Questions tagged [sobolev-spaces]

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

28
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806 views

Why are functions with vanishing normal derivative dense in smooth functions?

Question Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm? Here I define $...
11
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329 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function. A sufficient condition for the integral $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^...
11
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212 views

Question about boundedness of a sequence in $ W^{3,q} $ for any $ 1\leq q < \frac{N}{N-1} $

I have asked this question several months ago, I have understood every thing and there are good comments and they have helped me , but only I have a question about tomas comment, how can Calderón-...
10
votes
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244 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
8
votes
0answers
131 views

Is the normalized derivative of a holomorphic function Sobolev?

Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$ be the closed unit disk, and let $f:B \to \mathbb{C}$ be holomorphic. More precisely, I assume that $f$ is holomorphic on the interior $\text{int}(B)$, and ...
8
votes
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397 views

Sobolev spaces on Riemannian manifold and the Laplacian

Craioveanu, Puta and Rassias define the Sobolev space $H_k(M)$ (pg. 106) on a compact Riemannian manifold $M$ as the completion of $C^\infty(M)$ with respect to the norm $$ \|f\|_{H_k(M)} = \|f \|_{H_{...
8
votes
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236 views

Half Solved: A problem on the heat operator not being elliptic with a weakened version of elliptic regularity

I should first mention this: in my studies of Sobolev spaces I have completed all the questions of chapter 9 from Folland's real analysis with the help of this site and this is my last one, which is ...
8
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190 views

Sobolev space on $M \times [0,\infty)$, $M$ compact closed manifold

Consider a manifold of the form $M \times [0,\infty)$, where $M$ is a closed compact manifold without boundary. So $M \times [0,\infty)$ is a semi-infinite cylinder. I want to know information about ...
8
votes
0answers
866 views

Showing that smoothing operators are compact

Suppose I have a bounded, linear map $T: H^1(X) \to H^1(X)$ such that $T(H^1(X)) \subset C^\infty(X)$. Is $T$ a compact operator? I'm guessing this depends on whether or not $X$ is (pre)compact, and ...
7
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61 views

Heuristic on Sobolev and BV functions

Let $f: \Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a Sobolev or BV vector field. A heuristic that I've heard frequently is the following: $f$ is almost Lipschitz on a large "good" set but ...
7
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367 views

The equivalent definition of $W_0^{1,\infty}(\Omega)$

Usually, for $1\leq p<\infty$, we define $W_0^{1,p}(\Omega)$, where $\Omega$ is open bounded smooth boundary, by taking the closure of $C_c^\infty(\Omega)$ under $W^{1,p}$ norm. However, we don't ...
7
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384 views

Moduli of smoothness, Besov spaces, and Sobolev spaces

For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is \begin{equation} \omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})} \end{equation} where $\Omega_{rh}=\{...
6
votes
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65 views

Gagliardo–Nirenberg–Sobolev inequality for weighted Sobolev space with exponential weights

Consider the weighted $L^p_\omega(\mathbb{R}^d)$ space on $\mathbb{R}^d$ be the set of Lebesgue measurable functions such that $$\|f\|_{L^p_\omega}=\int_{\mathbb{R}^d}|f|^p\omega_\mu(x)\,dx< \...
6
votes
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136 views

Sanity check: self-adjoint operator on Sobolev space

I just wanted to check if the conclusion below is true, and whether the following reasoning works: Let $H^i$ be the Sobolev spaces on a compact manifold $M$ and $D$ a self-adjoint (in the $H^0$-inner ...
6
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369 views

Question about the proof of General Sobolev Inequality in P.D.E. by Evan

I have been reading the chapter of Sobolev Space in Partial Differential Equations by Lawrence C. Evan, and I came across the General Sobolev Inequality stated as follows: Theorem (General Sobolev ...
6
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130 views

How to proof $C_0^\infty(\mathbb{R}^n)$ is dense in $H^s(\mathbb{R}^n)$ by using mollifier

Since the definition of $u\in H^s(\mathbb{R}^n)$ is $\left(1+|\lambda|^2\right)^{s/2}\hat{u}(\lambda)\in L^2(\mathbb{R}^n)$ I find it difficult to give an constructive prove that use mollifier. let $...
6
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218 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 ...
6
votes
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517 views

Regularity of a weak solution

The problem is formulated as follows: Let $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. For $u\in L_2(\Omega)$ one can define the one dimensional weak dirivative $\partial_1 u$. Now define ...
6
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86 views

Trace theorems for arbitrary differentiability $k$, with embedding constants under control as $k\to\infty$

The usual trace theorem (with non-optimal exponents, but I don't care for those at the moment) says that $$ W^{1,p}(\Omega)\hookrightarrow L^p(\partial\Omega) $$ for Lipschitz domains. When $\...
6
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216 views

Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ (...
6
votes
0answers
347 views

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
6
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208 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. $\frac{\...
6
votes
0answers
359 views

Gradient on Sobolev spaces

Suppose I have a function $\Phi \in L_p(\Omega)$ and $\nabla\Phi \in W^{-\alpha}_p(\Omega)$, $1<p<\infty$ and where $\Omega\subset\mathbb{R}^n$ is a bounded domain. Can I conclude that $\Phi\in ...
5
votes
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66 views

How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
5
votes
0answers
41 views

Is the embedding of $W^{2,p}$ onto $C^1(\overline{I})$ compact?

We know that when $I$ is a bounded interval and $1<p\leq \infty$ that the injection $W^{1,p}\subset C(\overline{I})$ is compact. The proof of this fact uses the Arzela-Ascoli theorem on the unit ...
5
votes
0answers
96 views

Is being in the Sobolev space of power $\frac{d}{2}$ necessary for having well defined point evaluations?

From the Sobolev embedding theorem we know that for $\alpha = \frac{d}{2}$, $W^{\alpha, 2}(\mathbb{R}^d)$ is continuously embedded in $C^0(\mathbb{R}^d)$. Especially the point evaluations are in the ...
5
votes
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29 views

When Sobolev maps are localizable?

Let $M,N$ be smooth $d$-dimensional Riemannian manifolds. A Sobolev map $f \in W^{1,p}(M,N)$ is called localizable if for every $x_0 \in M$ there exists a neighbourhood $U$ of $x_0$ in $M$ and a ...
5
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114 views

Compact embedding in weighted Sobolev spaces

I have a question concerning Sobolev's embedding. Let (for simplicity) $\Omega=\left( 0,1\right) $. Then it is well known by Rellich's theorem that $H^{1}\left( \Omega\right) $ is compactly ...
5
votes
0answers
85 views

Sobolev spaces for conformal metrics on a Riemannian manifold

Suppose we have a compact Riemmanian manifold $(M^n,g)$ without boundary, and some class $\{g_u\}$ of conformal metrics $g_u:=e^{2u}g$, where $\{u\}\subset C^\infty(M^n)$. For some $k,p$ suppose there ...
5
votes
0answers
91 views

Does the following minimum exist?

Let $g \in L^{p}(\Omega ; \mathbb{R}^n)$ , $p>1$ and $\Omega\subset \mathbb{R}^n $ an open and bounded set with smooth boundary and consider the following functional:$$W_{0}^{1,p}\left(\Omega ; \...
5
votes
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189 views

R.H.S. of Poisson equation localized $\Rightarrow$ Solution localized

Let $\Omega\subset\mathbb{R}^d$ a connected open set (which is not necessarily bounded). Assume that $f\in C_0^\infty(\Omega)$ with $\operatorname{supp}(f)\subset K,$ with $K$ compactand let $u$ be ...
5
votes
0answers
96 views

Equivalence of definitions of Sobolev space

I'm reading through Steinbach's book on Eliptic Boundary Value Problems and struggling with understanding of the definition of Sobolev space He defines it as $$H^s(\mathbb R^n) := \{u \in \mathcal S^*...
5
votes
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177 views

What is a distribution in $H^{-1}(\Omega)$?

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $\partial \Omega$ being $C^2$. Suppose $u\in H^1_0(\Omega)$, $f\in L^2(\Omega)$ and $\nu>0$. It is said in the Navier-Stokes Equations ...
5
votes
0answers
109 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 ...
5
votes
0answers
102 views

Sobolev type embedding

Consider a compact manifold $M$ and a point $q \in M$. Let us say that that the following inequality holds: $$ \Vert \varphi u\Vert_{L^p} \leq C\Vert \varphi u\Vert_{H^1},$$ where $\varphi \in C^\...
5
votes
0answers
2k views

The best constant of Poincare inequality can be determined by eigenvalue of Laplace operator

Given $\Omega\subset \mathbb R^N$ be open bounded and smooth boundary. Then for $u\in H_0^1(\Omega)$ we have well-known Poincare inequality $$ \int_\Omega \lvert u\lvert^2dx\leq C\int_\Omega \lvert\...
5
votes
0answers
158 views

$H^{1/2}$ function but not better

I am looking for an example of a function $f: [0, 1] \longrightarrow \mathbb{R}$ that is in the Sobolev space of order $1/2$, $H^{1/2}([0, 1])$, but not in the Sobolev space of order $1/2 + \...
5
votes
0answers
248 views

Extension of Rellich's theorem: Embedding “sort of” compact in the limit case?

Let $\Omega$ be a domain with a nice boundary (i.e., smoothness of the boundary shall not be central to my question). Now it well-known that the embedding $\iota \colon W^{1,p}(\Omega) \to L^q(\Omega)...
5
votes
0answers
128 views

What is $H^1([0,1]) \otimes H^1([0,1])$?

Let $H^1([0,1])$ denote the Sobolev space $H^1$ on the interval $[0,1]$. What is $H^1([0,1]) \otimes H^1([0,1])$? Here, $\otimes$ the tensor product of Hilbert spaces. In particular, how is that ...
5
votes
0answers
167 views

Is this function in the Sobolev space $H^{2,-s}(\mathbb{R}^3)$?

I have the function $$f(x)=\frac{e^{iz|x-y|}}{4\pi|x-y|}$$ with $y\in\mathbb{R}^3$ and $\Im z>0$. Let $s>\frac{1}{2}$. Clearly it is not in $H^{2,-s}(\mathbb{R}^3)$ for the singularity of order $...
5
votes
0answers
182 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
5
votes
0answers
138 views

Can we do some scaling argument in the presence of inhomogeneous norms?

Notation: $B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$. $\hat{f}$ stands for the Fourier transform of $f$. Question. The following inequality holds true for all $f\in H^s(\mathbb{R}...
4
votes
0answers
22 views

weak solution to non-homogeneous initial-boundary value problem

Let $\Omega\subset \mathbb{R}^n$ be a bounded Lipschitz domain, and consider a general linear parabolic equation of the divergence form: $$ u_t-\partial_i(a^{ij}\partial_ju)+b^i\partial_i u+cu=f, \...
4
votes
0answers
47 views

Weak convergence in $H^1(\mathbb R^3)$ implies convergence of integrals

Suppose that $f_n \rightharpoonup f$ in $H^1(\mathbb R^3)$ (weak convergence). Then $$\int_{\mathbb R^3} \frac{\lvert f_n(x) \rvert^2}{\lvert x \rvert} dx \stackrel{n\to \infty}{\longrightarrow} \int_{...
4
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0answers
100 views

Fractional Hardy inequality

From classic literature, I know the following result. Let $\Omega\subset\mathbb{R}^d$ be a bounded open set of class $C^1$. Then there exists $C>0$ such that \begin{equation}\label{1} \|\frac{...
4
votes
0answers
48 views

Definition of $H^{s}(\mathbb{R^{+}})$ and it's norm

What's the definition of $H^{s}(\mathbb{R^{+}})$(classical Sobolev space on the half line) and its norm in terms of Fourier Transform? I'm aware of the definition of classical Sobolev Space $H^{s}(\...
4
votes
0answers
74 views

$C_0^\infty(\overline \Omega)$ is dense in $H(\operatorname{div};\Omega)$

I've been looking for a while in different Functional Analysis books such as Luc Tartar, Jean Pierre Aubin and Brezis but couldn't find the proof of the density of $C_0^\infty(\overline \Omega)$ in $H(...
4
votes
0answers
86 views

Compact embedding between $H^{m+1}(\Omega)$ and $H^{m}(\Omega)$ for $\Omega$ bounded

I know that we have Rellich-Kondrachov Theorem that says that there is a compact embedding between $H^{1}(\Omega)$ and $H^{0}(\Omega)$, or more generally as Adams states (pag 168 theorem 6.3) we have ...
4
votes
0answers
113 views

Gagliardo-Niremberg inequality on annuli (proof)

Let $$\frac{1}{\tau}=a\left(\frac{1}{p}-\frac{1}{d}\right)+\frac{1-a}{q},$$ $\tau>0, p\geq 1, a\in [0,1]$, $q\geq 1$ and $d\geq 1$. I know that $$\|u\|_{L^\tau(\mathbb R^d)}\leq C\|\nabla u\|_{L^p(\...
4
votes
0answers
72 views

Are $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}~~~~and~~\| u \|_{W^{2,p}(\Bbb R^d)}$ equivalents norms?

Do we have that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}~~~~and~~\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms This results is pretty easy and straightforward for $p=2$ using ...