Questions tagged [fractional-sobolev-spaces]
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17 questions
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On compact embeddings in weighted Riesz potential spaces
I wonder if there is any references for the study of the following type of spaces
$$ X_{\delta,\alpha}=\{ u\in L^2_\delta(\mathbb{R}^n):\, u= (-\Delta)^\alpha f \quad\text{for some}\quad f\in L^2_{\...
- 4,125
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Sobolev's PDE Scottish Book Problem (Problem 188)
In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution.
In 2015, when the second edition of the Scottish Book with updates and commentary on ...
- 12.9k
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Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains
On a bounded domain $\Omega \subset \mathbb R^d, d\geq 2$ with smooth boundary, it is well known that for the Dirichlet Laplacian $\Delta_D$, $D((-\Delta_D)^\frac12) = H^1_0(\Omega)$.
I'm interested ...
- 153
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Whether the fractional Sobolev seminorm of any smooth function with compact support is finite
Let $n\geqslant 1$, $p\in [1,\infty)$, and $s\in (0,1)$. Define the fractional Sobolev seminorm
\begin{equation*}
[f]_{\dot{W}^{s,p}(\mathbb{R}^n)}
:=\Bigl[\frac{f(x)-f(y)}{|x-y|^{\frac{n}{p}+s}}\...
- 3
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Characterization of the dual of intersection of Banach spaces
Let $U,V$ Banach spaces and define $X = U\cap Y$ endowed with the norm $\|u\|_X = \|u\|_U + \|u\|_V$. If we take $\varphi \in U'$ and $\psi \in V'$, I can prove that $\varphi|_X + \psi|_X \in X'$, ...
- 187
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Equivalence of Sobolev norms for smooth functions with compact support
Let $f\in C^\infty_c([0,1]^n)$, then we can extend it to a $1$-periodic smooth function $\tilde f$. We define the fourier transform (series) of $f$ ($\tilde f$):$$
\hat f(\xi):=\int e^{2\pi i x\cdot \...
- 435
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Whether the constant of a fractional Sobolev inequality is universal for all cubes
Let $D\subset \mathbb{R}^d$ with $d\geqslant 1$ be a bounded open connected Lipschitz set, $\eta\in (0,1)$, $p>0$, and $\alpha>0$. Then the paper (On comparability of integral forms written by ...
- 3
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3/2 Sobolev Norm on the boundary of a bounded open subset of $\Bbb R^n$
Let $\Omega\subset\mathbb{R}^{n}$ be a open bounded set and $\partial\Omega$ be the boundary of $\Omega$.
Following the reference text by Alois Kufner, Oldřich John and Svatopluk Fučík, Function ...
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Compact embeddings RKHSs into Sobolev Spaces
Let $\mathcal{H}$ be an RKHS over an open domain $\Omega \subseteq \mathbb{R}^d$. Are there conditions under which $\mathcal{H}$ can be compactly embedded in a Sobolev space $W^{s,p}(\Omega)$ for ...
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Existence of $H^{1/2}(\partial\Omega)$-regular unit tangent field on smooth surface
Suppose that $\Omega$ is a bounded, smooth, simply connected domain in $\mathbb{R}^3$. My goal is to show that there is a $p(x) \in H^1(\Omega,\mathbb{S}^2)$ such that $p(x)$ lies on the tangent plane ...
- 54
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103
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Strong convergence of a sequence in $L^2((0,T); H^{s,2}(\Omega)) \cap C([0,T];H^{-s,2}(\Omega))$, $0<s<1$
Let $u_n$ be a sequence with $u_n \in L^2((0,T);H^{1,2}(\Omega))$ and $\frac{\partial u_n}{\partial t} \in L^2((0,T);H^{1,2}(\Omega)^*)$. Then, how could one get a subsequence of $u_n$ that strongly ...
- 101
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$H^{s,\infty}$ via Triebel Lizorkin spaces
I know that
$F^s_{p,2} = H^{s,p}$
for $p \in (1, \infty)$, but what about $H^{s,1}$ and $H^{s,\infty}$? I know one extension for $F^0_{\infty,2}$ which should be BMO. But how do I get the $H^{s,\infty}...
- 11
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Proof of the equivalence between Triebel Spaces and Bessel Potential
I've encountered a question regarding the relationship between Triebel-Lizorkin spaces and Bessel potential spaces. Specifically, I understand that
$F^s_{p,2} = H^{s,p}$, for $p \in (1,\infty)$.
...
- 11
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For any $p, q \in [1,\infty]$ and $s \in (0,\infty)$, can we find some $f \in L^q - W^{s,p}$?
Sobolev inequalities show us when we can embed a Sobolev space into another.
However, I wonder if these inclusions are always proper.
More specifically, let $\Omega \subset \mathbb{R}^n$ be a bounded ...
- 3,477
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Dual of homogeneous Triebel-Lizorkin
Let $ p, q \in (1,\infty)$ and consider the homogeneous Triebel- Lizorkin space $\dot{F}^{s}_{p,q}$ to be the space of all tempered distributions (modulo polynomials) with
$$
[f]^{p}_{\dot{F}^{s}_{p,q}...
- 31
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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
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Does the union of fractional Sobolev spaces fills $L^p$?
Let $\Omega\subset \Bbb R^d$ be any open set. Recall that for $s\in (0,1)$, the fractional Sobolev space $W^{s,p}(\Omega)$ is the collection of function in $L^p(\Omega$ such that
\begin{align*}
\iint_{...
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