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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15
votes
2answers
563 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 ...
12
votes
1answer
351 views

$\tilde{u}=0,\ a.e.\ x\in\Gamma$?

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 ...
11
votes
2answers
387 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 ...
11
votes
1answer
489 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 ...
10
votes
3answers
332 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 ...
10
votes
2answers
716 views

Sobolev space is an algebra

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 ...
10
votes
1answer
344 views

Why no trace operator in $L^2$?

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 ...
10
votes
2answers
188 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 } ...
10
votes
1answer
105 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 ...
10
votes
1answer
487 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 ...
9
votes
1answer
385 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 $\Omega = \mathbb{R}^d$ or $\mathbb{R}^d_+$, then $C^\infty_{0}(\bar\Omega)$ is dense in $W^{k,p}(\Omega)$. However, in some literatures, ...
9
votes
2answers
561 views

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 $v \in H^1(U)$ $u \rightarrow (u,v)_{H^1}$ is an element of $H^{-1}$. Moreover for every $v \in L^2(U)$ $u ...
9
votes
1answer
147 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 ...
9
votes
1answer
259 views

Regularity of elliptic PDE with coefficients in some Sobolev space

Is there a regularity theory for elliptic equations "optimized" for coefficients in a Sobolev space $W^{k,r}$? By this, I mean a result (and the according elliptic estimates) that gives you (for $k$ ...
8
votes
2answers
517 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?
8
votes
1answer
144 views

Is $W_0^{1,p}(\Omega)$ complemented in $W^{1,p}(\Omega)$?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $p\in (1,\infty)$. It is know that there exist a unique bounded surjective linear map $T: W^{1,p}(\Omega)\to W^{1-1/p,p}(\partial\Omega)$ with ...
8
votes
1answer
205 views

General formula needed for this product rule expression (differential operator)

Let $D_i^t$, $D_i^0$ for $i=1,\dots,n$ be differential operators. (For example $D_1^t = D_x^t$, $D_2^t = D_y^t,\dots$, where $x$, $y$ are the coordinates). Suppose I am given the identity $${D}_a^t ...
8
votes
1answer
134 views

Calculus on the Sobolev space valued function of one real variable $t$?

Now I am interested in the calculus on Banach space valued function, especially the function with value in a certain Sobolev space. I want to prove that $$\bigcap_{k=0}^m ...
7
votes
1answer
967 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?
7
votes
2answers
872 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 ...
7
votes
2answers
182 views

Definition of weak solutions from geometrical point of view

Why are weak solutions defined like: A function $u \in H^1(\Omega)$ is a weak solution of $$ Lu:=\operatorname{div}(A\nabla u)+b\cdot\nabla u+cu=f+\operatorname{div}F, \;\text{in } \Omega $$ if ...
7
votes
1answer
843 views

Elliptic Regularity Theorem

I want to collect some results on elliptic regularity. The problem I consider is \begin{aligned} Lu&=f,&in\,\,\,U,\\ u&=g,&on\,\,\, \partial U.\tag{1} ...
7
votes
1answer
227 views

Function in $H^1(\Omega)$ which cannot be extended to a greater Sobolev Space

The problem is like this: Consider the open set $\Omega \in \Bbb{R}^2$ by $\Omega=\{(x,y) : 0<x<1, 0<y<x^2 \}$ Is $\Omega$ with Lipschitz boundary? (i.e. the boundary is ...
7
votes
1answer
215 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 ...
7
votes
2answers
175 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
7
votes
1answer
129 views

The constant in the Sobolev trace theorem inequality

The trace theorem for nice enough domains states that there is a operator $T:H^1(\Omega) \to L^2(\partial \Omega)$ such that $$|Tu|_{L^2(\partial \Omega)} \leq C|u|_{H^1}.$$ My question, is there an ...
7
votes
1answer
208 views

Help me understand this proof (showing that something is a norm).

I am reading the following paper: Takáč, Peter On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51 (2002), no. 1, 187–237. I need help to understand the ...
7
votes
1answer
267 views

Do the two limits coincide?

Let $a$ be a non negative (positive almost everywhere) weight in $L_{loc}^1(\Omega)$, $\Omega\subseteq\mathbb{R}^n$ is open. For $\varphi\in C_c^{\infty}(\Omega)$ define $$ ...
6
votes
4answers
339 views

Survey papers for PDE?

I want to know if there is a good website which allows you to download survey papers on PDEs? The "survey" should include a summary of methods, skills, developments etc. I wish to get some basic (or ...
6
votes
3answers
83 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 ...
6
votes
2answers
286 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 ...
6
votes
1answer
60 views

Is a Sobolev function absolutely continuous with respect to a.e.segment of line?

Let $p\in [1,\infty]$ and take $u\in W^{1,p}(\mathbb{R}^N)$. It is a well know result that $u$ is absolutely continuous (A.C) on a.e. segment of line parallel to the coordinate axes. It seems to me ...
6
votes
1answer
220 views

Surjectivity of the trace operator in Sobolev spaces

Suppose $U$ is an open bounded set with $C^1$ boundary. It is a well-known result in the theory of Sobolev spaces $W^{1,p}$ that there is a continuous linear operator $T:W^{1,p}(U)\rightarrow ...
6
votes
2answers
70 views

Decay of $H^1(\mathbb{R}^n)$ functions

Is it true (is there a commonly known theorem) that says: $f \in H^1(\mathbb R ^n)$ $\Rightarrow$ $\displaystyle \lim_{|x| \to \infty} f(x) = 0$ pointwise (where $H^1$ denotes the Sobolev space ...
6
votes
1answer
700 views

Schwartz Space is a subspace of Sobolev Space, but how can I show that?

How can I see that $S(\mathbb{R}) \subset H^s(\mathbb{R})$, where the former is Schwartz and the latter is Sobolev space ? This should be obvious according to my notes but unfortunately I can't make ...
6
votes
2answers
529 views

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 ...
6
votes
1answer
871 views

Why is the dual space of $H_0^1(\Omega)$ denoted $H^{-1}(\Omega)$?

Why is the dual of the Sobolev space $H_0^1(\Omega)$ denoted $H^{-1}(\Omega)$ ? For a positive integer $k$, $H^k(\Omega)=W^{k,2}(\Omega)$. What is the motivation behind the $-1$ exponent?
6
votes
1answer
83 views

Proof of equivalence of Sobolev Space and Lipschitz functions

The attachment is a proof from Evans book "Measure Theory and Fine Properties of Functions" pg 132 Theorem 5. The statement of the theorem is: Let $f:U \rightarrow \mathbb{R}$. Then $f$ is locally ...
6
votes
1answer
134 views

Caccioppoli-Leray Inequality for De Giorgi's regularity theorem

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
6
votes
1answer
208 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 ...
6
votes
2answers
231 views

Trace regularity result $\lVert n \times u\rVert_{H^{-1/2}}$

There is a result in a paper I am reading : Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that $$\lVert n \times u\rVert_{H^{-1/2}(\partial ...
6
votes
1answer
45 views

Prove that energy functional cannot be minimized

As an exercise I want to show that for $$E(v)=\int_0^1 \frac 1 2 x^5 (v'(x))^2 - v(x) dx $$ and $H_0^1(0,1) = \{ v \in H^1(0,1) : v(0)=v(1)=0 \}$ there exists no solution to the problem ...
6
votes
1answer
189 views

Natural question about weak convergence.

Let $u_k, u \in H^{1}(\Omega)$ such that $u_k \rightharpoonup u$ (weak convergence) in $H^{1}(\Omega)$. Is true that $u_{k}^{+}\rightharpoonup u^{+}$ in $\{u\geqslant 0\}$? You can do hypothesis on ...
6
votes
0answers
93 views

Proposed proof of analysis result

Hi please advise on my proof of the following result: Assume that $I \subset \mathbb{R}^{n}$ is convex, bounded open set with Lipschitz boundary and let $u_{m},u$ be such that $$u_{m} ...
6
votes
0answers
98 views

Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to ...
5
votes
2answers
67 views

Examples where $1 \in W_0^{k,p}\left( U \right)$

$M$ is a Riemannian manifold, $U$ is a domain in $M$. Consider the Sobolev space $W_0^{k,p}\left( U \right)$: the closure of $C_0^\infty \left( U \right)$ (smooth functions with compact support) in ...
5
votes
1answer
158 views

Spectrum of 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 $\sigma(\Delta)=]-\infty,0]$?
5
votes
2answers
133 views

PDE uniqueness by energy method contradicts non-uniqueness???

Consider $$u_t - \Delta u + u = 0$$ $$\frac{\partial u}{\partial \nu} = 0$$ $$u(T) = u(0)$$ on a domain $\Omega$ (the BC is obviously on $\partial\Omega$. If $u$ solves this PDE, clearly as does ...
5
votes
2answers
135 views

Sobolev spaces - about smooth aproximation

Consider $\Omega $ a open and bounded set of $R^n$ . Let $u \in H^{1}(\Omega)$ a bounded function. I know that exists a sequence $u_m \in C^{\infty} (\Omega) $ where $u_m \rightarrow u$ in ...
5
votes
1answer
201 views

Why is $H^1 \neq H_0^1$?

I've been doing some reading on Sobolev-spaces and one remark said that $H_0^1$, i.e. the space of $H^1$-functions with zero-boundary values, is not the same as $H^1$. This seems clear to me, but when ...