All Questions
77 questions
12
votes
3
answers
2k
views
Reference request: Simple facts about vector-valued Sobolev space
Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\...
- 29.8k
9
votes
1
answer
1k
views
Traces of Sobolev spaces
Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...
- 28k
8
votes
3
answers
1k
views
Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains
I am not really familiar with the topic, thus I am looking for some references about the following problem.
Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...
- 81
8
votes
0
answers
260
views
Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$
I have found the following claim made very clearly at least once in the published literature (see below):
Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...
- 503
7
votes
1
answer
652
views
Extending Hölder functions
I originally asked this question on MathStackExchange some time ago, but it seems that MathOverflow would be more appropriate. Essentially, I would like to find references for extension theorems for (...
- 820
6
votes
1
answer
320
views
Stability of fractional Sobolev spaces under diffeomorphisms
Let $H^s_p(\mathbb{R}^n)$ be the fractional Sobolev space of fractional order $s\in \mathbb{R}$, for $1<p<\infty$, and let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Assume that the ...
- 83
6
votes
1
answer
380
views
Sobolev embedding theorems on manifolds
I had asked the following question on math.stackexchange but did not get any response:
I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional ...
- 131
5
votes
3
answers
1k
views
Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain.
How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
I define $H^{\frac 1 2}(\...
- 307
5
votes
2
answers
364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
5
votes
1
answer
453
views
Seeking for references on some PDEs
This is not a technical mathematical question. I came across some PDEs with no references nor their names.
$$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$
The above ...
- 1,101
5
votes
1
answer
2k
views
Density of smooth functions in Sobolev space, respecting nonnegative traces
I am looking for a reference for the following result (if it is true, which I would expect):
Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. Let $\Gamma_0\subset\partial\Omega$ be ...
- 53
5
votes
2
answers
1k
views
Rates of convergence of mollifiers with Sobolev norms on manifold
Let $M$ be a smooth compact Riemannian manifold of dimension $n$, and let $H^s_p(M)$ for $s\in \mathbb{R}$ be the fractional Sobolev space of order $s$ on the manifold (defined for instance through ...
- 83
5
votes
1
answer
365
views
Does the Nash inequality hold on manifolds with Lipschitz boundary?
Let $N$ be a smooth manifold without boundary of dimension $n$. $M$ is a manifold with Lipschitz boundary if $M \subset N$, $M$ and $N$ are of the same dimension, and in the charts of $N$, the ...
- 687
5
votes
0
answers
445
views
Why are functions with vanishing normal derivative dense in smooth functions?
Question
Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm?
Here I define $...
- 881
5
votes
0
answers
340
views
Real interpolation of weighted Sobolev spaces with different weights
Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
4
votes
2
answers
391
views
Lebesgue differentiation theorem at boundary points for Sobolev traces
$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...
- 5,380
4
votes
1
answer
2k
views
Homogeneous fractional Sobolev spaces
Given $s\in (0,1)$ and a measurable function $u:\mathbb{R^n}\to\mathbb{C}$, let us define $$\|u\|_{\dot H^s(\mathbb{R}^n)}^2:=\iint\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy$$
and let $\dot H^s(\...
- 3,146
4
votes
1
answer
1k
views
Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$
I want to show:
Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align}
H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N)
\end{align}
is compact.
I was able to show ...
- 437
4
votes
1
answer
1k
views
$H^s(\mathbb T)$ is a Banach algebra for $s>1/2$
I have not managed to find a reference for the following fact:
$H^s(\mathbb T)$ is a Banach algebra for $s>1/2$.
In particular, I need reference for the following inequality:
$$
\|uv\|_{H^s} \,\...
- 2,933
4
votes
1
answer
255
views
Regularity of Nemitskii maps on Sobolev spaces
Let $\Omega\subset \mathbb R^N$ be a bounded smooth domain, and $f\colon\Omega\times \mathbb R\to \mathbb R$ be a smooth function (let's say $C^2$).
Let $X=W^{1,p}(\Omega)$ with $p>1$ be the ...
- 41
4
votes
0
answers
68
views
Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$
For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
- 61
4
votes
0
answers
176
views
If $u \in W^{\alpha,p}(0,T;X)$ for $\alpha \in (0,1)$, then $f(u) \in W^{\alpha,q}(0,T;Y)$ for some good $f:X \to Y$
Let $u \in W^{\alpha,p}(0,T;X)$ for some reflexive Banach space $X$ (you can also take a Hilbert space if it helps) for $\alpha \in (0,1)$ and $p \geq 2$. I am fine with both the Sobolev-Slobodeckij ...
- 51
4
votes
0
answers
102
views
$W^{k,p}$ and Holder regularity for linear elliptic systems with Neumann boundary data
I'm looking for a text or paper that discusses regularity in the Sobolev and Holder sense for general linear elliptic systems of PDEs on bounded domains with Neumann boundary data.
The book by ...
4
votes
0
answers
111
views
A reference for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$
Let $\Omega$ be an open domain with nice boundary and $u\in W^{1,p}(\Omega)$. I believe that $|u|^p\in W^{1,1}$ with
$$
\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u
$$
but couldn't find a good ...
- 1,245
3
votes
1
answer
296
views
Weighted Lebesgue space with exponential weights: smoothing effect and properties
I am researching whether there are weighted Lebesgue spaces of the type
$$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$
...
- 677
3
votes
1
answer
579
views
What's the predual of Holder continuous function spaces?
Given Holder space $C^{\alpha}(\mathbb{R}^n)$, does there exist a Banach space $X$ such that the dual of $X$ is $C^{\alpha}(\mathbb{R}^n)$?
What I can imagine is that such $X$ must contain the ...
- 1,350
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
3
votes
1
answer
1k
views
Gagliardo-Nirenberg inequality for bounded domain
For concreteness let's assume that $u\in W^{1,2}(\Bbb R^2).$ It is well known that
$$
\|u\|_4\le C \|u\|_2^{\frac 12} \|\nabla u\|_2^{\frac 12}.
$$
This is also true if $u\in W^{1,2}_0(\Omega)$ for a ...
- 1,245
3
votes
3
answers
228
views
References for well-posedness of weak solutions to Stefan problem
Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)?
...
- 307
3
votes
1
answer
1k
views
Friedrichs mollifiers and Sobolev spaces
$\renewcommand{\epsilon}{\varepsilon}$The following is from John Roe's book Elliptic operators, topology and asymptotic methods. $S$ is a vector bundle on a compact manifold $M$, but I think for my ...
- 1,141
3
votes
1
answer
695
views
Continuous embedding between parabolic Sobolev spaces
I was wondering whether the following continuous embedding theorem for parabolic Sobolev space is correct?
Let $I=[0,T]$ and $\Omega$ be a sufficiently smooth domain in $\mathbb{R}^n$, we consider ...
- 503
3
votes
0
answers
181
views
Variational problems living in two different Sobolev spaces
Is there a general reference concerning variational problems living in $W^{h,p}\times W^{k,p}$, with $h, k\in\mathbb{N}_0$ not coinciding? I'm thinking to problems of type:
$$\inf_{u,v}\int_{\Omega} ...
- 4,459
3
votes
0
answers
185
views
Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)
Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse.
Does there ...
- 93
3
votes
0
answers
140
views
convergence of $e^{it\Delta}f$
I heard of a conjecture that
$e^{it\Delta}f\rightarrow f$ a.e. as $t\rightarrow 0$ for $f\in H^{1/4+\epsilon}$
but couldn't find a proper reference.
- 31
2
votes
2
answers
235
views
$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?
Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
- 128k
2
votes
2
answers
197
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
- 128k
2
votes
2
answers
450
views
The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$
Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...
2
votes
1
answer
232
views
Generalization of maximum principle to other norms
Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ ...
- 2,286
2
votes
1
answer
1k
views
Mixed (anisotropic) Sobolev spaces
Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, ...
- 23
2
votes
1
answer
766
views
reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$
I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
- 5,380
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
2
answers
339
views
List of tensor product spaces with uniform crossnorms
Let $H^{(j)}$ and $G^{(j)}$ be Banach spaces for $j\in\{1,\dots,n\}$. Call norms $\|\cdot\|_{H}$ and $\|\cdot\|_{G}$ on the algebraic tensor products
$H:=\bigotimes_{j=1}^n H^{(j)}$ and $G:=\...
- 302
2
votes
2
answers
357
views
Composition operators on fractional-order (periodic) Sobolev spaces
(The question was originally posted on MSE.)
Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication ...
- 63
2
votes
2
answers
342
views
compact inclusion of domains of unbounded operators
Let $L$ be a positive self-adjoint operator defined densely on $L^2(M)$ where $M$ is a compact manifold.
Also, let $\mathcal{D}(L) \subset H^1(M)$. It is known that $\mathcal{D}(L) \subset \mathcal{D}(...
- 21
2
votes
1
answer
192
views
Nonlocal Stefan problems
Has there been much work in the setting of Stefan (or general free boundary) problems with some type of nonlocality?
A search on Google and MathSciNet give me only a handful of results which greatly ...
- 43
2
votes
0
answers
54
views
Distance between a Hölder function and a Sobolev ball
Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively.
My question ...
- 400
2
votes
0
answers
124
views
Uniqueness in interpolation of Hilbert spaces
I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...
- 21
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
2
votes
0
answers
559
views
Multiplication in Sobolev space with negative exponent
My initial problem is the following: I would like to estimate $\lVert f^2\rVert_{H^{-2}}$ in the sense of Sobolev embeddings, where $f:\Omega \rightarrow \mathbb{R}$ is a function defined on a bounded ...
- 914
2
votes
0
answers
125
views
Regularity up to boundary of a solution $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ to $\Delta^2 u = -\text{div}\, F$
Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system
...
- 1,245