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

learn more… | top users | synonyms

2
votes
1answer
406 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 ...
11
votes
2answers
973 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 ...
4
votes
1answer
675 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 ...
5
votes
2answers
150 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 ...
2
votes
1answer
141 views

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 $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
3
votes
1answer
750 views

Acting of a dual pairing in Sobolev Spaces

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 $H^1_0$ ...
3
votes
2answers
135 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 ...
3
votes
1answer
434 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 ...
2
votes
1answer
103 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 ...
1
vote
2answers
100 views

Discontinuous Sobolev Function

I'm trying to show that there's an $f \in H^1(\mathbb{R}^2)$ which is not ae equal to a continuous function. Per a couple of suggestions, I've decided to look at the function $f(x) = ...
5
votes
1answer
321 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
2answers
218 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 ...
10
votes
3answers
377 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 ...
1
vote
1answer
374 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 ...
10
votes
1answer
501 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 ...
4
votes
1answer
109 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 ...
3
votes
1answer
87 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 ...
3
votes
1answer
188 views

Gauss–Ostrogradsky formula for Distributions

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
2
votes
1answer
250 views

Sobolev inequality

If $f\in H^2(\mathbb R^2)$, I want to show that $||f||_{L^\infty}\le c||f||_{H^2}$ $||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$ For 1, I use $||f||_{L^\infty}\le \sup_{x\in \mathbb ...
2
votes
3answers
959 views

Help with Evans PDE problem

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 Poincare inequality but I still cannot ...
1
vote
3answers
683 views

Weak convergence and strong convergence

Suppose $f_i$ is uniformly bounded in $W^{1,p}$ for some $+\infty>p>1$, then by passing to a sub-sequence, we can suppose $f_i$ is weakly convergent to $f$ in $W^{1,p}$. Assume furthermore that ...
0
votes
1answer
365 views

sobolev spaces - product of two functions

I am working in a exercise, to my solution works I need the following affirmation is true: Let $\varphi : R \rightarrow R$ a convex and smooth function. Let $u \in H^{1}(U)$ a bounded function and $v ...
7
votes
3answers
184 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 ...
5
votes
2answers
195 views

How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms?

How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms on a bounded domain? I hear there is a way to do it by RRT but any other way ...
4
votes
2answers
225 views

How to show that $p-$Laplacian operator is monotone?

Define $$\langle -\Delta_p u, v \rangle_{(W^{1,p})', W^{1,p}} = \int_{\Omega}|\nabla u |^{p-2}\nabla u \nabla v.$$ How do I show that this operator is monotone? I get $$\langle-\Delta_p u_1 - ...
3
votes
1answer
83 views

Is $C^\infty_0$ dense in $C^\infty$ w.r.t. $\|\cdot\|_{L^p}$ and $\|\cdot\|_{W^{1,p}}$?

Is the space $C^\infty_0(\Omega)$ of smooth functions with compact support, dense in the set of smooth functions $C^\infty(\Omega)$ with respect to the norms $\|\cdot\|_{L^p}$ and the Sobolev-Norm ...
3
votes
2answers
169 views

Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
3
votes
1answer
128 views

variational question

Let $\Omega$ a bounded domain, connexe and regular, and let $f \in L^2(\Omega).$ Let the variational problem: Find $u \in H^1(\Omega)$ such $$\int_{\Omega} \nabla u \nabla v dx + (\int_{\Omega} u ...
3
votes
1answer
748 views

Poincare Inequality

In page 290 of this book, Evans prove the Poincare inequality (Theorem 1) arguing by contradiction. Is there a direct proof of this theorem (Theorem 1) without arguing by contradiction?
2
votes
1answer
27 views

Trace on $H^1(\Omega_1\cup\Omega_2)$ (one little question in the conclusion of my proof)

Let $\Sigma$ a smooth surface that separates $\Omega_1$ and $\Omega_2$ (open and bounded sets) and let $(q^n_1)_{n\in\mathbb{N}}$ and $(q^n_2)_{n\in\mathbb{N}}$ sequences in $H^1(\Omega_1)$ and ...
1
vote
1answer
38 views

Are all functions in the Sobolev space $W_0^{1,2}(\Omega)$ continuous and bounded?

Are all function in $W_0^{1,2}(\Omega)$, $\Omega$ being a bounded domain in $\mathbb{R}^n$, $n \geq 2$, continuous and bounded w.r.t. $|.|$?. In other words, given $u\in W_0^{1,2}$ can one say that ...
1
vote
2answers
213 views

Eignvalues of Laplacian operator and Sobolev spaces

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od ...
1
vote
2answers
364 views

Question about limits of weakly convergent sequence in $H^1_0(\Omega)$

Let $H = H_{0}^{1}(\Omega)$ where $\Omega$ is a bounded domain in $R^N$ whose boundary $\partial\Omega$ is a smooth manifold. We know that the embedding $$H\hookrightarrow L^s(\Omega)$$ is compact for ...
-1
votes
2answers
122 views

finite elements-exercice

We consider in $\mathbb{R}^2$ the set of points $$\{M_1(-1,1),M_2(0,1), M_3(2,1),M_4(-1,0),M_5(1,0),M_6(2,0)\}$$ Let $\Omega$ a rectangular structure consisting of the heads $\{M_4(-1,0),M_6(2,0), ...
9
votes
1answer
424 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, ...
8
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?
10
votes
1answer
435 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 ...
7
votes
1answer
142 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 $(u_n)\subset C^\infty(\Omega)$ ...
6
votes
1answer
417 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 $\sigma(\Delta)=]-\infty,0]$?
2
votes
1answer
1k views

question 9 - chap 5 evans PDE

The question is : Integrate by parts to prove : $$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$ for $ 2 \leq p < \infty$ ...
4
votes
0answers
77 views

$u\in W^{1,p}(B)$ implies $u\in W^{1,p}(\partial B_t)$ for almost $t\in (0,1]$?

Suppose that $u\in W^{1,p}(B)$ where $B=B(0,1)\subset\mathbb{R}^N$ and $p\geq 1$. It was showed on this post (as an application of Fubini's theorem) that for almost $t\in (0,1]$ $$u_{|\partial ...
2
votes
1answer
682 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 $u\in W^{1,p}(0,1)$ is equal a.e. to an absolutely continuous function and that $u'$ exists a.e. and ...
13
votes
1answer
374 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 ...
8
votes
2answers
556 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?
7
votes
1answer
259 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 ...
6
votes
0answers
115 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} ...
4
votes
2answers
81 views

Bounded data means bounded solution to parabolic PDE

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$ Consider $$u_t - \Delta u = f$$ $$u|_{\partial\Omega} = 0$$ $$u(0) = u_0$$ or more generally replace $\Delta$ with a suitable ...
4
votes
2answers
293 views

The dual space of the Sobolev space $H_0^1$

I am slightly confused about the properties of the dual space of the Sobolev space $H_0^1$ as outlined on page 299 in Evans. In particular, following the notation in the book, item 3 says that ...
4
votes
1answer
164 views

What can be said about the eigenvalues of the Laplace operator in $H^k(\mathbb{T}^2)$

Consider the Laplace operator $$\Delta: H^{k+2}(\mathbb{T}^2) \to H^k(\mathbb{T}^2)$$ where $\mathbb{T^2}$ is the two-dimensional torus (which is a compact manifold without boundary), so that $$ ...
4
votes
1answer
161 views

Need help understanding this proof of regularity of traveling wave solutions to the Gross-Pitaevskii equation

These are actually 4 question about a proof given in this paper. Any hint to solutions for any of these questions would be much appreciated! Lemma 1. Assume $v$ is a solution to the equation ...