All Questions
Tagged with reference-request sobolev-spaces
151 questions
11
votes
0
answers
317
views
+50
Sobolev's PDE Scottish Book Problem (Problem 188)
In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution.
In 2015, when the second edition of the Scottish Book with updates and commentary on ...
- 13k
1
vote
0
answers
84
views
Does sets of positive capacity rule out constant functions?
Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by
\begin{align*}
\text{Cap}_{p}(K, U) :=
\inf \left\{
\int_U |\...
- 1,101
2
votes
1
answer
93
views
Reference needed: estimate of the second order derivatives
In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions)
$$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
1
vote
1
answer
85
views
$H^2$-elliptic regularity (up to the boundary) for operators with lower order terms for Lipschitz/convex domains
Let $\Omega$ be a bounded domain which is Lipschitz or convex. Given an elliptic operator of the form
$$\langle Au, v \rangle = a_{ij}u_{x_i}v_{x_j} + b_i u_{x_i}v + cuv$$
are there any elliptic ...
- 93
2
votes
0
answers
54
views
Distance between a Hölder function and a Sobolev ball
Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively.
My question ...
- 410
3
votes
2
answers
349
views
Reference for proof about a result concerning Sobolev spaces and exponential growth
I'm reading an article and I saw the following affirmation without proof:
Let $u \in H^1(\mathbb{R}^2)$ and $\alpha>0$, then
$$\int_{\mathbb{R}^2}(e^{\alpha u^2}-1)dx<+\infty.$$
Is this claim ...
- 213
7
votes
1
answer
580
views
Sobolev spaces are smooth? Their dual is strictly convex?
Do you know any reference which says something about the:
Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton.
...
- 1,759
5
votes
1
answer
173
views
Properties of Poisson Integral and Traces of functions in Sobolev space $W^{1,2} (\mathbb D)$
Let $W^{1,2}(\mathbb D)$ be the complex valued Sobolev space on $\mathbb D$ where $\mathbb D $ is the open unit disk of the complex plane. By definition, $W^{1,2} (\mathbb D)$ is the set of all ...
- 151
5
votes
2
answers
364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
1
vote
0
answers
222
views
Poincaré-Wirtinger inequality for more general "means"
Let $\Omega$ be a ball of radius $r$. It is well known that given a function $f \in W^{1,p} (\Omega)$, it holds the Poincaré-Wirtinger inequality
$$ \left(\int_\Omega |f - f_\Omega|^p dx \right)^{1/p} ...
- 697
1
vote
0
answers
78
views
Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
3
votes
3
answers
252
views
Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?
If $\Omega$ is an open Lipschitz domain with bounded boundary then their exists a continuous operator $E: W^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$ where $s\in (0,1)$ such that $Eu=u$ on $\...
- 163
0
votes
1
answer
77
views
$L^\infty$ estimate for elliptic PDE with mixed boundary conditions
Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint.
Consider the problem
$$\Delta u = f \quad\text{in $\...
- 93
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
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
2
votes
0
answers
93
views
Sobolev inequalities in weighted Sobolev spaces
My definition for "weighted Sobolev space" $W^{1,p} _w (\Omega)$ is just a set of functions with the property that they admit distributional derivatives and that
$$ \int_\Omega |f|^p (x) w(x)...
- 697
2
votes
0
answers
124
views
Uniqueness in interpolation of Hilbert spaces
I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...
- 21
6
votes
0
answers
202
views
Reference request: Elliptic regularity estimate in domains with $C^{1,\alpha}$ boundary
I'm wondering if there is a reference for the following (or if it's not true). Let $\Omega$ be a bounded domain with $C^{1,\alpha}$ boundary, where $0<\alpha<1$. For the inhomogeneous Dirichlet ...
7
votes
1
answer
652
views
Extending Hölder functions
I originally asked this question on MathStackExchange some time ago, but it seems that MathOverflow would be more appropriate. Essentially, I would like to find references for extension theorems for (...
- 820
2
votes
0
answers
56
views
Inequality for a weighted bilinear form in Fourier variables
Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$.
Consider the ...
- 1,101
2
votes
2
answers
451
views
The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$
Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...
1
vote
0
answers
84
views
First-order interpolation inequalities with weights by L.Caffarelli, R.Kohn and L.Nirenberg
L. Caffarelli, R. Kohn and L. Nirenberg showed in this article that, under some conditions, the following weighted interpolation inequality is valid
THEOREM: There exists a positive constant $C$ such ...
- 677
6
votes
1
answer
342
views
The Cauchy problem in general relativity, hyperbolic PDEs, and Sobolev spaces on manifolds
(I apologize in advance if this question is ill-posed or not suitable for Math Overflow, I am not yet a research mathematician, just a student.)
Let $(\Sigma,\bar{g})$ be an $n$-dimensional Riemannian ...
- 221
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
$$
u(x)...
- 5,380
4
votes
0
answers
99
views
Interpolation on Sobolev space on $[0, 1]^d$ over finite meshes
Let $\Omega = [0, 1]^d$ and suppose that $f \colon \Omega \to \mathbb{R}$ lies in order $m > d/2$ Sobolev space; i.e.,
$$
\|f\|_{H^m(\Omega)}^2 = \sum_{|\alpha| \leq m} \|D^\alpha f\|_{L^2(\Omega)}^...
- 410
20
votes
3
answers
2k
views
Convergence of convex functions
I can prove the following result.
Theorem 1. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions
that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$.
Then ...
- 28k
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
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
150
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
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
2
votes
0
answers
92
views
Reference for an extension theorem for Neumann boundary data
$\DeclareMathOperator\Tr{Tr}$Let $\Omega \subset \mathbb{R}^d$ be a smooth bounded domain (we denote by $n$ the normal to $\partial\Omega$) and $p\in(1,\infty)$. Do you know where I can find (book or ...
- 489
0
votes
0
answers
162
views
Compact embedding of anisotropic Sobolev space
I am wondering if the embedding from $W^{2,1}_p(\Omega \times [0,T])$ to $C^{\alpha,\alpha/2}(\Omega \times [0,T])$ is compact, for some suitable domain, $p$ and $\alpha$. I have found some results. I ...
- 54
6
votes
1
answer
382
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
0
answers
159
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 ...
- 505
3
votes
3
answers
196
views
Non convex optimization problem in $W_0^{1,2}$
Let $0< \alpha \ll 1$. I'm trying to minimize $\int_0^\pi |f'|^2 dx$ over the functions $f \in W_0^{1,2}([0,\pi])$ (or at least find "good" lower bound in terms of $\alpha$) such that ...
- 393
1
vote
0
answers
79
views
Reference for smoothness of Nemytskii operator on fractional Sobolev spaces
Let $\varphi:\mathbb{R}\to\mathbb{R}$ be smooth and bounded (together with all of its derivatives). Define the operator
$$
\big(N_\varphi x\big)(t)=\varphi\big(x(t)\big)
$$
for $x\in H^s(T^d)$, the ...
- 93
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
1
vote
0
answers
56
views
Smooth approximation in Sobolev spaces for surfaces with boundary
Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ with closure $\overline{\mathbb{D}}$, and let $\varphi:\partial \mathbb{D}\to \partial \mathbb{D}$ be any continuous homeomorphism. Let $\mu$ be a ...
- 163
1
vote
1
answer
173
views
Definition of Euler-Lagrange equation and properties, where can I find?
I'm studying a paper and in the introduction appears the following:
It is well known that existence of critical points and solvability of Euler-Lagrange equations are related, and there is and ...
- 63
4
votes
0
answers
802
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 ...
1
vote
1
answer
116
views
uniform convergence of $H^r$ projectors on compact sets?
Let $\Omega\subset \mathbb R^d$ be a smooth, bounded domain. Let $(e_n)_{n\geq 0}\subset L^2(\Omega)$ be the Hilbert basis generated by the Dirichlet-Laplacian eigenfunctions, i-e $-\Delta e_n=\...
- 5,380
6
votes
1
answer
585
views
$H_0^1(\Omega, D) \hookrightarrow L^2(D)$ is compact, for $\Omega$ quasi-open in $D$ - Proof verification
In their paper The spectral drop problem, Buttazzo and Velichkov state that the embedding $H_0^1(\Omega, D) \hookrightarrow L^2(D)$ is compact, where $\Omega \subset D$ is quasi-open, $D \subset \...
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
3
votes
0
answers
216
views
integration by parts on a Lipschitz domain as $\epsilon\to 0$
For a fixed, bounded, smooth domain $\Omega\subset \mathbb R^d$ and any $u\in W^{1,1}(\Omega)$ with trace $u|_{\partial\Omega}=g\in L^1(\partial\Omega)$ one can prove that
$$
\lim\limits_{\epsilon\to ...
- 5,380
0
votes
1
answer
96
views
Interpolated Sobolev norm inequality
Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz set, and let $W^{k,p}(\Omega)$ denote the usual Sobolev space with $k \in \mathbb{N}$ being the order of the derivatives and $p \in [1, \infty)$...
- 13
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
1
vote
0
answers
76
views
While doing Lp estimates, is the constant C monotonically increasing with respect to the parameters it depends on?
For example, consider the third boundary value problem:
\begin{align}
&\frac{\partial u}{\partial t}-\sum_{i,j=1}^n a_{ij}(x,t) \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{i=1}^n ...
- 51