All Questions
Tagged with fa.functional-analysis sobolev-spaces
652 questions
0
votes
0
answers
78
views
Sobolev trace inequality with exterior domains
Let $x_1\in \mathbb{R}^n$, $n\geq 3$, $\Omega=\mathbb{R}^n\backslash B_1(x_1)$, define $D_{\Omega}$ by taking the closure of $C_{c}^{\infty}(\overline{\Omega})$ under the norm
\begin{align*}
\|u\|_{D_{...
- 471
3
votes
0
answers
102
views
Can Sobolev space be characterized by spectral decomposition?
Consider a homogeneous Carnot group $\mathbb{G}$ with step $r$. Let $X_1,\cdots,X_m$ be the first layer of its Lie algebra. Denote by $\mathcal{L}=-\sum_{i=1}^m X_i^2$ the sub-Laplacian on $\mathbb{G}$...
- 561
1
vote
1
answer
89
views
Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?
The following startles me. Let $\Omega \subseteq \mathbb R^n$ and write $W^{\sigma,1}(\Omega)$ for the fractional Sobolev space with norm
$$|u|_{W^{\sigma,1}(\Omega)} := \iint \frac{|u(x) - u(y)|}{|x-...
- 5,327
5
votes
1
answer
177
views
Sobolev inequality with holes
Classical Sobolev inequality says, $n\geq 3$, we have
\begin{equation}
\left(\int_{\mathbb{R}^n}|u|^{2 n /(n-2)}\right)^{(n-2) /(2 n) } \leq C(n)\left(\int_{\mathbb{R}^n}|\nabla u|^{2}\right)^...
- 471
2
votes
2
answers
450
views
The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$
Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...
1
vote
0
answers
89
views
Domain where the fractional Laplacian operator is a closed operator
Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$
Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\...
- 111
2
votes
0
answers
78
views
Zero trace Sobolev space on Carnot group
Let $\mathbb{G}=(\mathbb{R}^{n},\circ)$ be a Lie group on $\mathbb{R}^n$ and $\mathfrak{g}$ be the corresponding Lie algebra of $\mathbb{G}$. Let $X_{1},\ldots,X_{m}$ be the left invariant vector ...
- 561
1
vote
1
answer
154
views
Dense properties of weighted Sobolev space define on $\mathbb{R}^n$
Consider the usual Sobolev space $H^1(\mathbb{R}^n)$ and $H^1_0(\mathbb{R}^n)$, where $H^1_0(\mathbb{R}^n)$ is the closure of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm of $H^1(\mathbb{R}^n)$....
- 561
5
votes
1
answer
216
views
Bounds on dimension of a subspace
Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that:
$$ \| u\|_{...
- 4,135
0
votes
0
answers
142
views
Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
- 1
4
votes
2
answers
391
views
Lebesgue differentiation theorem at boundary points for Sobolev traces
$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...
- 5,380
3
votes
0
answers
88
views
Using a maximum principle to deduce regularity
Suppose $\Omega \subset \mathbb{R}$ is an bounded domain and that $u \in C(0,T; H^{2}) \cap L^{2}(0,T; H^{3})$ where $T >0$.
Consider the PDE on $\Omega \times [0,T]$
$$ \partial_{t}u = a_{1}(x,t) \...
- 131
4
votes
0
answers
140
views
Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$
I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...
- 1,075
3
votes
0
answers
209
views
Interpolation between Sobolev spaces
In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by
$$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$
where $D^sf$ is defined by the Fourier transform
$$(D^...
- 71
4
votes
2
answers
364
views
Equivalence between two Sobolev norms on manifolds
On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following.
Use pseudo-differential operators on $M$...
- 71
1
vote
0
answers
52
views
A question of interpolation space on homogeneous Carnot group
Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows:
A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
- 287
4
votes
2
answers
158
views
A ball with slit at the radius is not $W^{1,1}$-extension domain
Recall that: A domain $\Omega\subset \mathbb{R}^d$ is an $W^{1,1}$-extension domain if there exists an operator $E:W^{1,1}(\Omega)\to W^{1,1}(\mathbb{R^d})$ and a constant $c= c(d,\Omega)>0$ such ...
- 1,101
2
votes
0
answers
259
views
Research in analysis of PDEs
I am currently figuring out what topic to work on for my undergraduate thesis and was able to narrow it down to mathematical analysis. As of now, I have two main options: a thesis working on Sobolev ...
- 149
0
votes
0
answers
103
views
Extension for fractional Sobolev spaces, s>0
In their paper, Fractional Sobolev extension and imbedding, the author describes all extension domains for $s \in (0,1)$ -- meaning spaces functions in which are not required to have weak derivatives. ...
- 93
4
votes
0
answers
308
views
Fourier characterization of weighted Sobolev space $W^{1,2}(\mathbb R^n, \gamma_n)$
For integers $n \ge 1$ and $m \ge 0$, the Sobolev space $W^{m,2}(\mathbb R^n)$ is characterized by
$$
f \in W^{m,2}(\mathbb R^n) \text{ iff } \tilde f_m \in L^2(\mathbb R^n),
\label{1}\tag{1}
$$
where ...
- 6,853
2
votes
1
answer
176
views
Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?
Let $\Omega \subseteq \mathbb{R}^M$ be a measurable subset of positive measure. R. A. Adams and J. Fournier in their article have proven that if $\Omega$ satisfies the so-called weak cone property, ...
- 820
3
votes
1
answer
606
views
On the domain of the Neumann Laplacian
Let $U$ be a bounded domain of $\mathbb{R}^d$, and write $m$ for the Lebesgue measure on $U$. For $k=1,2$, we denote by $H^k(U)$ the set of all locally $m$-integrable functions $u\colon U \to \mathbb{...
- 721
3
votes
1
answer
443
views
Endpoint Calderon-Zygmund inequality of nonlocal fractional laplacian
For $s\in(0,1],$ consider the following non-local fractional laplacian:
$$(-\Delta)^sv= f ~~\text{on } \mathbb{R}^n.$$
Then how to use "the standard elliptic estimate" to obtain:
for $p\in[...
- 705
3
votes
0
answers
59
views
Sufficient conditions for the weight function to have compact embedding of a weighted Sobolev space
Let $\rho$ be a smooth density function on $\mathbb{R}^N$, that is, $\rho(x)\ge 0$ for all $x\in\mathbb{R}^N$ and $\int_{\mathbb{R}^N}\rho(x) dx=1$. Let $L^p_\rho(\mathbb{R}^N)=\{f: \int_{\mathbb{R}^N}...
- 407
1
vote
0
answers
157
views
When can we characterize Sobolev space $W^{2k,p}(\Omega)$ only via the Laplacian-like terms
One of the characterizations of the fractional Sobolev space $W^{s,p}(\mathbb{R}^n)$ uses the Fourier transform $\mathcal{F}$:
$f \in W^{s,p}(\mathbb{R}^n)$ iff $f$ is a tempered distribution such ...
- 820
2
votes
1
answer
154
views
Function monotony between [0,T] and $L^2$
Let $\Omega\subseteq\mathbb{R}^N$ be a bounded and smooth domain. If $z:[0,T]\to L^2(\Omega)$ is a function in $H^1([0,T],L^2(\Omega))$ with the property that $z'(t)(x):=z'(t,x)>0$ a.e. on $\Omega$ ...
- 1,759
5
votes
0
answers
235
views
Proof of density of smooth functions in $H(\operatorname{curl})$ with $L^2$ tangential trace
I am trying to understand the proof of the following statement that is presented in the book “Finite Element Methods for Maxwell's Equations” by Peter Monk. The original source of the proof is a 1997 ...
0
votes
0
answers
152
views
Concentration compactness lemma and the best Sobolev constant
It is well known that the best Sobolev constant can be achieved on $\mathbf{R}^n$. More precisely, we have the following theorem (A):
Let $\frac{1}{q}=\frac{1}{2}-\frac{1}{n}$, $$S=\inf\limits_{{u\in ...
- 705
5
votes
1
answer
1k
views
Chain rule in Sobolev space
In the theory of Sobolev space, we have the following chain rule:
For a uniformly Lipschitz function $F : \mathbf{R}\to \mathbf{R}$ such that $F(0)=0$,
and $u\in W^{1,1}(\mathbf{R}^n)$, then we have ...
- 705
3
votes
1
answer
466
views
Equivalence between two fractional Sobolev spaces
For $s \in (0,1)$, we consider the spectral fractional Laplacian
\begin{align}
(-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k
\end{align}
where
\begin{align*}
\begin{cases}
...
- 161
4
votes
0
answers
68
views
Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$
For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
- 61
6
votes
1
answer
173
views
Sobolev space is spanned by distributions supported on half-lines?
I asked this question on Mathematics Stack Exchange previously.
This seems to be a very basic property of Sobolev spaces, but I wasn't able to find a proof for it.
For any $s \leq 1/2$,
$$H^s(\mathbb{...
- 86
1
vote
0
answers
353
views
$H^{1/2}(0,1)$ and continuous functions
It is known that $H^s(0,1)$ embeds into Hoelder continuous functions for $s>1/2$. I am not interested in Hoelder continuity, but merely in continuity: do I get the continuous embedding $H^{1/2}(0,1)...
- 235
-1
votes
1
answer
153
views
Sobolev estimates $\|\nabla\phi\|_{\infty}\leq C\|\phi\|_{H^2}$
This is a cross post in continuation to this question on Mathematics Stack Exchange. I wanted to know if this inequality holds true in two or three dimensions,
$\|\nabla\phi\|_{L^{\infty}(\Omega)}\leq ...
- 101
2
votes
1
answer
576
views
Riesz potential and homogeneous Sobolev spaces
Consider the Bessel potential and Riesz potential spaces $$L^{s,p}=\{f:f=(I-\Delta)^{-s/2}g,\, g\in L^p\},$$ $$\dot{L}^{\alpha,p}=\{f:f=(-\Delta)^{-s/2}g,\, g\in L^p\},$$
where $(I-\Delta)^{s/2}$ and $...
- 101
2
votes
1
answer
351
views
References on duality of fractional order Sobolev spaces
I would like to ask you for any good references regarding fractional order Sobolev spaces. I know Hitchhiker's guide to the fractional Sobolev spaces is a very popular one, and I found it to be quite ...
- 101
2
votes
0
answers
149
views
Reference for weighted Sobolev spaces
I'm looking for a comprehensive reference illustrating, from the ground up, the basics of weighted Sobolev Spaces on Lipschitz domains (this case should be included, but I don't need less than it). ...
- 235
1
vote
1
answer
404
views
Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$? [closed]
Can someone kindly confirm that the Schwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all ...
- 6,853
1
vote
0
answers
128
views
Self-ajointness of the Laplacian over a Riemannian manifold with boundary
I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf).
Let
$(M,g)$ be a Riemannian manifold with boundary;
$E\to M$ be an hermitian fiber bundle;
$\Delta$ ...
- 141
1
vote
1
answer
329
views
Hölder continuity of Radon transform of smooth function
Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by
$$
R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
- 6,853
1
vote
0
answers
254
views
Sobolev variant of Wasserstein space
Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
- 5,405
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
1
vote
0
answers
110
views
Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space
$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set.
For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
- 2,533
4
votes
1
answer
490
views
ODE in Banach space
Have I understood this correctly:
So originally we consider the following partial differential equation:
$$u_t= \frac{u_{xx}}{1+w}-\frac{1}{\epsilon}(1+w)u^3+\frac{wu}{\epsilon(1+w)} \text{ in } \...
- 41
6
votes
2
answers
1k
views
Exercise 8.13 - Brezis
Let $1 \leq p < \infty$ and $u \in W^{1,p}(\mathbb{R}$). Set
$$
D_{h}u(x) = \frac{1}{h}(u(x+h) - u(x)), \ \ x \in \mathbb{R}, h> 0
$$
Show that $D_{h}u \to u'$ in $L^{p}(\mathbb{R}$) as $h \to ...
- 241
3
votes
1
answer
1k
views
Possible way to define $H_0^1(\Omega)$ Sobolev spaces
Let $\Omega$ be an open set of $\Bbb R^d$: consider the following function spaces
$H_0^1(\Omega)$, i.e. the closure of $C_c^\infty(\Omega)$ in $H^1(\Omega)$
$H_*(\Omega)$, i.e. the closure of $C_c^\...
- 1,101
4
votes
1
answer
423
views
Real interpolation for vector-valued Sobolev spaces
I am having trouble finding any results concerning real interpolation of vector-valued Sobolev spaces. Namely, I would like to know if a continuous embedding of the type,
$$
L^p(0,T;X_1)\cap W^{1,p}(0,...
- 213
3
votes
1
answer
741
views
About radial Sobolev inequality (Strauss Lemma)
As shown in Strauss: Existence of solitary waves in higher dimensions, Strauss introduces the Stauss lemma. Precisely speaking, we have the following theorem:
Theorem Let $N \ge 2$, every radial ...
- 429
2
votes
1
answer
128
views
How to connect the functions in spaces $H^1$ and $H_r$?
\begin{align*}
L^2 (\mathbb{R}^3)& {}=\{ u : \int_{\mathbb{R}^3} \lvert u\rvert^2 dx<+\infty \}. \\
H^1(\mathbb{R}^3) & {}=\{ u\in L^2 (\mathbb{R}^3):\, \lvert\nabla u\rvert\in L^2(\mathbb{...
- 41
2
votes
1
answer
911
views
Weak convergence in $H^1_{\mathrm{loc}}$
$\newcommand{\loc}{\mathrm{loc}}$Let $\Omega$ be a bounded open set (smooth as we wish if necessary) in $\mathbb{R}^n$, $(\omega_k)$ a sequence of open subsets whose closure is contained in $\Omega$ ...
- 364