Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [fractional-sobolev-spaces]

This tag is devoted to any problem concerning fractional Sobolev spaces which are approximation of the classical Sobolev spaces

0
votes
0answers
28 views

compute $H^{3/2}(\partial\Omega)$-norm for smooth $u$ and $\Omega$

I am a little bit confused about different definitions of the trace space $H^{3/2}(\partial \Omega)$, and I hope I can find some simple examples on how to explicitly compute these norms for simple ...
1
vote
0answers
33 views

Gagliardo-Nirenberg inequality for fractional Sobolev spaces

Wikipedia states two versions of the Gagliardo-Nirenberg inequality for nonfractional Sobolev spaces. I'm interested in generalizations to fractional (Slobodeckij) Sobolev spaces. Such a ...
1
vote
0answers
24 views

Reference request: Laplace-Beltrami eigenfunction bases for Sobolev spaces

I'm working on a smooth $(d-1)$-dimensional surface $M\subset \mathbb{R}^d$. Let $(\phi_k)_{k\in\mathbb{N}}$ be an orthonormal basis of $L^2(M)$ consisting of the eigenfunctions of the Laplace-...
1
vote
1answer
33 views

Question about local Sobolev spaces $H^s_{loc}(\mathbb{R}^n)$

We define the local Sobolev space $H^s_{loc}(\mathbb{R}^n)$ as $$ H^s_{loc}(\mathbb{R}^n)=\big\{f\in\mathcal{D}^{\prime}(\mathbb{R}^n): \forall \Omega\Subset\mathbb{R}^n \ \exists g_{\Omega}\in H^s(\...
1
vote
0answers
14 views

Is there a french version for this paper “Critical exponents of Fujita type for certain evolution equations ..”?

I seek many times in the web to get the French version of this paper entitled " Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional ...
0
votes
0answers
10 views

Principal value definition of the fractional Laplacian on the torus

I have a periodic function $u$ defined on the torus $\mathbb{T}^2= \mathbb{R}^2/\mathbb{Z^2}$ and want to find the corresponding definition of the fractional Laplacian $(-\Delta^{s})u$ in terms of the ...
1
vote
0answers
17 views

General continuous embedding in fractional Sobolev space

Let $\Omega$ be a bounded domain. Then there exists a continuous embedding between the fractional Sobolev spaces $W^{s_1,p}(\Omega)\rightarrow W^{s_2,p}(\Omega)$ for $s_1>s_2$. But does there exist ...
8
votes
1answer
76 views

Heat semigroup norm between fractional Sobolev and $L^p$ spaces

What is the actual inequality that holds for the heat semigroup between fractional Sobolev space $W^{2\alpha,p}$ and classical Lebesgue space $L^q$? I am trying to derive an inequality $$ \lvert\...
2
votes
0answers
45 views

Can we extend the Riesz potential convolution operator for the Laplacian to a continuous operator from $L^p$ to $\mathcal{S}'$ if $p\ge\frac{n}{2}$?

If $n\ge3$ and $\omega_n$ is the $n-1$-dimensional Hausdorff measure of the unit sphere in $\mathbb{R}^n$, define: $$K_n(x):=\frac{1}{(n-2)\omega_n} \frac{1}{|x|^{n-2}}.$$ Then $K_n$ is locally ...
2
votes
0answers
34 views

$H^s(\mathbb R^d) \subset \bigcap_{2<p<\infty} L^p(\mathbb R^d)$ $0<s<1/2$?

Consider Sobolev spaces $$ H^s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^2(\mathbb R^d) \}$$ where $\langle \cdot \rangle = (1+ |\...
6
votes
1answer
137 views

Sobolev embedding for the $L^q$ norm.

Suppose $f \in H^1(\mathbb R^2)$, where $H^1$ is the Sobolev space, then how to use this information to bound $\Vert f \Vert_{L^q}$, where $q>2$? It seems like Sobolev embedding, but it's not.
1
vote
0answers
19 views

Density in fractional Sobolev space

Suppose $s\in (0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1-\Delta)^{-s/2} L^2(\mathbb{R}^d), $$ $$ H_D^s=\{f\in H^s: f=0 \ \ a.e. on \ D^c\}. $$ Q: Is $C_c^\infty(D)$ dense in $...
1
vote
0answers
43 views

On boundedness in fractional Sobolev norm

Let us asssume that a sequence $(v_n)$ satisfies (i) $v_n\in{\rm W}^{2,\infty}(0,1)$, (ii) $||v_n||_{{\rm H}^{\theta}(0,1)}$ is bounded for some $0<\theta<1$. Is it true that then the ...
2
votes
1answer
53 views

About pseudo-differential operators

Let $\Omega$ be an open and connect subset of $\mathbb{R}^2$,we denote by $\partial \Omega$ its boundary the latter is supposed to be smooth ($\mathcal{C}^\infty)$, its outword normal vector is ...
1
vote
1answer
37 views

Uniform boundedness of weak solution

Let $u_n\in W_{0}^{1,p}(\Omega)$ be a positive weak solution of the equation: $$ -\Delta_p u=\frac{f_n(x)}{(u+\frac{1}{n})^\delta}\text{ in }\Omega. $$ Let $p=N$ and $f\in L^m(\Omega)$ for some $m&...
2
votes
0answers
40 views

Composition with Lipschitz map is Lipschitz on Sobolev spaces

Suppose that $F: \mathbb{R}^d \rightarrow \mathbb{R}^d$ is Lipschitz with some constant $L$ and that $F(0)=0$. Then it is clear that $F$ defines a Lipschitz continuous map $L^2(\mathbb{R}^d) \...
0
votes
0answers
18 views

Unable to determine Sobolev space

I have the following function: $T ( \textbf{r}) = \frac{q}{4 \pi k} \int_{s_i=0}^{L_i} \frac{1}{\left| \textbf{r}-\textbf{r}'\right|} ds_i = \frac{q}{4 \pi k} \log\left( \frac{\tan(\theta_2/2)}{\tan(\...
0
votes
1answer
48 views

Use trace theorem to define $H^2_0$ space and the requirement of the boundary?

For homogeneous biharmonic problems, the solution space is in general defined as $$H^2_0(\Omega):=\{u\in H^2(\Omega): u=\frac{\partial u}{\partial \mathbf{n}}=0\text{ on }\partial \Omega\}.$$ With ...
3
votes
1answer
40 views

A inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator?

Does someone know any inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator? I need to find something like $||u||_{p}\leq C||\gamma (u)||_{p, \...
3
votes
0answers
40 views

Fractional embedding inequality with $L^{\infty}$ norm

Here we consider the fractional Sobolev spaces and suppose $u$ is a vector function in $\mathbb R^2$. Is the following always true? $$\Vert Du \Vert_{L^{\infty}(\mathbb R^2)} \leq C\Vert Du \Vert^{1-\...
4
votes
0answers
96 views

Fractional Hardy inequality

From classic literature, I know the following result. Let $\Omega\subset\mathbb{R}^d$ be a bounded open set of class $C^1$. Then there exists $C>0$ such that \begin{equation}\label{1} \|\frac{...
1
vote
0answers
44 views

Definition of homogeneous Sobolev spaces $\dot W^{-s,q}$

For the homogeneous Sobolev spaces $\dot H^{-s}$, we can define its norm like this: $$\Vert f \Vert_{\dot H^{-s}}=\Vert \Lambda^{-s} f\Vert_{L^2}$$, i.e., we use fractional derivatives to define it. ...
0
votes
0answers
34 views

Bounding $H^{-1/2}(\Gamma)$ by $H^{1/2}(\Gamma)$

Let $\Omega\subset\mathbb{R}^2$ a bounded domain with polygonal boundary $\Gamma$ (for example, $\Omega$ could be a triangle). Let $f\in H(div,\Omega)$. There exist some result bounding the $H^{-1/2}(...
4
votes
0answers
42 views

Definition of $H^{s}(\mathbb{R^{+}})$ and it's norm

What's the definition of $H^{s}(\mathbb{R^{+}})$(classical Sobolev space on the half line) and its norm in terms of Fourier Transform? I'm aware of the definition of classical Sobolev Space $H^{s}(\...
3
votes
1answer
111 views

Interpolation inequalities

Let $\Omega$ be a regular domain of $\mathbb{R}^d$, $d=2,3$. Let $\mathcal{T}_h$ be a triangulation of $\Omega$ of size $h>0$. Assume we can prove \begin{equation*} \begin{aligned} \|v\|_{L^2(\...
0
votes
1answer
48 views

Sum of two functions is in $L^2(\mathbb{R}^n)$, does that imply each of them is in $L^2(\mathbb{R}^n)$

Let $u \in L^2(\mathbb{R}^n)$ be a measurable function such that $\Delta u - (-\Delta)^{\lambda} u \in L^2(\mathbb{R}^n)$ for $\lambda \in (0,1)$. Does that imply each of $\Delta u$ and $(-\Delta)^{\...
1
vote
0answers
26 views

$s-$fractional Laplacian

Let $$u(x) = \dfrac{1}{|x|^{N-2s}},N \neq 2s.$$ I want to show that $(-\Delta)^s u(x) = 0,\forall x \neq 0$, where $(-\Delta)^s $ is the $s-$fractional Laplacian, defined by $$(-\Delta)^s u(x) := C(N,...
0
votes
0answers
34 views

Palais-Smale Condition of a functional

Palais-Smale condition: Defined as in the strong formulation wiki link: https://en.wikipedia.org/wiki/Palais%E2%80%93Smale_compactness_condition I am unable to check how the following functional $J$ ...
2
votes
0answers
30 views

Morrey embedding for Potential Spaces

I'm trying to prove that for $f\in H^{s,p}:=\{f/f=G_s*g,\,\,g\in L^p\}$ where $G_s$ is the bessel potential: $G_s(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}(1+|w|^2)^{-s/2}e^{iw\cdot x}dw$ the following ...
0
votes
0answers
20 views

A doubt in the proof of a proposition about the constant of fractional laplacian

I was reading about the asymptotic behavior of the constant $C(n,s)$ in the paper "Hitchhiker's guide to the fractional Sobolev spaces". In the proposition 4.1 I don't understand the part where they ...
1
vote
0answers
51 views

Sobolev spaces on boundary and higher order traces

I'd like to clarify the definitions of the spaces involved in the theory of traces of Sobolev functions. Actually I'm referring to the books of Kufner and Necas. In these books the boundary integral ...
1
vote
1answer
65 views

Definitions of fractional Sobolev Spaces

In a paper I read that for a bounded domain $W^{s,p}(\Omega)=\{ u \in L^p(\Omega), (id-\Delta)^{s/2}u \in L^p(\Omega) \}$ and $s$ is not assumed to be an integer, $p \neq 2$ in general. If $p=2$ the ...
5
votes
0answers
94 views

Is being in the Sobolev space of power $\frac{d}{2}$ necessary for having well defined point evaluations?

From the Sobolev embedding theorem we know that for $\alpha = \frac{d}{2}$, $W^{\alpha, 2}(\mathbb{R}^d)$ is continuously embedded in $C^0(\mathbb{R}^d)$. Especially the point evaluations are in the ...
1
vote
0answers
34 views

Fractional reaction diffusion equation

Consider the linear fractional reaction diffusion equation: $\begin{align} \frac{\partial u}{\partial t} + \alpha(-\Delta)^{s} u = 0, \\ \label{pr: fractionallineal2} u(x,0)=u_{0}(x) \end{align} ...
1
vote
1answer
147 views

Why this is called semi norm?

How to prove the following is semi-norm $$[u]_{s,p}=\Bigg(\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy\Bigg)^{1/p}$$ where $\Omega$ open set in $\mathbb R^n$, $1\leq p<\infty$, $...
0
votes
0answers
29 views

Practical applications of intermediate spaces and interpolation theorems

I am reading the book of Lions and Magenes about intermediate spaces (see Def. 2.1 in p. 10) and interpolation theorems (see p. 27) and I wish to know what are the practical applications of these ...
4
votes
0answers
70 views

About multiplication operators on fractional Sobolev space

Let $\Gamma$ be a regular boundary of a $C^{k,1}$ domain $\Omega$ and $H^s(\Gamma)$, $s\in(0,1)$, denote the fractional Sobolev space on $\Gamma$. Suppose I define a multiplication operator $M_\phi:H^...
3
votes
0answers
92 views

A Generalized Hölder Inequality for Sobolev Spaces

One has the following "generalized version" of Hölder's inequality: $$ \| u v \|_{L^1} \leq \| u \|_{W^{-s , p}} \| v \|_{W^{s , p'}} $$ where $s \geq 0$ and $\frac{1}{p} + \frac{1}{p'} = 1$, $1 <...
2
votes
0answers
21 views

Does $\|Tf\|_{H^s} \ge C \|f\|_{L^2}$ for all $f \in L^2$ imply $\mathrm{range}(T) \supset H^s$?

Let $T \in B(L^2(\mathbb{R}^n))$ be a bounded linear operator and $0 \le t \le s$. Suppose there exists $C_1, C_2 > 0$ such that for all $f \in L^2$ \begin{align} \| Tf \|_{H^s} &\le C_1 \| f \|...
2
votes
0answers
134 views

Dual of the fractional Sobolev space $W^{s,p}(\mathbb{R}^n)$

For $s\in\mathbb{R}$ and $1<p<\infty$, one can define the fractional Sobolev space $W^{s,p}(\mathbb{R}^n)$. Every element $f$ in $W^{s,p}(\mathbb{R}^d)$ is a tempered distribution such that the ...
1
vote
1answer
37 views

Domaine of an adjoint operator

Let $A$ be the derivative operator which defined on $L^2(0,1)$ with domain $$D(A) = \left\{ {v \in {H^1}(0,1),v(0) = 0} \right\}.$$ It is obvious that with an integration by part we find that $A^*=A$ ...
2
votes
2answers
114 views

Second variation corresponding to the functional

I am facing difficulty to calculate the second variation to the following functional. Define $J: W_{0}^{1,p}(\Omega)\to\mathbb{R}$ by $J(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^p\,dx$ where $p>1$. ...
0
votes
0answers
32 views

Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space

It is known that for $\alpha\in(0,1)$ and $p>1$, the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by $$ W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^...
1
vote
1answer
130 views

Bi Laplacian operator explicitly form

let $u \in H^2(\mathbb{R}^n)$. My question is: please how we can write explicitly the operator $$ u- \Delta u + \Delta^2 u? $$ We can write it into the form $\sum_{|\alpha|\leq 2} (-1)^{|\alpha|} D^{...
0
votes
0answers
68 views

comparing two Sobolev spaces:$W_0^{1,p}(\Omega)$ and $\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{1,p}(\Bbb R^d)}$

Given $\Omega$ a bounded regular open set in $\Bbb R^d$ we consider $C_c^\infty(\Omega)$ the space of smooth functions compactly supported in $\Omega$. For $1<p<\infty $ Let's denote by $W^{1,...
0
votes
1answer
61 views

$H^s$ norm of a Fourier transform

I have to evaluate the $H^s$ norm of the Fourier transform of a function, $\hat f(\lambda,t)=\mathscr{F}(f(\xi,t))(\lambda,t)$. According to the definition (that I know) of the norm of a Sobolev ...
0
votes
0answers
52 views

Counter example for analogous Poincare inequality does not hold on Fractional Sobolev spaces

Let $\Omega$ be open bounded set in $\mathbb R^n$,$0<s<1$, $1\leq p<\infty$. Does the following inequality holds, $$||u||_p\leq c [u]_{s,p}, \forall u\in W^{s,p}_0(\Omega)$$ Where $[u]_{s,p}$...
2
votes
0answers
62 views

Localization of Sobolev functions is continuous

I'm reading the book Grigis, Sjöstrand, Microlocal Analysis for differential operators. Introducing fractional Sobolev spaces, it says that for all test functions $\phi\in C^\infty_0$ and any $u\in H^...
3
votes
0answers
54 views

For which functions $u$ is $\partial_\nu u \in H^{-1/ 2}(\partial\Omega)$?

Let $\Omega$ be a smooth bounded domain, with outward normal vector $\nu$. Under what conditions on a function $u$ do we get that $$ \partial_\nu u \in H^{-1/ 2}(\partial\Omega). $$ That is, under ...
3
votes
1answer
443 views

Embeddings between Hölder spaces $ C^{0,\beta} \hookrightarrow C^{0, \alpha} .$

Let $ \Omega \subset \mathbb R^n $ be an open subset and let $ 0 < \alpha < \beta \leq 1.$ We consider the space of Hölder continuous functions $C^{0, \alpha}$ which is a Banach space endowed ...