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.
1,037
questions
4
votes
0
answers
133
views
Sobolev space of maps between manifolds with boundary
Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds with non-empty smooth boundary.
If we consider the Sobolev space $W^{1,p}(M,N)$, is there a reference
on how to model this as a manifold?
If ...
- 65.1k
2
votes
0
answers
181
views
Interpolation of embedded Hilbert spaces and intersection
I'm wondering under what hypothesis it is true a property like
$$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$
where $\mathcal{H}...
- 121
5
votes
2
answers
625
views
Is an $H_0^1$ function continuous to the boundary if it is continuous in the interior?
Suppose $\Omega$ is a bounded domain in $\mathbb R^3$ with Lipchitz boundary $\partial\Omega$, and $u\in H_0^1(\Omega)\cap C(\Omega)$. Is $u$ continuous to the boundary i.e. do we have $u \in C( \...
- 51.6k
4
votes
0
answers
150
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
2
votes
1
answer
364
views
Trace of a function
Let $T,L> 0$ two real numbers and we consider the Sobolev space $X := L^2(0,T; H^1(0,L))\cap H^{1}(0,T;H^{-1}(0,L))$. My question is:
Given $f \in X$, the trace $ t \mapsto f(t,L)$ belongs to what ...
- 37.6k
1
vote
0
answers
156
views
Does a sequence of Jacobians converge to the 'correct' continuous part plus some controlled singular part?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \in W^{1,...
- 6,621
0
votes
1
answer
108
views
Functions for which $|f^{(k)}|_{C^{0,\alpha}(0,1)} \le \Vert f \Vert_{L^1(0,1)}$
Let $f \in C^k(0,1)$ and assume that the $k$-th derivative is $\alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) \subset (0,1)$. Can we characterize (or at least find some ...
- 118k
1
vote
1
answer
73
views
Is there any quantitative relationship between the two terms of a Helmholtz decomposition?
Let $\Omega \subset \mathbb R^3$ denote an open, bounded and simply connected set with smooth boundary. The Helmholtz decomposition
$$ L^2(\Omega) = \nabla H^1_0(\Omega) \oplus L^2(\operatorname{div}=...
- 847
4
votes
1
answer
214
views
Regularity of Nemitskii maps on Sobolev spaces
Let $\Omega\subset \mathbb R^N$ be a bounded smooth domain, and $f\colon\Omega\times \mathbb R\to \mathbb R$ be a smooth function (let's say $C^2$).
Let $X=W^{1,p}(\Omega)$ with $p>1$ be the ...
- 60.2k
1
vote
1
answer
193
views
Does weak continuity of Jacobians hold for non nondegenerate maps?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).
Let $f_n \...
- 1,976
8
votes
2
answers
275
views
Averaging the mass of a Sobolev function $f\in W^{1,p}(\Omega)$ near $\partial\Omega$
Recently, I asked a somewhat related question here. In the comment section, I found the formula
$$
\lim_{r\to 0}\frac{1}{r}\int_{\Omega_r} f(x)\,dx = \int_{\partial \Omega}f(\sigma)\,d\mathcal{H}^{n-1}...
- 37.6k
2
votes
0
answers
1k
views
Compact embedding of the Sobolev space $H^m(\Omega)$ and $L^2(\Omega)$ from Rellich-Kondrachov theorem
From the Rellich-Kondrachov theorem we know that $H^m(\Omega)\hookrightarrow_c L^2(\Omega)$ when $\Omega$ is bounded of class $C^1$ and $m\geq 1$ is an integer. Also this is not true if $\Omega:=\...
- 647
1
vote
0
answers
80
views
Compact imbedding for weight space
We begin with some definitions. Let $\gamma \geqslant 1,\,p \in \left[ {1,\infty } \right)$, we define
$$L_\gamma ^p\left( {0,1} \right) = \left\{ {v:\left( {0,1} \right) \to \mathbb{R}:{{\left\| v \...
- 183
4
votes
0
answers
136
views
Uncertainty principle, Sobolev embedding, and norm estimates
Terence Tao offers a nice discussion of different function spaces in this blog. In the blog there is an explanation of the tradeoff between regularity $s$ and integrability $p$, where $s,p$ are ...
- 1,641
2
votes
1
answer
146
views
Minimum solution over closed ball of $H_0^1(\Omega)$
Since more than 4 months ago, I have posted a question on Mathstack and I haven't recieved any concrete answers. The link to the original post with the problem and my attempts are here.
To summarize, ...
- 39
2
votes
2
answers
332
views
Regularity of the quasi-linear PDE $-\Delta u + c(u) = f$
Let $K$ be a compact set in $\mathbb R^n,$ $f\in C^\infty(K),$ and $c:\mathbb R \to \mathbb R$ be a smooth function. Consider an element $u\in H_0^1(K),$ satisfying the PDE
$ -\Delta u + c(u) = f$ in ...
- 1,641
1
vote
0
answers
110
views
Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
$\DeclareMathOperator\rad{rad}$
Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ be compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
In $H_{\rad}^1(\mathbb{R}^3)$, by Struass estimate $|f(x)| \lesssim |x|^{-1} ...
- 429
2
votes
1
answer
194
views
Estimates for an elliptic PDE
Let's say I have an equation of the form $\Delta A = J$ where $J=u\nabla u + A|u|^2$ (Clarification: We are on $\mathbb{R}^3$ and $u$ is assumed to be in $H^1(\mathbb{R}^3)$). Then I could simply ...
- 5,255
3
votes
1
answer
395
views
Explicit example $f_k \to f$ converging strongly in $L^6(R^3)$, but only weakly in $H^1(R^3)$
Can we find an explicit example of a sequence of functions $f_k \in H^1({\mathbf R}^3)$ such that, $f_k \rightharpoonup f$ weakly converges in $H^1({\mathbf R}^3)$ and $f_k \to f$ strongly converges ...
- 543
1
vote
0
answers
74
views
Fourier transform of a Sobolev function dependent on a "parameter"
Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that
$$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$
and ...
- 2,758
2
votes
0
answers
42
views
Generalized Hardy operator and Lorentz gamma spaces
I would like to find an inequality which would 'place' the generalized Hardy operator $\int_0^th(y)dy\int_y^tk^*(s)ds$ in between two Lorentz gamma spaces.
Any literature or ideas would be greatly ...
- 109
1
vote
1
answer
139
views
Is a locally invertible weak limit of injective maps injective almost everywhere?
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 Lipschitz injective maps ...
- 1,976
6
votes
1
answer
331
views
Optimal constant in Sobolev embedding
It is well-known that the Sobolev space $H^1(0,s)$ embeds continuously in the space of continuous functions $C[0,s]$; in fact, Marti has found in 1983 that the optimal embedding constant is $\sqrt{\...
- 5,255
1
vote
0
answers
384
views
Stein's extension operator for fractional Sobolev spaces
In his book Singular Integrals and Differentiability Properties of Functions,
Stein constructs an extension operator $\mathcal{E}:W^{m,p}(\Omega)\rightarrow
W^{m,p}(\mathbb{R}^{N})$, $m\in\mathbb{N}$, ...
- 2,758
4
votes
1
answer
203
views
Is $L^1$ strong convergence of Jacobians valid for maps between manifolds?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\Vol}{\operatorname{Vol}}$
$\newcommand{\Det}{\operatorname{Det}}$
$\newcommand{\Volm}{\...
- 27.3k
2
votes
1
answer
163
views
A question about series involving a Sobolev functions
Let $\Omega\subset\mathbb{R}^n$ open, bounded and smooth. Let $\lambda_j$ and $e_j$, $j\in\mathbb{N}$, be the eigenvalue and the corresponding eigenfunctions of the Laplacian operator $-\Delta$ in $\...
- 241
1
vote
0
answers
87
views
A bilinear estimates involving critical Sobolev norms
Given $q>1$, consider the critical Sobolev space $W^{n/q,q}(\mathbb{R}^n)$, which fails to embed in $L^{\infty}(\mathbb{R}^n)$. I'm wondering if we can recover some critical estimate by considering ...
- 923
1
vote
0
answers
185
views
Uniformly local Sobolev spaces and interpolation
Let $d\in\mathbb{N}^+$, $s\geq 0$, and consider the uniformly local Sobolev space
$$H^s_{u,loc}(\mathbb{R}^d):=\{f\in H^s_{loc}(\mathbb{R}^d)\,s.t.\,\|f\|_{H^s_{u,loc}}:=\sup_{x\in \mathbb{R}^d} \|f\|...
- 923
10
votes
1
answer
386
views
Sobolev inequalities on manifolds: dependence of the constants on the Riemannian metric
Let $g$ be a smooth Riemannian metric on the 2-torus $T^2$. $g$ induces the Sobolev space $W^{2,2}_g(T^2)$ via the norm
$$
\|f\|_{W^{2,2}_g}^2 = \int_M |f|^2 + g(\nabla^2 f,\nabla^2 f)\, \text{vol}_g,
...
- 829
3
votes
0
answers
198
views
Hardy Littlewood maximal function bounds
Let $u \in W^{1,p}(\mathbb{R}^n) \cap L^{\infty}(\mathbb{R}^n)$ be a given function for some $1<p< \infty$ and let $k \in \mathbb{R}$ be any number and consider the following maximal function
$$
...
- 483
2
votes
0
answers
360
views
A question in Sobolev spaces involving time
Let $X$ be a Banach space, we understand $L^1(0, T, X)$ is the space of strongly measurable functions from $[0, T]$ valued in $X$, that is integrable. Assume ${\bf u}\in L^1(0, T, X)$, we say ${\bf v}\...
- 637
1
vote
1
answer
181
views
Example when Kantorovich condition would not hold
Let $K \in M_+(R_+^2), f \in M_+(R_+)$. Consider operator
$$
(T_k)(x)=\int_{R_+}K(x,y)f(y)dy, \quad y\in R_+.
$$
Denote by $f^*(t)=\inf\{\lambda>0: \alpha x \in R_+: \mu_f(y)>\lambda\}$ the non-...
- 1,849
6
votes
2
answers
2k
views
Arzelà-Ascoli theorem and Hölder spaces
Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist ...
- 153
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)=...
- 27.3k
2
votes
0
answers
131
views
Conditions on the inequality with a gauge norm
Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm
$$
\rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
- 343
3
votes
0
answers
89
views
Sobolev embedding if $p=3/2$, $n=3$ and $k=2$
Usually, Sobolev's embedding theorem goes as follows. Suppose $1\leq p<q < \infty$ and $k> \ell$. If
$$ \frac{1}{p} - \frac{k}{n} = \frac{1}{q} - \frac{\ell}{n} ,$$
then $W^{k,p}(\mathbb{R}^n)...
- 5,169
5
votes
1
answer
584
views
Showing integrability of a locally integrable function on a bounded domain under some additional assumptions
Suppose $\Omega\subset \mathbb{R}^3$ is a smooth and bounded domain, and $f:\Omega\to[0,\infty]$ is a given function which is finite almost everywhere and satisfies
Assumption A: For all $g\in C_0^1(\...
- 401
5
votes
0
answers
130
views
Is Sobolev limit of bijective maps surjective?
This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries. Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be $C^1$ be bijective maps ...
- 6,621
2
votes
0
answers
317
views
Analogue of Lipschitz continuity of $W^{1,\infty}$ for Hölder continuity and Sobolev-Slobodeckij spaces
A function $u : U \rightarrow \mathbb R$ is an element of the Hölder space $C^{\alpha}(U)$ if
$\sup\limits_{x \in U} |u(x)| < \infty$
$\sup\limits_{x,y \in U} \dfrac{|u(x) - u(y)|}{|x-y|^\alpha} &...
- 4,776
7
votes
2
answers
510
views
"Reversion" of class $J(\theta)$ interpolation property for Besov spaces
In (function space) interpolation theory, a Banach space $E$ is of class $J(\theta)$ (for $0 < \theta < 1$) if $$X \cap Y \subseteq E \subseteq X+Y,$$ where $(X,Y)$ are Banach spaces and form an ...
- 101
2
votes
0
answers
149
views
On sup over boundaries of Sobolev functions
In Gilbarg-Trudinger's section on the maximum principle for weak solutions, the sup of a boundary of a Sobolev function defined as follows:
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. For $u, ...
- 303
3
votes
1
answer
410
views
Is the maximum of derivatives of a function in (s,2)-Sobolev space (an RKHS) bounded by their norms?
Let $f(x) \in W^{s,2}(\Omega) \equiv H^s$, where $\Omega \subseteq \mathbb{R}^d$, $s > d/2$ and $W^{s,2}$ is a $(s,2)$-Sobolev space. Clearly, $W^{s,2}$ is an Reproducing Kernel Hilbert Space (RKHS)...
- 31
10
votes
1
answer
516
views
discontinuous functions on the Sobolev borderline
The Sobolev embedding theorem implies that every function of class $W^{k,p}$ on a reasonable $n$-dimensional domain is continuous if $kp > n$. Cases with $kp=n$ are known as "borderline" ...
- 27.3k
2
votes
2
answers
240
views
inequality involving the fractional Sobolev space
Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality
$$|u(x)...
- 407
2
votes
0
answers
141
views
Uniform bounds for a coupled parabolic system of PDE (linear)
Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon \...
9
votes
1
answer
373
views
Extending Sobolev function on Riemannian manifold
Let $(M, \mu, d)$ be a geodesically complete non-compact Riemannian manifold such that measure $\mu$ is volume doubling, i.e. \begin{equation}\label{VD}\mu(B(x, 2r))\leq C\mu(B(x, r))\end{equation} ...
- 60.2k
5
votes
1
answer
412
views
improved Sobolev embedding
This is probably not a research level question but I am struggling with the geometry. My question is related to whether some monotonicity can increase the range of exponents in the Sobolev embedding.
...
- 4,734
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 $f_k \to f$ ...
- 201
4
votes
0
answers
113
views
If $f:U_1\to\mathcal L^p(\mu;E_2)$ is Fréchet differentiable, can we say anything about the Fréchet differentiability of $u\mapsto f(u)(\omega)$?
Let $(\Omega,\mathcal A,\mu)$ be a $\sigma$-finite measure space, $p\ge1$, $E_i$ be a $\mathbb R$-Banach space, $U_1\subseteq E_1$ be open and $f:U_1\to L$ be Fréchet differentiable at $x\in U_1$, ...
- 161
2
votes
0
answers
143
views
Approximation in fractional Sobolev space
Assume $\Omega\subset \Bbb R^d$ is Lipschitz open set. Let $p\geq 1$ and $0<s\leq 1/p$.
How to prove that $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$?
Recall that,
$$|u|^p_{W^{s,p}(\Omega)}= ...
- 1,033