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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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 ...
2
votes
1answer
560 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 ...
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
2answers
183 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
242 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
890 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 $H^1_0$ ...
4
votes
1answer
125 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
2answers
147 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
502 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
158 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 ...
2
votes
1answer
185 views

A technical step in proving Hardy's inequality

A technical step in proving Hardy's inequality $$ \int_{B(0,r)}\frac{\mu^{2}}{|x|^{2}}dx\le C\int_{B(0,r)}(|D\mu|^{2}+\frac{\mu^{2}}{r^{2}})dx $$ where $n>3, r>0, \mu\in H^{1}(B(0,r))$ is to ...
1
vote
2answers
114 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) = ...
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 ...
5
votes
2answers
306 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 - ...
2
votes
2answers
226 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 ...
7
votes
1answer
810 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
2k 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$ ...
12
votes
3answers
447 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 ...
2
votes
1answer
268 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
1answer
880 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 ...
1
vote
1answer
414 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
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 ...
3
votes
1answer
148 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
218 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 ...
3
votes
4answers
1k 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
1answer
911 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 ...
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 ...
1
vote
3answers
819 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
238 views

If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

Provide details for the following alternative proof that if $u \in H^1(U)$, then $$Du = 0 \text{ a.e. } \, \text{ on the set } \{u=0\}. $$ Let $\phi$ be a smooth, bounded and nondecreasing ...
0
votes
1answer
428 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
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 ...
5
votes
2answers
218 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
1answer
1k 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?
3
votes
1answer
96 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
190 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
129 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 ...
2
votes
1answer
30 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
65 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
227 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
1answer
98 views

$C_c^{\infty}$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
1
vote
2answers
379 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 ...
0
votes
0answers
21 views

About the compact embedding $W^{1,p}(U)\subset\subset L^p(U)$

How can I get this compact embedding $W^{1,p}(U)\subset\subset L^p(U)$ for a open, bounded and $C^1$ subset $U$ of $\mathbb R^n$? Basicly This is the Evans' Remark on pg. 289 2nd edition. But I ...
0
votes
1answer
114 views

How to prove Poincaré-like inequality for the integral over the boundary?

Suppose $u\in W^{1,1}$ and $\partial u$ is $C^1$. I want to prove the following: $$\int_{\partial\Omega}|u-\bar u|\leq A\int_{\Omega}|\nabla u|$$ where $\bar u=\frac{1}{|\Omega|}\int_{\Omega}u$ and ...
-1
votes
2answers
123 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
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

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
564 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
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 $(u_n)\subset C^\infty(\Omega)$ ...
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 ...
4
votes
0answers
81 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 ...