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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16
votes
2answers
184 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 ...
15
votes
2answers
596 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
votes
1answer
391 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 ...
13
votes
2answers
1k 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 ...
13
votes
1answer
177 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 ...
12
votes
3answers
453 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 ...
12
votes
2answers
444 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
680 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
1answer
566 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
404 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
509 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
1k 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?
9
votes
1answer
471 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
1answer
1k 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
205 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
360 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
611 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
2answers
1k 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 ...
8
votes
1answer
161 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
224 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
2answers
855 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 ...
8
votes
1answer
280 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 ...
8
votes
1answer
204 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 ...
8
votes
1answer
144 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 ...
8
votes
0answers
172 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 ...
8
votes
0answers
125 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 ...
7
votes
1answer
815 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 $\sigma(\Delta)=]-\infty,0]$?
7
votes
3answers
234 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 ...
7
votes
2answers
199 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
1k 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
276 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
2answers
86 views

Why is Existence and Uniqueness for Navier-Stokes Easier in 2-D than in 3-D?

I know that existence and uniqueness for incompressible viscous flow in the 2-D case has already been established$^1$, and that doing the same for the 3-D case has yet to be shown. Not only that, but ...
7
votes
2answers
236 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
266 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 ...
7
votes
1answer
174 views

Use $C^\infty$ function to approximate $W^{1,\infty}$ function in finite domain

This is exercise 10.21 from Leoni's book. The exercise asks me to prove that for any $u\in W^{1,\infty}(\Omega)$ where $\Omega$ is open FINITE, there exists a sequence $(u_n)\subset C^\infty(\Omega)$ ...
7
votes
1answer
278 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
272 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
418 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
2answers
419 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
881 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 ...
6
votes
1answer
355 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 ...
6
votes
2answers
393 views

An inequality of J. Necas

The above inequality belongs to J. Necas. Sur les normes equivalentes dans $W^k_p$ et sur la coercivite des formellement positives. Sem. math. sup. Universite Montreal, 102-128 (1966). But I can't ...
6
votes
2answers
184 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 ...
6
votes
1answer
137 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
379 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
87 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
312 views

Is the Sobolev embedding $W^{l,2}(\mathbb{R}^d) \rightarrow C_0(\mathbb{R}^d)$ compact?

In p. 508 of the paper : http://www.jstor.org/stable/2243484 , it is mentioned that if $2l \geq d$, the embedding $W^{l,2}(\mathbb{R}^d) \rightarrow C_0(\mathbb{R}^d)$ is compact, where ...
6
votes
1answer
847 views

Rellich–Kondrachov theorem for traces

Let $W^{1,p}(\Omega)$ be the Sobolev space of weakly differentiable functions whose weak derivatives are $p$-integrable, where $\Omega \subset \mathbb R^n$ is a domain with Lipschitz boundary. Let ...
6
votes
1answer
713 views

An inequality in Evans' PDE

In Section $9.2$ Theorem $5$ of Lawrence Evans' Partial Differential Equations, First Edition the author proves that for a large enough $\lambda$, the equation $$\begin{array}-\Delta u+b(\nabla ...
6
votes
1answer
359 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 ...