Questions tagged [sobolev-spaces]
A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.
433
questions with no upvoted or accepted answers
15
votes
1
answer
2k
views
Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain
Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space
$$X = \left\{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{...
14
votes
0
answers
626
views
Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
12
votes
0
answers
446
views
Are Sobolev trace spaces equal from both sides of the boundary?
Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
9
votes
1
answer
2k
views
Chain rule for distributional derivative
Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$).
Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
8
votes
0
answers
441
views
Regularity result for the boundary value problem for the heat equation
Let $\Omega$ be an open bounded subset of $\mathbb R^N$.
Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$
Consider the following boundary value problem for the heat equation:
...
8
votes
1
answer
1k
views
Noncompactness of the Sobolev embedding in the critical exponent case
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$.
It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the ...
7
votes
0
answers
253
views
An open problem in Sobolev spaces
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Suppose that there there is a bounded extension operator
$$
E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n)
\quad
\text{and}
\quad
E:W^{1,q}(\Omega)\to ...
7
votes
0
answers
250
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 \...
7
votes
0
answers
130
views
Different definitions of fractional sobolev spaces
Let $\Omega$ be a bounded and smooth domain in $\mathbb R^d$. For any $s\in (0,1)$ we can define $H_s(\Omega)$ to be the space of functions $u\in L^2(\Omega)$ such that $$(x,y)\mapsto \frac{|u(x)-u(y)|...
6
votes
0
answers
111
views
A function whose derivatives belong to $BMO(\mathbb{R}^n)$
I am reading the paper "Bounded Mean Oscillation and Sobolev Spaces" by Robert
s. Strichartz, Indiana University Mathematics Journal , 1980, Vol. 29, pp.
539-558. In this paper he defines ...
6
votes
0
answers
190
views
Reference request: Elliptic regularity estimate in domains with $C^{1,\alpha}$ boundary
I'm wondering if there is a reference for the following (or if it's not true). Let $\Omega$ be a bounded domain with $C^{1,\alpha}$ boundary, where $0<\alpha<1$. For the inhomogeneous Dirichlet ...
6
votes
0
answers
181
views
Factorization of metric space-valued maps through vector-valued Sobolev spaces
Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that
$$
\int_{x\in X}\,d(y_0,f(x)...
6
votes
0
answers
97
views
A continuity argument for a dispersive $gKdV$ estimate
I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at
$$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$
where $F(u) = u^5$ (for example). The ...
6
votes
0
answers
244
views
Do asymptotically conformal maps converge to a weakly conformal map?
$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, ...
6
votes
0
answers
170
views
The distributional gradient of the closest isometry to the differential of a smooth map
The setting-a "linear algebra" fact:
Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \...
6
votes
0
answers
107
views
Interpolation of some Sobolev spaces
Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$
I am wondering what is
$$(X_0,X_2)_{1/2,2}=?$$
Would it be $H^2_0(0,...
6
votes
0
answers
113
views
Positive splitting of Sobolev convergence
Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...
6
votes
0
answers
254
views
Is a Sobolev map with invertible smooth minors smooth?
$\newcommand{\Cof}{\text{cof}}$
Let $k,d$ be even integers, such that $d\ge3$ and $2 \le k \le d-1$. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for ...
6
votes
1
answer
1k
views
Regularity of 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 =...
6
votes
0
answers
507
views
Which functions can be approximated by piecewise constant functions?
Let $\Omega \subset \mathbb{R}^d$ be a connected and bounded domain. We call a function $f\in L^\infty(\Omega)$ nice if for each $\epsilon>0$ there exist $n\in \mathbb{N}, a_1,\dots,a_n \in \mathbb{...
5
votes
0
answers
203
views
Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$
My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
5
votes
0
answers
104
views
Varifold convergence of images of Sobolev maps
Suppose I have a sequence of maps $\{f_k:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^{n+1}\}$ such that:
$f_k\rightharpoonup f_*$ weakly in $W^{1,p}(\Omega,\mathbb{R}^{n+1})$,
The images $\Sigma_k:...
5
votes
0
answers
202
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 ...
5
votes
0
answers
167
views
Convergence of the best constant in the $s$-fractional $L^p$ Sobolev inequality
It is known that the fractional $L^p$ Sobolev inequality
$$
\|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy
$$
...
5
votes
0
answers
111
views
$L^p$ estimates for linear parabolic pdes
Let $u$ solve the linear parabolic equation
$$
u_t - \Delta u = f \text{ on } \Omega \times (0,T)
$$
with initial condition $u(0)=u_0$ and homogeneous Dirichlet boundary condition on $\partial \Omega ...
5
votes
0
answers
144
views
On Sobolev spaces on domains in Riemannian manifolds
There is extensive literature on Sobolev spaces on complete Riemannian manifolds but are there any standard references regarding the definition and properties of Sobolev spaces on domains (possessing ...
5
votes
0
answers
220
views
How far can a continuous, almost everywhere differentiable function be from being a Sobolev function?
Let $\Omega$ be the open unit ball in $\mathbb R^n$. Consider the set $\mathcal D$ of continuous functions $f:\Omega \to \mathbb R$ that are differentiable a.e, and with $|\nabla f| \leq 1$ wherever $...
5
votes
0
answers
417
views
Dual norm for weighted Sobolev space
Consider a "weight" function $\rho: \mathbb{R}^{d} \rightarrow \mathbb{R}$ and the following weighted Sobolev norm:
\begin{equation}
\|h\|_{\dot{H}^{1}(\rho;\mathbb{R}^{d})}:=(\int_{\mathbb{...
5
votes
0
answers
107
views
Does there exist an injective Lipschitz map on the disk whose gradient switches between two given matrices?
While solving a problem in calculus of variations, I came to the following question:
Let $A,B$ be two real $2 \times 2$ matrices with positive determinants, and suppose that $\operatorname{rank}(A-B)=...
5
votes
0
answers
130
views
Is Sobolev limit of bijective maps surjective?
This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries. Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be $C^1$ be bijective maps ...
5
votes
0
answers
132
views
Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$
Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where
\begin{align}\label{eq:bounded-...
5
votes
0
answers
96
views
A bounded extension operator
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
5
votes
0
answers
94
views
Stability and capacity, error in the book of Adams-Hedberg?
I am struggling to understand the proof from the book of Adams and Hedberg, "Function spaces and potential theory". It seems to me that there is a serious flaw, and moreover, the statement is ...
5
votes
0
answers
382
views
A subspace of the homogeneous Sobolev space $\dot{W}^{1,p}% (\mathbb{R}^{N})$
Given $1\leq p<\infty$, the homogeneous Sobolev space $\dot{W}^{1,p}
(\mathbb{R}^{N})$ is defined as the space of all functions $u\in
L_{\operatorname*{loc}}^{1}(\mathbb{R}^{N})$ such that the ...
5
votes
0
answers
234
views
Is polar decomposition of a smooth map Sobolev?
Motivation:
Let $\mathbb{D}^2$ be the closed unit disk. I am studying the "elastic energy" functional $E(f)=\int_{\mathbb{D}^2} \text{dist}^2(df,\text{SO}_2)$, where $f \in C^{\infty}(\mathbb{D}^2,\...
5
votes
0
answers
400
views
Green's formula and traces in weighted Sobolev spaces
Let $B_1$ denote the unit ball in $\mathbb{R}^d$, let
\begin{equation}
\rho(x) = 1-|x| \quad \text{ for } x \in B_1,
\end{equation}
and let $\sigma >0$ be given. As per the comments, notice that $\...
5
votes
0
answers
417
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 $...
5
votes
0
answers
320
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 ...
5
votes
0
answers
278
views
The Spectrum of certain differential operators
We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.
We consider the following polynomial vector field on ...
5
votes
0
answers
876
views
Trace Theorem for $p=\infty$
I am considering the Sobolev space $W^{1,\infty}(\Omega)$ on a bounded Lipschitz domain $\Omega \subseteq \mathbb{R}^2$. I am wondering whether the trace theorem holds in this case with constant one (...
5
votes
0
answers
412
views
Is there an appropriate weighted Sobolev space to include exponential map and projection map?
Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: \...
5
votes
0
answers
518
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}=\{...
4
votes
1
answer
238
views
If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
4
votes
0
answers
128
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 $\...
4
votes
0
answers
98
views
Interpolation on Sobolev space on $[0, 1]^d$ over finite meshes
Let $\Omega = [0, 1]^d$ and suppose that $f \colon \Omega \to \mathbb{R}$ lies in order $m > d/2$ Sobolev space; i.e.,
$$
\|f\|_{H^m(\Omega)}^2 = \sum_{|\alpha| \leq m} \|D^\alpha f\|_{L^2(\Omega)}^...
4
votes
0
answers
261
views
Fourier characterization of weighted Sobolev space $W^{1,2}(\mathbb R^n, \gamma_n)$
For integers $n \ge 1$ and $m \ge 0$, the Sobolev space $W^{m,2}(\mathbb R^n)$ is characterized by
$$
f \in W^{m,2}(\mathbb R^n) \text{ iff } \tilde f_m \in L^2(\mathbb R^n),
\label{1}\tag{1}
$$
where ...
4
votes
0
answers
68
views
Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$
For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
4
votes
0
answers
591
views
Reproducing kernel Hilbert space of Matérn kernels
I am trying to read a recent paper titled "Interpolation and learning with scale dependent kernels" by Pagliana, Ruidi, De Vito, and Rosasco. (The paper can be found on ArXiv)
On the top of ...
4
votes
0
answers
304
views
Reference Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173
I have been searching without success for the reference:
Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173
It is cited in many related works. In ...
4
votes
0
answers
270
views
Sobolev spaces and spectral theorem
Consider a generalised harmonic oscillator of the form
$$
A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n,
$$
where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The ...