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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.

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801 views

Why are functions with vanishing normal derivative dense in smooth functions?

Question Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm? Here I define $...
27
votes
2answers
5k views

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 ...
25
votes
2answers
758 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 ...
17
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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 $\sigma(\Delta)=[-\infty,0]$?
17
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1answer
483 views

Suppose that there exist a set $\Gamma$ of positive measure such that $\nabla u=0, a.e.\ x\in\Gamma$.

Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain. Let $u\in H_0^1(\Omega)$ and $\tilde{u}\in L^2(\Omega)$ satisfies $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in C_0^{\...
16
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3answers
1k views

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)...
16
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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 (...
16
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1answer
855 views

Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form $$ \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} $$ It is well known that if $f$ is ...
15
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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 $...
15
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2answers
2k views

Why do we need semi-norms on Sobolev-spaces?

I have been studying Sobolev spaces and easy PDEs on those spaces for a while now and keep wondering about the norms on these spaces. We obviously have the usual norm $\|\cdot\|_{W^{k,p}}$, but some ...
15
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2answers
811 views

Smoothing a Sobolev function

Let $u \in H^1({\mathbb R}^n)$, $n \geq 2$. Let $\varphi \in C^\infty_0({\mathbb R}^n)$ with $\varphi \geq 0$. Let $\eta$ be a smoothing kernel with $\eta \in C^\infty_0({\mathbb R}^n)$, $\eta \geq ...
14
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1answer
601 views

Intuition behind losing half a derivative via the trace operator

This is an informal question, but here goes: For a function $f \in H^s(\Omega)$ ($s > 1/2$), there is a well-defined operator (the trace) $T$ such that $Tf = f\vert_{\partial \Omega}$ if $f \in C^\...
14
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2answers
7k views

What does “test function” mean?

I am trying to learn weak derivatives. In that, we call $\mathbb{C}^{\infty}_{c}$ functions as test functions and we use these functions in weak derivatives. I want to understand why these are called ...
14
votes
2answers
676 views

Elliptic regularity in Sobolev spaces of negative order

I am having some trouble with Sobolev spaces of negative order. More precisely I am considering the space $W^{-1,p}(\mathbb{R}^2),$ considered as 'the' dual space of $W^{1,q}(\mathbb{R}^2).$ Question ...
13
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1answer
2k views

Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
13
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2answers
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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 $...
13
votes
1answer
254 views

Pointwise estimate for a sequence of mollified functions

In the answer to Characterisation of one-dimensional Sobolev space Tomás wrote ... let $\eta_\delta$ be the standard mollifier sequence. Let $u_\delta=\eta_\delta\star u$ and note that for any $c\...
12
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2answers
1k views

Sobolev meets Wiener

Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point. Does it at least have weak derivatives?
12
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4answers
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dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...
12
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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 ...
12
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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 ...
12
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1answer
177 views

Are Sobolev spaces $W^{k,1}(\mathbb R^d)$ and $H^{k,1}(\mathbb R^d)$ the same?

We consider the following spaces $H^{k,p}(\mathbb R^d)$, $k \geq 1$ is integer, $p \geq 1$ (Bessel potential spaces): $$ H^{k,p}(\mathbb R^d) = \bigl\{ f \in L^p(\mathbb R^d) \colon \mathcal F^{-1}[...
12
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1answer
428 views

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
11
votes
2answers
2k views

Some basics of Sobolev spaces

Let $W^{m,p}(\Omega) = \{ f \in L^p(\Omega): D^\alpha f \in L^p(\Omega) \text{ for multi-indices } |\alpha| \leq m\}$, where $D$ denotes the weak derivative. Let $W_0^{m,p}$ denote the closure of $C_c^...
11
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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 $L^{p'}$-...
11
votes
2answers
1k views

$C^\infty_{0}(\bar\Omega)$ dense in $W^{k,p}(\Omega)$: the closure is necessary?

It is a fundamental result of Sobolve space that Let $\Omega \subset \mathbb{R}^d$ or $\mathbb{R}^d_+$, then $C^\infty_{0}(\bar\Omega)$ is dense in $W^{k,p}(\Omega)$. However, in some ...
11
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1answer
4k views

Elliptic Regularity Theorem

I want to collect some results on elliptic regularity. The problem I consider is \begin{align} Lu&=f,&in \quad U,\\ u&=g,&on \quad \partial U.\tag{1} ...
11
votes
1answer
2k views

Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
11
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2answers
1k views

Extension and trace operators for Sobolev spaces

Given that $\Omega \subset \mathbb{R}^{n}$ is an open, convex, Lipschitz bounded set. Let $O \subset \Omega$ be open bounded set then consider. $$u_{m} \rightharpoonup^{*} u \text{ in } W^{1,\infty}_{...
11
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1answer
496 views

Short and elegant introduction to Sobolev spaces

I am preparing a course on Nonlinear Analysis, and I need to teach the most important facts about Sobolev spaces to my students. I know most books on this subject, from Brezis' to Adams', from Mazya's ...
11
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1answer
1k views

Hille Yosida theorem application

Disclaimer: pretty long and specific (contraction semi groups involved). I have fourth order parabolic equation $$ u_t + \Delta^2 u = 0 $$ on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
11
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0answers
329 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function. A sufficient condition for the integral $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^...
11
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0answers
212 views

Question about boundedness of a sequence in $ W^{3,q} $ for any $ 1\leq q < \frac{N}{N-1} $

I have asked this question several months ago, I have understood every thing and there are good comments and they have helped me , but only I have a question about tomas comment, how can Calderón-...
11
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1answer
281 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
10
votes
2answers
3k views

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{...
10
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4answers
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Product rule of weak derivatives

I am working on proving the following proposition: If $u,v\in {W^1(\Omega)}$ and $uv,uDv+vDu\in L^1_{\operatorname{loc}}(\Omega)$, then we have the product formula $$D(uv)=uDv+vDu.$$ The definition I ...
10
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1answer
3k 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<\...
10
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1answer
585 views

Why are weak solutions to PDEs good enough?

Obviously solutions in $C^k$ are nicer but it seems people are happy with obtaining weak solutions in some Sobolev space that only satisfy the weak formulation. Why should this, in the real world, be ...
10
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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{...
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2answers
3k views

Sobolev spaces fourier norm equivalence

I am reading about Sobolev spaces and I have a question regarding Sobolev spaces and the Fourier transform. So by defining the Fourier transform, $F(\cdot)$ as an isometry we get $||f||_{L^2}=||\...
10
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2answers
214 views

$W^{s,p}(\mathbb{R}^{n})$ Is Not Closed Under Multiplication when $s\leq n/p$

For $s\in\mathbb{R}$, $1<p<\infty$, and $n\geq 1$, define the Sobolev space $W^{s,p}(\mathbb{R}^{n})$ by $$W^{s,p}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}(\mathbb{R}^{n}) : \|(\langle{\xi}\...
10
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1answer
149 views

Proof Nehari manifold of semilineal subcritical $-\Delta u = f(u)$ in $\Omega$ is not empty.

Given the problem $$ \left\{ \begin{array}{rll} -\Delta u& = f(u) & \text{in }\Omega \\ u & = 0 & \text{in } \partial\Omega \end{array} \right. $$ In a bounded domain $\Omega\subset ...
10
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1answer
596 views

Understanding a theorem concerning Sobolev spaces

I have two doubts in the proof of the theorem below. If you want the detaIls can be found here. Theorem A Let $q$ a given real number $q>p$. Let $u \in W^{1,p}$ be a solution to (E2). Assume ...
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0answers
244 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
9
votes
3answers
5k views

$C^\infty$ dense in Sobolev spaces

I'm reading a theorem which says that $C^\infty$ intersection with $W^{k,p}$ is dense in $W^{k,p}$. I don't understand why they take the intersection. Isn't it $C^\infty$ a subspace of $W^{k,p}$? ...
9
votes
3answers
716 views

For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...
9
votes
1answer
4k views

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\...
9
votes
2answers
2k views

Dual space of $H^1(\Omega)$

I'm a bit confused, why do people not define $H^1(\Omega)^*$? Instead they only say that $H^{-1}(\Omega)$ is the dual of $H^1_0(\Omega).$ $H^1(\Omega)$ is a Hilbert space so it has a well-defined ...
9
votes
1answer
591 views

Is there a Rellich-Kondrachov theorem for manifolds with boundary?

As special case, consider the cylinder $C=[0,T]\times S^n$. Is there a compact embedding $H^1(C)\subset\subset L^2(C)$? The Wikipedia entry to the Rellich-Kondrachov theorem claims that such an ...
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. ...