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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Trace inequality for a family of domains.

I'm curious to know if the following is true. Suppose $\{\Omega_{\alpha}\}_{\alpha}$ is a family of smooth, bounded domains in $\mathbb{R}^n$ (or more generally $C^1$ domains) with $\Omega_{i}\cap\...
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A inclusion of sobolev space $W^{2,p}$ in a Holder space

Let $\Omega \subset \mathbb{R}^{N}$ be a bounded smooth domain and $L$ a uniformly elliptic operator given by $$ Lu = -div(A(x) \nabla u) + \langle b(x), \nabla u\rangle + c(x) u, $$ where $b = (b_{1},...
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Integration by parts involving $\Delta^\frac{-1}{2}$

Let $f$ and $g$ are well behaved functions. Using the Laplacian and the gradient, is the following equality true? $$\nabla(f \Delta^\frac{-1}{2}g)= \nabla f \Delta^\frac{-1}{2}g+ f \nabla \Delta^\frac{...
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Definition of space $\mathcal{D}^{1,2}$

My question is the regular definition of the space $\mathcal{D}^{1,2}(\Omega)$, for some open $\Omega \subset \mathbb{R}^3$. I know this space is usefull for the studies of solution for PDE problems. ...
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Integration by parts for Bessel's operator.

I am having the following integral: $$I = \int u\, J^s(\partial_x \overline{u})- \overline{u}\, J^s(\partial_x u))dxdy$$ where $J^S= (I-\Delta)^\frac{2}{2}$, $\mathbb{R} \ni s \geq 1$ and $u=u(x,y)$, $...
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Uniform norm of eigenfunctions of laplacian

Let $\{\phi_i\}$ the sequence of the eigenfunctions of laplacian operator on a domain $\Omega$, that is, considering $\{\lambda_i\}$ the respective eigenvalues, we have $$ \int \nabla \phi_{i} \cdot \...
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Prove or disprove the compactness of an operator?

Consider $X=L^{2}(0,\pi, \mathbb{R})$. Let $X_{\frac{1}{2}}$ be the domain of $(\Delta)^\frac{1}{2}$ where $\Delta$ is the laplacien operator. We define the operator $K:C([0,a],X_{\frac{1}{2}})\...
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Derivative of Riesz Transform

I am trying to find a bound for the $L^p$ norm, $1<p\leq \infty$ of the Riesz transform of $f^2$, where $f \in S$. $$\mathcal{F}[{\mathcal{R}_x f}](\xi,\eta) = -i \frac{\xi}{|(\xi,\eta)|}\hat{f}(\...
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Elliptic regularity with right-hand side in $H^{-1/2}$

If $\mathbf{f} \in H^{1/2}(\Omega)$ for a bounded domain $\Omega$ with smooth boundary, does the elliptic regularity for the Laplacian guarantee that the solution to the elliptic problem in divergence ...
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Weak convergence in $H^1(\mathbb{R}^N)$ implies weak convergente in $H^1(B)$

Let $\{u_n\}$ be a bounded sequence in the sobolev space $H^1(\mathbb{R}^N)$ converging weakly to some $u \in H^{1}(\mathbb{R}^N)$. How can I prove that some subsequence converges to the same limite ...
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Sobolev inequality for cubes

Consider the fallowing result from Evans, 2010, page 279: Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose $\partial U$ is $C^1$. Assume $1 \leq p < n$, and $u \in W^{1,p}(U)$. Then ...
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Relationship between Sobolev-Slobodeckij spaces and Besov spaces

I am trying to understand these two different ways of defining fractional Sobolev spaces. In particular, I want to determine embeddings or equality between the Besov spaces $B^{s}_{p,p}$ and the ...
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Function in $H^1(\mathbb{R})$

Consider the function $$f=|x|^pe^{-x^2},$$ where $p$ is a real constant. The function $f$ is in $L^2(\mathbb{R})$ iff $p>-1/2$. The function $f$ is in $H^1(\mathbb{R})$ iff $p>1/2$ or $p=0$. I ...
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A coercive bilinear form

Let $\alpha > 0$ and $X = [H^{2}(\Omega)\cap H^{1}_0(\Omega)] \times H^{1}_{0}(\Omega)$. Find $\lambda_0 > 0$ for which the bilinear form $B: X \times X \rightarrow \mathbb{R}$ given by $$ B((u,...
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Power of Sobolev function: if $ u \in H^2(\Omega) $, will $ |u|^{1+p} \in H^s(\Omega) $ for some $ s>1 $? Assume that $ p \in (0, 1) $.

Here, $ \Omega \subset \mathbb{R}^n $ is a bounded domain which can be as nice as you want (such as $ \mathbb{T}^n $ or $ B_r(0) $). I have an $ H^2 $ funciton $ u $ and I want to know if there is ...
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Can we use Newton-Leibnitz for $W^{1,p}$ function

For a function $u\in W^{1,p}$, we always use a sequence of $C^1$ function to approach it and derive the consequences. However, can we just claim for a.e. x, y: $$u(x)-u(y)=\int_0^1 Du(y+t(x-y))\cdot (...
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Interpolation of Bochner-Sobolev spaces

Does anyone know how to prove or has an explicit reference for interpolation inequalities of the form $$ \|f\|_{H^{l}(0,T;H^{(1-l)}(\Omega))} \leq C \|f\|_{H^{1}(0,T;L^2(\Omega))}^l\|f\|_{L^2(0,T;H^1(\...
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About the continuity of the integral on the boundary of a ball of a $H^1$ function

I’m considering a $H^1$ function u on a open domain D. Is the integral: $$ \int_{\partial B_r(x)} u \hspace{2pt}dH^{n-1}$$ continuous with respect to x? I tried to prove that it’s differential by ...
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Every $H^1_0$ function is bounded?

Let $U$ be a bounded open set with smooth boundary. We know that a function $u$ in $C^{\infty}_0(U)$ is bounded, because it has compact support and is continous. But is it true that a function $u \in \...
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The trace theorem for functions in $H^{1/2}(\Omega)$

In my textbook, it said that we have the trace operator on Sobolev space like this: (Suppose $\Omega$ is a nice domain in $R^d$) \begin{equation*} H^{s}(\Omega) \hookrightarrow H^{s-\frac{1}{2}}(\...
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Inequality $(\alpha f,g)_{L^2}\leq C\|\alpha\|_{W}\|f\|_{H^{1/2}}\|g\|_{H^{3/2}}$

Let $f\in H^{1/2}(S)$, $g\in H^{3/2}(S)$, $\alpha\in L^\infty(S)$, where $S$ is a closed smooth $n$-dimensional surface (e.g, $n$-dimensional sphere). Problem: to find such Banach space $W$ that the ...
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Application for the trace inequality.

I know two theorems about the trace inequality. Suppose that $\Omega$ is a bounded domain with smooth boundary. One is that: $$ \gamma_0(H^1(\Omega)) = H^{\frac{1}{2}}(\partial \Omega) $$ where $\ \ ...
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Primitive of a function

Let $\delta>0$. Then for $k>0$, let $g_k(l)=\min\{l^{-\delta},k\}$ if $l>0$ and $g_k(l)=k$ if $l\leq 0$. Let $G_k$ be the primitive of $g_k$ such that $G_k(1)=0$. Then we have $$ \int_{\Omega}...
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Cauchy principal in the definition of fractional laplacian

Let $\alpha \in \mathbb{R}$,$\ 0<\alpha <2 \ $, the fractional laplacian is defined as: \begin{equation}\label{def} (-\Delta)^{\frac{\alpha}{2}} u (x) := C_{n,\alpha} \lim\limits_{\epsilon \...
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$W^{2,2} \subset W^{1,p}$ for same $p > 2$?

Consider the following problem: Let $U \subset \mathbb{R}^{n}$ be an bounded open set. Find conditions on $p$ for which $W^{2,2}(U) \subset W^{1,p}(U)$. What I have done: I have proved that $W^{2,2}(U)...
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Elliptic lifting theorem of Laplace-Beltrami Operator.

We know that the Laplace-Beltrami Operator is widly used in many areas, whose definition can be found here. I have learned basic knowledge about PDE and Sobolev spaces, but I know little about the ...
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1 answer
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Fractional Sobolev spaces in 1D and the function $f(x) = x^{\alpha}$

If we consider the function $f(x) = x^{\alpha}$ for $\alpha \in (1/2,1)$ and $x \in \Omega = (0,1)$, this function should be in the Sobolev-Slobodeckij space $H^{1+\sigma}(\Omega) (= W^{1+\sigma,2}(\...
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Fractional Sobolev regularity of the solution to the fractional Dirichlet problem on a ball

While reading some notes due to Markus Faustmann, I came across the following exercise regarding the fractional Dirichlet problem on the unit ball. \begin{equation*} \left\{ \begin{aligned} (-\Delta)^\...
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4 votes
1 answer
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How to define real fractional Sobolev space?

For $s>0 $ define $H^s(\mathbb{R}^n)=\lbrace u\in L^2(\mathbb{R}^n): (1+\vert y\vert^2)^\frac{s}{2}\hat{u}\in L^2(\mathbb{R}^n)\rbrace$. This is a Hilbert space with the inner product given by $$\...
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2 votes
0 answers
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Fractional Sobolev space norm given by Laplace Beltrami operator

I'm currently reading Lions and Magenes Non-Homogeneous Boundary Value Problems and Applications and I'm stuck at one point. Let $\Gamma$ be the smooth boundary of an open bounded subset $\Omega\...
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2 answers
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Why is dirac delta in $H^{-1/2-\epsilon}$ and why is multiplication not well-defined in that space?

I read on this Sobolev space with negative index question that the dirac delta distribution belongs to $H^{-1/2-\epsilon}(\mathbb{R})$. I was trying to figure out why that is the case but it wasn't ...
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Does there exist a function in $B_{\infty,1}^{1/2}\setminus H^{1/2}$?

Does there exist a function in $B_{\infty,1}^{1/2}([0,1])\setminus H^{1/2}([0,1])$? If so can it be constructed explicitly? A reference will be much appreciated.
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Sobolev embedding theorem on Manifolds

I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional Sobolev spaces. The references I know either don't deal with fractional spaces, or don't ...
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Does $\int_{E^c}\int_{E}\frac{dy\, dx}{|x-y|^{n+\alpha}}$ converge for any $E$?

Question: Given an $\alpha\in(0,n)$, does there exist a measurable set $E\subset\mathbb R^n$ with $\mu(E)\in(0,\infty)$ such that $$\int_{E^c}\int_{E}\frac{dy\, dx}{|x-y|^{n+\alpha}}<+\infty$$ ? (...
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3 votes
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Bounding the Laplacian of a function using its boundary data

Let $U \subset \mathbb{R}^n$ be a bounded domain with smooth boundary, which consists of two smooth disjoint pieces $\partial U = \Gamma_1 \cup \Gamma_2$. Let $f \in H^2(U)$ be a given function and ...
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1 answer
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Sobolev spaces are Hilbert spaces

Define $H^s(\mathbb{R}^n)=\lbrace u\in \mathcal{S}'(\mathbb{R}^n): (1+\vert y\vert^2)^\frac{s}{2}\hat{u}\in L^2(\mathbb{R}^n)\rbrace$ where $\mathcal{S}'(\mathbb{R}^n)$ is the space of tempered ...
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1 vote
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Intersection of the kernel with the interpolation space

Given two Banach spaces $X$ and $Y$ with a continues inclusion $X\subset Y$, and another couple $X’ \subset Y’$ with the same properties. Take $f : Y \longrightarrow Y’$ linear continues, such that $...
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How to show two norms are equivalent on periodic Sobolev space

Let $f\in H^s[0,2\pi]$, be Sobolev space of periodic functions then we have the following norm $$ \Vert f\Vert_{s}=\{ \sum_{m\in \mathbb{Z}} (1+m^2)^s\vert \widehat{f_m}\vert^2 \}^{1/2}$$ (Given in ...
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2 answers
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techniques to know whether given function is in $H^s(\Omega)$ or not?

Suppose $f$ be a function defined as $f_m(x)=x^m(1-x)^m$ if $x\in [0,1]$ and then we extend it by zero in $[0,3]$. Since $f_1(x)$ is not differntiable at $x=1$. I would like to know how to calculate ...
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  • 107
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Rotation invariance of fractional p-Laplacian

Let $(-\Delta_p)^s$ is the fractional $p$-Laplace operator defined in the usual principal value sense. In the local case, $s=1$, the p-Laplace operator is rotationally invariant under an orthogonal ...
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High-order Sobolev function on sphere

For a bounded $\Omega\subset \mathbb{R}^n$ with Lipschitz boundary, there are various definitions of fractional Sobolev spaces (a.k.a. Sobolev-Slobodeckij spaces) on $\partial \Omega$, either by using ...
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1 vote
1 answer
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Show that Fourier transform is isomorphism between $H^m_r (\mathbb R)$ and $H^m_r (\mathbb R)$.

Let $H^m_r (\mathbb R) = \{u \in L^2(\mathbb R): \| \rho^r u \|_{H^m} < \infty \}$ and $H^r_m (\mathbb R) = \{u \in L^2(\mathbb R): \| \rho^m u \|_{H^r} < \infty \}$ where $\rho(x) = (1+|x|^2)^{\...
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  • 1,640
4 votes
1 answer
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Restriction of fractional Sobolev "function" of negative order to subset

Assume $ U\subset V\subset \mathbb{R}^n$ are bounded open subsets with smooth boundary. We define $H^{-s}(\Omega)=(H_0^{s}(\Omega))'$ for $s>0$. It is straightforward to show that $\left. v\right|_{...
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$H^{-3/2}$ control of the laplacian of a function by means of $H^{1/2}$ norm in $\mathbb{R}^2$

I am wondering for my research if, given a smooth bounded domain $\Omega\subset\mathbb{R}^2$ then, for a sufficiently regular function $u$ we can write $$ \Vert \Delta u\Vert_{H^{-3/2}(\Omega)}\leq C\...
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Proving $(1+|\xi|^2)^{-s/2}\widehat{u} \in L^2(\mathbb{R}^n)$

Show that $(1+|\xi|^2)^{-s/2}\widehat{u} \in L^2(\mathbb{R}^n)$ where $u$ is a tempered distribution ($u \in S^{\prime}(\mathbb{R}^n)$) and is a linear functional on $H^{s}(\mathbb{R}^n)$ for $s>0$....
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63 views

Norm on the Sobolev space

Let $0<s<1<p<\infty$ and $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$. Define the space $X=W^{1,p}(\Omega)\cap W^{s,p}(\Omega)$ endowed with the norm $\|\cdot\|_{X}$ defined by $$ ...
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  • 409
1 vote
1 answer
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If $u$ is a function between two functions in $H_{0}^{s}(\Omega).$ Then $u$ is in $H_{0}^{s}(\Omega)$ too?

Let $s\in(0,1),$ $\Omega\subset\mathbb{R}^{N}$ regular and limited, $v\in H_{0}^{s}(\Omega)$ and $u:\Omega\rightarrow\mathbb{R},$ such that, \begin{equation} \iint\limits_{\Omega\times\Omega}\frac{|u(...
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2 votes
0 answers
46 views

Showing that every weak solution in a pseudo differential equation with elliptic symbol is in Sobolev space.

Problem 16.3 Wong, An introduction to pseudo differential operators. Let $\sigma\in S^m,\, m>0$ be an elliptic symbol and let $f\in L^p,\, 1<p<\infty$. Prove that every weak solution $u$ in $...
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Is the embedding of $H^{1+s}(\Omega)$ (fractional Sobolev space) in $H^1(\Omega)$ compact?

Here, $s\in]0, 1]$ and $\Omega$ is a bounded open connected Lip. domain in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$. I know that $H^1(\Omega)$ is compact in $L^2(\Omega)$, but I have no idea how use it to ...
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1 vote
2 answers
86 views

Estimate of supremum of a function by an integral

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, $n\geq 2$. For $r>0$, denote by $B_r(x_0)=\{x\in\mathbb{R}^n:|x-x_0|<r\}$ whose closure is a proper subset of $\Omega$. Let $u\in W^{1,p}(\...
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