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

3
votes
0answers
46 views

Rellich-Kondrachov compacteness theorem for the Euclidean space with Gaussian measure

Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$ \gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}. $$ Let $d\gamma_n$ denote the following measure ...
3
votes
0answers
35 views

Equivalent descriptions of Sobolev spaces on compact manifolds

While reading through a set of lectures on the Laplacian on manifolds, I encountered two descriptions of Sobolev spaces. The first one, valid only for compact manifolds (because it needs to globalize ...
3
votes
0answers
44 views

How do I show that a distribution is a function?

While reading Grigor'yan's book on the heat kernel I have encountered the following definition of a Sobolev space on a Riemannian manifold $M$: $W^2 (M) = \{ u \in W^1 (M) : \Delta u \in L^2 (M) \}...
3
votes
0answers
48 views

Sobolev spaces and symmetric operators

I am slightly confused with regards to the way one obtains self - adjoint differential operators in spectral theory. The aspect that I'd like to understand better is the following: Suppose we are ...
3
votes
0answers
86 views

Is the space $\mathcal{H}=\{v\in H^1(\Omega):\Delta v\in H^1(\Omega)^*\}$ a Banach space?

Let $\Omega$ be a Lipschitz domain in $\Bbb R^n$, is the space $$\mathcal{H}=\{v\in H^1(\Omega):\Delta v\in H^1(\Omega)^*\}$$ a Hilbert space when endowed with the norm $\|\cdot\|_\mathcal{H} = \|u\|_{...
3
votes
0answers
85 views

Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
3
votes
0answers
51 views

The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E $, where $\mathcal E$ denotes the ...
3
votes
0answers
75 views

How can we show that $u$ as a weak solution has properties $u \in L^{\infty}(\Omega)$ , $ u>0 $

Let $\Omega$ be an open domain in $\mathbb {R^n}$ and $f \in C^{\infty}(\Omega)$ then how can we prove there is a weak solution $u \in ‎‎ W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \cap L^{\infty}(\...
3
votes
0answers
37 views

Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles.

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
3
votes
0answers
37 views

A linear functional of $H^k$

Define $H^k$ as usual: $$H^k=\left\{f; \int (1+|\xi|^2)^k|\widehat{f}(\xi)|^2d\xi<\infty\right\}$$ Define $l(u)=\int (1+|\xi|^2)^k\widehat{u}(\xi)d\xi$, how to show that $l\in (H^k)^*$. I tried ...
3
votes
0answers
147 views

Complicated convergence of nonlinear term

Let $1<p<\infty$, $\Omega\subset\mathbb{R}^m$ be open, bounded with $\partial\Omega\in C^1$. Assume that $u_k\to u$ weakly in $W^{1,p}(\Omega;\mathbb{R}^n)$. We know that $u_k\to u$ strongly in $...
3
votes
0answers
196 views

Prove difference quotient converges to weak derivative in $L^p$

I am trying to solve the following exercise: Let $U$ be an open set in $\mathbb{R}^n$ and let $V$ be a compact subset with Lipschitz boundary. Assume that $f$ is in $L^p(U)$ with $1<p<\infty$ ...
3
votes
0answers
79 views

Regularity of weak solution for $u_t - \Delta u = f$ with $u(0) = u_0 \in L^1(\Omega)$

Let $\Omega$ be a bounded domain, and consider the equation $$u_t - \Delta u = f$$ $$u(0) = u_0 \in L^1(\Omega)$$ with Neumann BCs (or Dirichlet if convenient) where $f$ is smooth. Using energy/...
3
votes
0answers
68 views

if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
3
votes
0answers
93 views

What is a good way to prove the weak form of the Sobolev embedding theorem?

From Richard Melrose, From microlocal to global analysis, Chapter 1, Problem 11. Suppose $u\in L^{2}(\mathbb{R}^{n})$ and that $$ D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) $$ ...
3
votes
0answers
101 views

Sobolev estimation of second derivative against Laplacian and higher terms

Given $u \in H^2(\Omega)$ (and $\Omega \subseteq \mathbb{R}^n$ with appropriate properties) is there a way to estimate the norm of the second derivative $\Vert D^2 u\Vert_{L^2(\Omega)}^2$ against the ...
3
votes
0answers
47 views

Proposed proof of continuous operator on Sobolev space

Hi I am interested in a question about continuity: Assume that $\Omega \subset \mathbb{R}^{n}$ is bounded and consider operator $$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow L^{q}...
3
votes
0answers
145 views

Function with divergence, curl and normal trace on boundary equals zero is zero

Let $u\in H^1(\Omega)$ with $\nabla\times u=0$ in $\Omega\subset\mathbb{R}^3$ (open bounded domain), $u\times n=0$ on $\partial\Omega$ (where $n$ is a a normal vector to $\partial\Omega$), $\...
3
votes
0answers
176 views

Weak subsolution and composition with convex smooth function

I'm trying to solve a problem in Partial Differential Equations by Lawrence C. Evans. It goes like this Assume $u\in H^1(U)$ is a bounded weak solution of \begin{equation}-\sum_{i,j=1}^n (a^{ij}u_{...
3
votes
0answers
131 views

Proving continuity on Sobolev space with weak topology

Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to ...
3
votes
0answers
24 views

Inverse continuous

According Dehman, Gérard and Lebeau in "Stabilization and control for the nonlinear Schrödinger equation on a compact surface" (Math. Z. (2006) 254:729–749, DOI 10.1007/s00209-006-0005-3) is claimed ...
3
votes
0answers
28 views

$\left\|f\right\|_{L^{\infty}(\mathbb R^d)} \leq K \left\|f\right\|_{H^{s}(\mathbb R^d)}$

The Swchartz, $\mathcal S(\mathbb R^d)=\left\{f\in C^{\infty}(\mathbb r^d): \sup_{x\in \mathbb R^d}(1+|x|^{2})^{\frac{k}{2}}\sum_{|\alpha\leq l|}|D^{\alpha}f(x)|< \infty\right\}$, for all $k, l \in ...
3
votes
0answers
50 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
3
votes
0answers
155 views

What domains of $\mathbb{R}^n$ have the property that $H^1(\Omega)=H^1_0(\Omega)$?

i wonder what are sufficient conditions on an unbounded domain of $R^n$ called $\Omega$ to get : $C_c^\infty (\Omega)$ dense in $H^1 (\Omega)$ ? where $C_c^\infty$ stands for the set of functions with ...
3
votes
0answers
150 views

Regularity of a Weak Solution to Fokker-Planck Equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t =...
3
votes
0answers
50 views

Missing step in Evan's PDE?

In the attached proof, I am having trouble justifying the presence of "$x_n^{p-1}$" in eh line immediately below "Thus" and "Letting $m\to\infty$". I understand he uses Minkowski's inequality, and ...
3
votes
0answers
62 views

Problem of convergence of characteristic functions

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
3
votes
0answers
59 views

estimations of solutions

Let $\Omega=\mathbb{R}^2_+=\{(x,y) \in \mathbb{R}^2; y > 0\}$, $f \in L^2(\Omega)$, $\lambda \in \mathbb{R}^*_+$, $A=(a_{ij})_{1\leq i,j \leq 2}$, $a_{ij} \in \mathbb{R}$ and there exist $\alpha &...
3
votes
0answers
78 views

Sobolev Spaces and Convergence

I have a question about one of my homework question. I have been struggling for a while and I really need some help. Assume $N>2$ and $u_k$ is a bounded sequence in $W^{1,2}(\mathbb{R}^N)$ ...
3
votes
0answers
111 views

A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times \...
3
votes
0answers
65 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, |f|f,...
3
votes
0answers
103 views

In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
3
votes
0answers
129 views

$W^2_p$ regularity of solutions of linear elasticity

I want to prove the following statement: Let $\Omega$ be bounded, polygonal, and convex, then for the solution of the linear elasticity (elliptic) equation with inhomogenous Dirichlet boundary ...
3
votes
0answers
156 views

Generalized chainrule for Sobolev functions with a cut-off

let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $f\in C^1(\bar\Omega \times (0,g(x)),[0,1])$ and $f(x,\cdot)$ increasing and $g(x)\in\mathbb{R}$ continuous (maybe better, Lipschitz?). I want ...
3
votes
0answers
98 views

Trace spaces of Orlicz-Sobolev spaces.

Recently I have the need of study trace spaces of Orlicz-Sobolev spaces. By looking in google I have discovered in this PDF page 14 (not only in the PDF), that the main contributions come from the ...
3
votes
0answers
83 views

A “straightforward” inequality.

In section 5.5 page page 117 of Fanghua, Quing's book, in order to obtain $W^{2,p}$ estimates the idea is to show that for $p \in (1,\infty]$ the condition $\theta \in L^p(\Omega)$ implies $D^2u \in L^...
3
votes
0answers
158 views

Checking initial condition of PDE is satisfied in Galerkin method

The following is an argument I read, it is part of a proof of checking that the solution given by the Galerkin method satisfies the initial conditions. From our PDE $$\langle u', v \rangle_{V',V} + a(...
3
votes
0answers
47 views

Estimation of $\|(1-\Delta)^{\gamma/2}f\|_1$ type

Let $\phi(\xi)\in C_c^\infty(\mathbb{R}^d)$ with value $1$ in a neighborhood of the unit ball and vanishing fast outside the ball. I want to estimate $$\|(1-\Delta)^d (\cdot)^\alpha \partial^\alpha\...
3
votes
0answers
319 views

Weak solution to PDE boundary value problems

Take $\lambda>0$ and let $\Delta$ be the Laplacian operator in dimension $n$ and let $\Delta^2:=\Delta\circ\Delta$. Consider the boundary value problem $\Delta^2 u+\lambda u=f$ on some open subset $...
3
votes
0answers
143 views

A finely open set, not open up to polar set?

Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, which is not open up to a polar set (i.e. zero capacity), i.e., there does ...
3
votes
0answers
138 views

invertible operator Sobolev space

Let $T:H_0^2(D)\rightarrow H_0^2(D)$ a linear bounded operator defined via the sesqulinear form $(Tu,v)=a(u,v):=\int_{D}\Delta u\Delta v dx$. I have shown that T is coercive and self-adjoint. How can ...
3
votes
0answers
300 views

When the weak derivative just is the strong (or classical) derivative?

When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).
3
votes
0answers
188 views

Is the functional $F:H^{1}_{0}(\Omega) \longrightarrow \mathbb{R}$ given below weak lower semicontinuous?

Is the functional $F:H^{1}_{0}(\Omega) \longrightarrow \mathbb{R}$ given by \begin{equation} F(u) = \int_{\Omega} \langle (A_1(x)\chi_{\{u>0\}}+A_2(x)\chi_{\{u\le0\}}) \nabla u, \nabla u \rangle \...
3
votes
0answers
111 views

Truncation in singular integrals

After some thinking, I have a terrible headache caused by the following problem. Imagine we have a function $u \colon \mathbb{R}^n \to \mathbb{R}$ such that $u \in L^2(\mathbb{R}^n)$ and$$\int_{\...
3
votes
0answers
335 views

Why is this functional coercive?

Let $h\in L^2(0,1)$, such that $|\int h \mathrm{d}x|<1$. On the space $H^1(0,1)$, consider $$J(v)=\frac{1}{2}\int_0^1 v'^2\,\mathrm{d}x+\int_0^1\sqrt{1+v^2}\,\mathrm{d}x-\int_0^1hv \,\mathrm{d}x.$$ ...
2
votes
0answers
37 views

The eigenfunction of modified 1-laplace equation

Let $\Omega\subset \mathbb R^2$ be a bounded, smooth boundary domain. I am interested in the following operator $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0. $$ ...
2
votes
0answers
13 views

Different definitions of Besov norm/space

I'm following two "different" approaches to the Besov Spaces, but I don't get if the two definitions given are equivalent. Victor I. Burenkov - Sobolev Spaces On Domains. Given $f:\mathbb{R}^n \...
2
votes
0answers
14 views

Fourier coefficients of $\;\log\log$

I was curious if there is an effective way to compute (the asymptotic of) the Fourier coefficients of $$ F(x)= \log\log\left(\frac{1}{\left\lvert x\right\rvert}\right) \cdot \chi\left(\left\lvert x\...
2
votes
0answers
21 views

Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality: $||...
2
votes
0answers
40 views

$W_0^{1,\:p}(\Lambda)$ is dense in $L^2(\Lambda)$

Let $d\in\mathbb N$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Lambda\subseteq\mathbb R^d$ be open with $\lambda(\Lambda)<\infty$ $p\ge 2$ $W^1(\Lambda)$ denote the set of weakly ...