Questions tagged [fractional-sobolev-spaces]

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

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fractional heat equation and spectral method

I want to apply a spectral method for the weak formulation of the equation $(-\Delta)^su=f$ $s>0$ with zero Dirichlet boundary conditions, where $(-\Delta)^s$ shall be the fractional laplacian on ...
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For $f \in H^s$, then $\exists g \in C_c$ such that $f=g$ a.e.

Consider the space $H^s(\mathbb R^d)$ ($f \in L^2$ not in Schwartz class), $s \in \mathbb R$. Apply Riemann-Lebesgue Lemma to $\hat{f}$ to show that for some $s>s_0$ then there is a continuous ...
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Comparing $(-\Delta)^{1/2}$ and $\nabla$ in $L^p$ space for $p \in (1,\infty)$

Let $\mathbb{T}^n := \bigl(\mathbb{R}/\mathbb{Z} \bigr)^n$ be $n$-dimensional torus. Then, I wonder if $(-\Delta)^{1/2}$ and $\nabla$ are "equivalent" in $L^p(\mathbb{T}^n)$ for each $p \in (...
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Extension of Sobolev functions on non-connected set [closed]

Let $D = D_1 \cup D_2 \cup...\cup D_m$ be a subset of $\mathbb{R}^d$, where each $D_j$ is a (connected and) bounded domain with Lipschitz boundary, and $\min_{i,j} dist(D_i,D_j) \geq c > 0$ for ...
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Continuous embedding of the fractional space $H^{\frac12}(\mathbb R)$ into $L^q(\mathbb R)$

My question is about the Theorem 6.5 in the Hitchhiker's guide to the fractional Sobolev spaces. Consider the case $n=1$ and $p=2$. Theorem 6.5 states that if $s<\frac12$, then $H^s(\mathbb R)$ is ...
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$\Delta (u^m)=\mathrm{div}(m u^{m-1} \mathrm{grad}(u))$?

How can one show that for a suitable $u$ the weak laplacian $\Delta (u^m)$ equals $\mathrm{div}(m u^{m-1} \mathrm{grad}(u))$ both in the weak sense ($m>1$) and what conditions need one to impose on ...
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Confusion about definition of Fractional Sobolev space II

I'm studying fractional Sobolev spaces and, similarly to a previous question of mine, I have some troubles to understand some definitions. Consider the Bessel potential spaces, defined as $$ H^{s,p}\...
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Confusion about Fourier transform and fractional Sobolev space definition

Consider the fractional Sobolev spaces defined as $$ W^{s,p}\left(\mathbb{R}\right):=\left\{ u\in L^{p}\left(\mathbb{R}\right):\int_{\mathbb{R}}\left(1+\left|\xi\right|^{sp}\right)\left|\widehat{u}\...
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Why the Sobolev norm is equivalent with the graph norm of the fractional Laplacian operator?

Why the Sobolev norm $\left\|u\right\|_{H^s}=\left\|(1+\xi^2)^{s/2}\widehat{u}\right\|_{L^2(\mathbb{R}^n)}$ is equivalent to the graph norm of the operator $(-\Delta)^{s/2}$, $$\left\|u\right\|_{(-\...
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Question on the definition of fractional sobolev norm

The Sobolev spaces with fractional order $H^s(\Omega)$ ($s>0,n\in\mathbb{N}, \Omega\subseteq \mathbb{R}^n$ open) in Hitchhikers guide is defined wrt. the norm $$\|u\|^2_{s}:= \|u\|_{H^m(\Omega)}^2+\...
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Definition of $H_0^s(\Omega)$ via Fourier transform

The Sobolev spaces with fractional order $H^s_0(\Omega)$ ($s>0,n\in\mathbb{N}, \Omega\subseteq \mathbb{R}^n$ open) can be defined as the closure of the test function space $C_0^\infty(\Omega)$ in $...
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Equivalent norms on $H^s$ $s>1$

Let $H^s(\mathbb{R}^n)$ be the Sobolev space of order $s>1$. I want to show that the norms $\|f\|_s^2:=\int\limits_{\mathbb{R}^d} (1+|\xi|^2)^s |\mathscr{F}f(\xi)|^2d\xi$ and $[f]_s^2:= \|f\|_{H^{m}...
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Connection between $H^s_0(\Omega)$ defined by Fourriertransform and $H^s_0(\Omega)$ defined canonically.

I am studying Sobolev spaces and have a couple of question to those. It is known that the space $H^s(\mathbb{R}^n):=\lbrace f\in L^2(\mathbb{R}^n) : (1+|\xi|^2)^s |(\mathcal{F}f)(\xi)|^2 \in L^2(\...
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Proving that a weighted Sobolev space is a Hilbert space

Consider $N = n+ k$ and in $\mathbb{R}^N$ and also take the variable $(x,y)$. In the article https://doi.org/10.1017/prm.2022.43 is defined for $\gamma \geq 0$ the space $$ H^{1,2}_{\gamma}(\mathbb{R}^...
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Functions in $H_0^s(U)$ disappear on their boundaries?

I have an issue in a proof I face. I have a function $f$ in $H^s_0(U)$, where $U$ is bounded with $C^1$-boundary. Then I would like to say that $f=0$ on $\partial U$ however this is not necessarily ...
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$L^2(G)$ convergence $\Rightarrow$ $H_0^s(G)$ convergence?

Under what conditions does a sequence $(f_n)$ in $H^s_0(G)$, that converges in $L^2(G)$, where $G\subseteq \mathbb{R}$ is bounded with smooth boundary imply convergence in $H_0^s(G)$ where $s>2$? ...
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Questions on Sobolev Spaces and the Laplacian Operator

In the following let $\Omega$ be an open and bounded domain in $\mathbb{R}^d$ with smooth boundary. As far as I understood, for the Laplacian $\Delta: H_0^1(\Omega)\to L^2(\Omega)$ with Dirichlet ...
Mathmaxis's user avatar
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Pre-compactness of $H^s(\mathbb R)$ when $s\in (\frac12, 1)$

Let $s\in (0, 1)$. By Corollary 7.2 in the paper https://arxiv.org/pdf/1104.4345.pdf, it is know that, if $s<1/2$, then $H^s(\mathbb R)$ is pre-compact in $L^p_{loc}(\mathbb R)$ for any $p\in [1, 2^...
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The space $H^s(\mathbb R)$ and the embedding in $L^p_{loc}(\mathbb R)$: for which $p$?

Let $s\in (0, 1)$ and consider the fractionl Sobolev space $H^s(\mathbb R)$ (see e.g. https://www.sciencedirect.com/science/article/pii/S0007449711001254). Let $(A, ||\cdot||)$ be a Banach space and ...
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Compactness criterion for subsets in a fractional sobolev space

The compactness criterion of frechet kolmogorof gives necessary and sufficient conditions on when a set in $L^p$ is compact. Given a set of function in a Sobolev space $W^{k,p}$ one can apply that ...
MackeyTopology's user avatar
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Is $H^s(\mathbb R^n)$ continuously embedded in $L^2(\mathbb R^n)$ when $2s>n$?

Let $1/2 < s<1$. I got a question about the fractional Sobolev space $H^s(\mathbb R^n)$. It is well known that, if $2s>n$, then $H^s(\mathbb R^n)$ is continuously embedded in $L^\infty(\...
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Compactness criterion on subsets of fractional Sobolev spaces

Are there sufficient conditions for a family of functions $F \subset H^s_0(\Omega)$ $(s>0)$ to be relatively compact in $H^s_0(\Omega)$, where $\Omega \subseteq \mathbb{R}^n$ is compact. $n$ is ...
MackeyTopology's user avatar
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Confusion with the definition of homogeneous Sobolev spaces

From Modern Fourier Analysis by Grefakos, the homogeneous Sobolev space $\dot{L}_s^p(\mathbb R^n)$ is the space of all $u \in \mathcal S'(\mathbb R^n) / \mathscr P(\mathbb R^n)$ for which the well-...
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Compact embedding of $D^{s,p}(\mathbb{R}^N)$ into $L_{loc}^{r}(\mathbb{R}^N)$ for every $r \in [1,p_{s}^*)$, where $p_{s}^{*}=\frac{Np}{N-sp}, sp<N$

We know that the embedding of $D^{s,p}(\mathbb{R}^N)$ into $L_{loc}^{r}(\mathbb{R}^N)$ is compact for every $r \in [1,p_{s}^*)$, where $p_{s}^{*}=\frac{Np}{N-sp}, sp<N$. Can anybody give me the ...
Rohit Kumar's user avatar
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Dense subspace of radial functions of $H^1(\mathbb{R}^N)$

By definition the space of radial functions in $H^1(\mathbb{R}^N)$ is $$ H^1_{rad}(\mathbb{R}^N) = \{u \in H^1(\mathbb{R}^N) : u = u \circ R, \forall R \in O(N)\}. $$ I'm trying to find a dense ...
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equivalence of Sobolev space in manifolds

Define the fractional Sobolev spaces (for $0<s<1$) in some Riemannian $N$-manifold $M$ as follows : $$W^{s,p}(M):=\left\{u\in L^p(M) \ \bigg| \ \iint_{M\times M}\frac{|u(x)-u(y)|^p}{(d_g(x,y))^{...
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Norm on $H^1/2(\partial\Omega)$ by first isomorphism theorem.

I want to show that $\operatorname{ran}(T)$ equipped with \begin{equation*} \lVert v\rVert:=\inf\{u\in H^1(\Omega):Tu=v\} \end{equation*} is a Hilbert-Space. I have seen that the definition of the ...
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Fourier multipliers on $L^2(\mu)$

On $L^2(\mathbb{R}^d)$, we have $T_m$ defined $\widehat{T_m f} = m \widehat{f}$ is a bounded operator on $L^2$ if and only if $m \in L^\infty$. What can be said about the same problem for more general ...
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Is $H(div; \Omega ) \cap H(curl; \Omega )$ compactly embedd in $L^2(\Omega)$?

I know a similar question says that $H_0^1(\Omega)=H_0(div;\Omega)\cap H_0(curl;\Omega)$, which was shown in Lemma~ 2.5 of the book "Finite Elements Methods for Navier-Stokes Equations" by ...
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$H^{2}_{0}(\Omega)$ is embedded in $L^{\infty}(\Omega)$?

In a research paper https://hal.science/hal-02891557/, the authors used the embedding of $H^{2}_{0}(\Omega)$ in $L^{\infty}(\Omega)$ (see page 16-line 1) to obtain some estimates in their research ...
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Relationship between Schwartz space and fractional Sobolev space

Let $u$ be an element of the Schwartz space $\mathcal{S}(\mathbb{R}^N)$. Given the fractional Laplacian operator $(-\Delta)^s:\mathcal{S}(\mathbb{R}^N) \to L^2(\Omega)$, with $s \in (0,1)$ , defined ...
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If $u\in D^{s, p}(\mathbb R^n)$, is that true that $u\in C(\mathbb R^n)$?

Let $0<s<1$, $p>1$ and $n>sp$. For $u\in C_0^{\infty}(\mathbb R^n))$ let $$ [u]_{s, p} =\left(\iint_{\mathbb R^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\right)^{\frac1p}$$ be the ussual ...
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maximum principle for compact manifolds

Are the maximum/minimum principles on $\mathbb{R}^n$ available for both local or nonlocal operators can be modified for a compact Riemannian manifold? For instance, we have a nonlocal maximum ...
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Estimate the fractionnal Sobolev norm of Hermite functions.

I consider the Hermite functions $$\psi_n(x) = (-1)^n (2^n n! \sqrt{\pi})^{-1/2} e^{x^2/2} \dfrac{d^n}{dx^n} e^{-x^2}$$ and I would like to bound above, for $r \in (0, 2)$: $$I_r(n) = \int_{\mathbb{R}}...
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How to prove$|\cdot|^{s}g\in\mathcal{S}'$? a question from Bahouri-Chemin-Danchin book "fourier analysis and nonlinear pde"

In page 27 and 28 of book "fourier analysis and nonlinear partial differential equations", proposition 1.36 the authors Bahouri-Chemin-Danchin give a proof of this proposition, but I really ...
monotone operator's user avatar
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1 answer
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What exactly is the "real interpolation space" $(L^q, W^{2,q})_{1-p^{-1},p}$ for $1<p,q<\infty$?

I came across the notation https://en.wikipedia.org/wiki/Interpolation_space#Real_interpolation the real interpolation space is discussed for Bessel potential spaces. However, I do not see exactly how ...
Keith's user avatar
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Show that a nonlocal operator with symmetric kernel is well defined

Assume that $J:\mathbb{R}^N\backslash \{0\} \to (0, \infty)$ is a symmetric function; that is, $J(x)=J(-x)$ for any $x \in\mathbb{R}^N$. Moreover, we assume that there is a constant $J_0 > 0$ and $...
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The integral $\int_{\mathbb{R}^N} \int_{|x - y| \geq 1} \dfrac{|u(x)|^p}{|x - y|^{N+sp}}dxdy$ is finite?

Let $s \in (0,1)$ and $p \in [1,\infty)$. If $u \in L^p(\mathbb{R}^N)$, it is possible to conclude that the integral $$\int_{\mathbb{R}^N} \int_{|x - y| \geq 1} \dfrac{|u(x)|^p}{|x - y|^{N+sp}}dxdy$$ ...
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How do I show that the definition of Sobolev–Slobodeckij spaces $W^{s,p}(\Omega)$ coincides with the classical one when $s$ is integer.

For any nonnegative real number $s$ and $1<p<\infty$, the Sobolev–Slobodeckij space $W^{s,p}(\Omega)$ is defined as in the following link: https://en.wikipedia.org/wiki/Sobolev_space#Sobolev%E2%...
Keith's user avatar
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3 votes
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Inequality involving the Gagliardo seminorm

Does the following estimate hold for any value of $p \in [1,+\infty)$? $$\int\int_{\mathbb{R}^N \times \mathbb{R}^N} \dfrac{|u(x) - u(y)|^p}{|x - y|^{N + sp}}dxdy \leq 2\left[\left(\int\int_{\mathbb{R}...
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Weak convergence in $W^{2, p}_{loc}(R^{N})$

If $(u_{n})$ is a sequance in $W^{2, p}_{loc}(R^{N})$ ($1< p < \infty$) such that $u_{n}(x)=0$ if $|x|\geqslant n$, $u_{n} = u_{n+1}$ in $B(0,n)$, $u_{n}\in C_{0}^{2}(B(0,n))$ and $\|u_{n}\|_{W^{...
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Proving that a $W^{1,2}$ type space is Hilbert

Split the $N-$dimensional Euclidian space as $\mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$. A vector in $\mathbb{R}^N$ will be denoted by $z = (x,y)$. Let $\alpha > 0$ and consider the ...
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Dirichlet and Neumann Traces in $H^{2}(\Omega)$, $\Omega$ Lipschitz and Unbounded

Let $\Omega$ (possibly unbounded) be a domain of $\mathbb{R}^{n}$ with smooth boundaries. Consider $$\gamma_{D}^{0}:C^{\infty}(\overline{\Omega})\to C^{\infty}(\partial\Omega)\\\gamma_{0}^{D}u=u|_{\...
SpuriousMatemagician's user avatar
1 vote
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Multiplication in fractional sobolev spaces

Assume that $f(t)$ belongs to $W^{s_1,2}(0,T)$ and $h(x,t)$ belongs to $W^{s_2,2}(0,T;H)$ for some $s_1,s_2<\frac12$ where $H$ is a Hilbert space. It is known that for any $s<s_1+s_2-\frac12$, $...
math's user avatar
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Functional convergence in $W^{s,p}(M)$

Let $M$ be a compact Riemannian $N$-manifold . Let $s\in(0,1)$ and $p\in(1,\infty)$ . Define $(u_n)$ a bounded sequence in $W^{s,p}(M)$ . I want to verify following statements (up to some subsequence) ...
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Proving the existence of a solution of the fractional heat equation using semigroup methods

I am trying to solve the following problem: $$u_t + (-\Delta)^su = 0$$ in $\Omega \subset \mathbb{R}^N$ with $N > 2s$, where $s \in (0,1)$ and Dirichlet Boundary conditions. Let my operator $A = (-\...
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Embedding in compact manifold

I want to verify the statement below : Let $(M,g)$ be a compact Riemannian $N$-manifold, $p\in[1,\infty)$ and $s\in(0,1)$. Then $$||u||_{W_0^{s,p}(M)}\leq C||u||_{W^{1,p}(M)}$$ for some suitable ...
am_11235...'s user avatar
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Using Semigroup Theory of Linear Operators to show that the operator $(-\Delta)^s$ is closed.

Consider the fractional Laplacian defined by $$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$ Also consider that $$D((-\Delta)^s) = \{u \in H^s(\...
José's user avatar
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A continuous function in $W^{1,p}(I)$ function with continuous weak derivative is a $C^1(\overline{I})$ function

In Theorem 8.2. of Brezis's book of Functional Analysis, says that a function $u \in W^{1,p}(I)$ has a continuous representante $\tilde{u}$ such that $$ \int_y^x u'(t) dt = \tilde{u}(x) - \tilde{u}(y)....
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Associating a non-local differential operator to its integral representation

It is known that, in $\mathbb{R}^2$, we can define the non-local operator $\frac{1}{\Delta}$ with the Green function of the Laplace operator $\Delta$. This provides the non-local operator with an ...
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