded. Suppose $U$ is bounded and $u \in W^{1,p}(U)$ for some $1 \le p \le \infty$. Show $$v :=F(u) \in W^{1,p}(U) \quad \text{...
6
votes
1answer
2k 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 $W^{1,n}(U).$...
5
votes
1answer
1k views

Poincaré inequality for a subspace of $H^1(\Omega)$

The following is the well known Poincaré inequality for $H_0^1(\Omega)$: Suppose that $\Omega$ is an open set in $\mathbb{R}^n$ that is bounded. Then there is a constant $C$ such that $$ \int_\...
4
votes
1answer
1k views

Evans PDE Chapter 5 Problem 11: Does $Du=0$ a.e. implie $u=c$ a.e.?

Let $W^{1,p}(U)$ be the Sobolev space. Suppose that $U$ is connected bounded domain in $\mathbb{R}^n$ and $u \in W^{1,p}(U)$ satisfies $Du=0$ a.e. in $U$. How can I prove that $u$ is constant a.e. in $...
4
votes
0answers
1k views

Stampacchia Theorem: $\nabla G(u)=G'(u)\nabla u$?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $G:\mathbb{R}\to\mathbb{R}$ a Lipschitz function with $G(0)=0$. Stampacchia's Theorem states that if $u\in W_0^{1,p}(\Omega)$, then $G(u)\in W_0^...
4
votes
1answer
3k views

Regularity of a domain - definition

What does it mean when we say that a domain is $C^k$, $C^{k,\alpha}$, Lipschitz, or smooth? Is there an intuitive understanding?
0
votes
0answers
48 views

Estimate on average with weight

Let $B_{2R}=B(x,2R)$ be the ball of radius $2R$ centered at $x$ and $v=log\,u$ for some positive function $u$ defined on $B_{2R}$. Denote by $v_{B_{2R}}=\frac{1}{w(B_{2R})}\int_{B_{2R}}v(x)w(x)\,dx$, ...
16
votes
2answers
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Dual space of $H^1$

It holds that $W^{1,2}=H^1 \subset L^2 \subset H^{-1}$. This is clear since for every $v \in H^1(U)$, $u \mapsto (u,v)_{H^1}$ is an element of $H^{-1}$. Moreover for every $v \in L^2(U)$, $u \mapsto (...
11
votes
1answer
4k views

The dual of the Sobolev space $W^{k,p}$

The dual of the Sobolev space if defined to be $$(W^{k,p}(\Omega))' = W_0^{-k,p'}(\Omega)$$ where $\frac 1 p + \frac 1 {p'} = 1$. Why makes this definition sense, especially why do we have $L^{p'}$-...
25
votes
2answers
760 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient ...
9
votes
2answers
4k views

$u\in W^{1,p}(0,1)$ is equal a.e. to an absolutely continuous function?

I have a simple question on Sobolev space theory. Let $1\le p \le \infty$. How can one prove that every $u\in W^{1,p}(0,1)$ is equal a.e. to an absolutely continuous function and that $u'$ exists a.e. ...
13
votes
2answers
2k views

Why no trace operator in $L^2(\Omega)$?

We have trace operator which allows us to define boundary values of an $H^1$ function. This is because of the fact that $C^\infty$ is dense in $H^1$ under the $H^1$ norm, I believe. I'm sure either $...
17
votes
3answers
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Are polynomials dense in Gaussian Sobolev space?

Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of $C_c^\infty(\mathbb{R}^n)...
5
votes
1answer
289 views

Can $u\in W_0^{1,p}\cap L^\infty$ be approximated by a sequence $u_k\in C_0^\infty $ with $\|u_k\|_\infty$ bounded?

Assume that $\Omega\subset\mathbb{R}^N$ is a bounded Lipschitz domain and let $p\in [1,\infty)$. Suppose that $u\in W_0^{1,p}(\Omega)\cap L^\infty (\Omega)$. Is it possible to approximate $u$ by a ...
7
votes
1answer
467 views

Explicit characterization of dual of $H^1$

Let's start by some well-known facts: $H^1(\mathbb{R})$ is a Hilbert space, hence there holds the Riesz representation theorem, stating that any linear functional on it can be represented as $L = \...
6
votes
1answer
135 views

Is a Sobolev map with smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$ Let $d>2$. Let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$ where $\Omega$ is an open subset of $\mathbb{R}^d$. Let $2 \le k \le d-1$ be fixed. Suppose that $\det df>0$...
5
votes
2answers
332 views

Boundedness of functions in $W_0^{1,p}(\Omega)$

Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and $p\in (1,\infty)$. Suppose that $u\in W_0^{1,p}(\Omega)$ and $u$ is locally essentially bounded. Does this implies that $u$ is globally ...
5
votes
1answer
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eqiuvalent norms in $H_0^2$

I have found that the $H^2(D)$ norm of a field with zero Cauchy data on $\partial D$ (i.e. in $H_0^2(D)$) is equivalent to the $L^2(D)$ norm of its Laplacian, where D is simply connected with smooth ...
4
votes
1answer
735 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 ...
2
votes
1answer
748 views

Delta Dirac Function

Show that $\delta$, function of Dirac, defined than $\left<{\delta_0,\phi}\right> = \phi(0)$ belongs to $W^{-1,p}(]-1,1[)$ and $\delta_0 \notin L^p(-1,1)$ $\forall p\geq 1$. How I will be able ...
7
votes
1answer
2k views

Does weak convergence in Sobolev spaces imply pointwise convergence?

I encounter a problem when reading Struwe's book Variational Methods (4th ed). On page 38, it is assumed that $\|u_m\|$ is a minimizing sequence for a functional $E$, i.e. $E(u_m)\rightharpoonup I$ ...
5
votes
2answers
300 views

Weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$

What does weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$ means? $\Omega$ is open, bounded, has boundary smooth and etc...
4
votes
1answer
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Function in $H^1$, but not continuous

By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
2
votes
2answers
303 views

About smooth approximation in a Sobolev space

I want to prove the following fact : Consider $\Omega \subset R^n$ a bounded and open set. Let $v \in H^{1}_{0}(\Omega)$ a nonnegative function. Then exists a sequence $v_m$ in $C^{\infty}_{0}(\Omega)...
1
vote
2answers
258 views

$\Delta u$ is bounded. Can we say $u\in C^1$?

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set. Let us say it has a Lipschitz boundary. Consider the Laplacian $\Delta$ in the classical sense. Suppose $\Delta u=\frac{\partial^2}{\partial x_1^...
6
votes
1answer
657 views

Question about definition of Sobolev spaces

I'm trying to understand the following definition: which can also be found here on page 136. Question 1: Closure with respect to what norm? It's not given in the definition. Question 2: Do I have ...
2
votes
1answer
2k views

Weak Derivative Heaviside function

I have to prove that the Heaviside function $$ H(x):=\begin{cases} 1 &\mbox{if } x \in [0,+\infty) \\ 0 &\mbox{otherwise}\end{cases} $$ doesn't admit weak derivative in $L^1_{loc}(\mathbb{R})$...
2
votes
2answers
255 views

variational problem-exercice

Let $\Omega$ an open bounded connexe and regular, and let $f \in L^2(\Omega)$ We consider the variational problem: find $u \in H^1(\Omega)$ such: $$\displaystyle\int_{\Omega} A \nabla u \cdot \nabla v ...
6
votes
3answers
3k views

Equivalent Norms on Sobolev Spaces

Let $u$ be in the Sobolev space $H^2$. Then the standard definition of the norm is $$ ||u||_{H^2}^2 = \sum_{|\alpha| \leq 2} ||D^{\alpha} u||_{L^2}^2. $$ However, I have also seen it defined this ...
10
votes
1answer
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Counterexample of Sobolev Embedding Theorem?

Is there a counterexample of Sobolev Embedding Theorem? More precisely, please help me construct a sobolev function $u\in W^{1,p}(R^n),\,p\in[1,n)$ such that $u\notin L^q(R^n)$, where $q>p^*:=\frac{...
8
votes
2answers
638 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 ...
3
votes
3answers
376 views

Definition of weak time derivative

My quesion involves the weak time derivative. In the book: 'Partial Differential Equations' by Evans the time derivative $u'$ of a function $u: [0,T] \rightarrow H^1_0(U)$ is defined by an element $...
5
votes
1answer
2k views

Poincaré Inequality

In page 290 of this book, Evans proves the Poincaré inequality (Theorem 1) arguing by contradiction. Is there a direct proof of this theorem (Theorem 1) without arguing by contradiction?
3
votes
1answer
1k views

Estimating Poincare constant for unit interval

I want to show that the Poincare constant for the $W^{1,2}_0(0,1)$ is smaller than $1$. More specifically, I want to show that there is a constant $C<1$ such that for any $f\in C^\infty_c(0,1)$ (...
2
votes
1answer
3k views

Prove Friedrichs' inequality

I'm trying to show that the theorem (Friedrichs' inequality) in my book: Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the ...
2
votes
2answers
660 views

Is $C_0^{\infty}(\Omega)\cap H^{m,p}(\Omega)$ dense in $H^{m,p}(\Omega)$?

Let $\Omega\subset\mathbb{R}^n$ be an open set. Let $H^{m,p}(\Omega)$ denote the Sobolev-space of at most $m$ times weakly differentiable functions whose weak derivatives are all $L^p$. Further let $C^...
7
votes
2answers
355 views

Decomposition of functionals on sobolev spaces

It is a well known theorem, that every signed measure can be split into its positive and negative parts (Hahn-Jordan-Decomposition). My question is, if something similar is possible for functionals on ...
5
votes
1answer
2k views

Evans's proof of the Leibniz's formula for the weak derivatives in Sobolev spaces

In PDE Evans 2nd edition, pages 261-263, there is a theorem and its proof which concerns the four properties of weak derivatives. Unfortunately, I do not understand the fourth property, which I will ...