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.

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Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$?

Can someone kindly confirm that the Schwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all ...
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Relation between the norm of Sobolev space $H^1$ and $L^p$ norm for non-increasing radial functions

I am interested to find $$\sup\|u\|_{p}^{p},$$ when $u$ are non-increasing radial functions on the unit ball $B_1$ of $\mathbb{R}^{n}$ such that $$\|u\|_{H1}^2 < r$$ for some $r > 0$. Since $u$ ...
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Self-ajointness of the Laplacian over a Riemannian manifold with boundary

I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf). Let $(M,g)$ be a Riemannian manifold with boundary; $E\to M$ be an hermitian fiber bundle; $\Delta$ ...
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Does Newton-Leibnitz apply to Sobolev space

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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Hölder continuity of Radon transform of smooth function

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
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Action of fractional Laplacian on Hölder / Besov spaces on Riemannian manifolds

Let $M$ be a compact Riemannian manifold (without boundary) and $\Delta$ be the corresponding (positive) Laplace-Beltrami operator. We also define the operators $I^s = (\mathrm{Id} + \Delta)^{-s}$ for ...
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Sobolev variant of Wasserstein space

Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
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Smoothness of Radon transform

Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by $$ R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
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About the continuity of the integral on the boundary of a ball

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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Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space

$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set. For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
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$f\in L^2_k(\mathbb R\times \mathbb S^1)$ implies that $t\mapsto f(t) \in L^2_a(\mathbb R, L^2_{k-a}(\mathbb S^1))$?

$\newcommand{\SS}{\mathbb{S}^1}$ $\newcommand{\R}{\mathbb{R}}$ Consider a function $f:\R\times \SS\to \R$ and suppose that $f$ is in the Sobolev space $L^2_k(\R\times\SS)$ for $k>1$ so that we can ...
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ODE in Banach space

Have I understood this correctly: So originally we consider the following partial differential equation: $$u_t= \frac{u_{xx}}{1+w}-\frac{1}{\epsilon}(1+w)u^3+\frac{wu}{\epsilon(1+w)} \text{ in } \...
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Exercise 8.13 - Brezis

Let $1 \leq p < \infty$ and $u \in W^{1,p}(\mathbb{R}$). Set $$ D_{h}u(x) = \frac{1}{h}(u(x+h) - u(x)), \ \ x \in \mathbb{R}, h> 0 $$ Show that $D_{h}u \to u'$ in $L^{p}(\mathbb{R}$) as $h \to ...
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Reference for an extension theorem for Neumann boundary data

$\DeclareMathOperator\Tr{Tr}$Let $\Omega \subset \mathbb{R}^d$ be a smooth bounded domain (we denote by $n$ the normal to $\partial\Omega$) and $p\in(1,\infty)$. Do you know where I can find (book or ...
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Second order differential operator with a Lipschitz coefficient

Let $a(x) \in W^{1, \infty}(\mathbb{R})$ be real-valued such that $a(x) \ge a_0 > 0$. Let $A^2$ denote the second order differential operator $A^2 : = -\partial_x (a(x) \partial_x) + 1 : L^2(\...
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Does the embedding $W^{2,1}(\mathbb R^2) \to L^\infty(\mathbb R^2)$ factor through some space that is "slightly better" than $W^{1,2}(\mathbb R^2)$?

Using the fundamental theorem of calculus, we can show that the Sobolev space $W^{2,1}(\mathbb R^2)$ embeds into $L^\infty(\mathbb R^2)$. If we attempt to prove this by applying Sobolev embedding ...
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Does the theorem in Sobolev spaces hold on the subset?

Here are some notations: $W^{1,2}(\mathbb{R}^N)=H^1(\mathbb{R}^N)=\{ u\in L^2(\mathbb{R}^N) \,|\,\nabla u\in L^2(\mathbb{R}^N) \}$ $D^{1,2}(\mathbb{R}^N)=\{ u\in L^6(\mathbb{R}^N) \,|\, \nabla u\in L^...
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2 votes
1 answer
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Possible way to define $H_0^1(\Omega)$ Sobolev spaces

Let $\Omega$ be an open set of $\Bbb R^d$: consider the following function spaces $H_0^1(\Omega)$, i.e. the closure of $C_c^\infty(\Omega)$ in $H^1(\Omega)$ $H_*(\Omega)$, i.e. the closure of $C_c^\...
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(0,1)-Extension operator of Sobolev spaces?

I'm struggling with the following problem (I would appreciate any comments or responses). Take $U\subset \mathbb{R}^n$ an open bounded subset (for my case $U$ is semialgebraic or subanalytic is enough)...
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3 votes
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Real interpolation for vector-valued Sobolev spaces

I am having trouble finding any results concerning real interpolation of vector-valued Sobolev spaces. Namely, I would like to know if a continuous embedding of the type, $$ L^p(0,T;X_1)\cap W^{1,p}(0,...
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3 votes
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About radial Sobolev inequality (Strauss Lemma)

As shown in Strauss: Existence of solitary waves in higher dimensions, Strauss introduces the Stauss lemma. Precisely speaking, we have the following theorem: Theorem Let $N \ge 2$, every radial ...
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How to connect the functions in spaces $H^1$ and $H_r$?

\begin{align*} L^2 (\mathbb{R}^3)& {}=\{ u : \int_{\mathbb{R}^3} \lvert u\rvert^2 dx<+\infty \}. \\ H^1(\mathbb{R}^3) & {}=\{ u\in L^2 (\mathbb{R}^3):\, \lvert\nabla u\rvert\in L^2(\mathbb{...
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3 votes
2 answers
185 views

A Sobolev embedding theorem for functions on spheres

$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds: $$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...
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Description of Sobolev functions with rapidly decaying wavelet expansion

Consider the Sobolev space $W^{k,p}((0,\infty))$ and fix a wavelet basis $\{\phi_i\}_{i=0}^{\infty}$ of for it (where $1\leq p<\infty$ and $0<k<\infty$). Since $\{\phi_i\}_{i=0}^{\infty}$ is ...
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Weak convergence in $H^1_{\mathrm{loc}}$

$\newcommand{\loc}{\mathrm{loc}}$Let $\Omega$ be a bounded open set (smooth as we wish if necessary) in $\mathbb{R}^n$, $(\omega_k)$ a sequence of open subsets whose closure is contained in $\Omega$ ...
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2 votes
0 answers
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Find weak approximation by smooth unit vector fields for Sobolev fields on manifold

I am considering the Sobolev space of unit tangent vector fields on a compact manifold: $Γ_{W^{1,2}}(M, UTM)$. I would like to approximate those weakly with smooth vector fields ($Γ_{C^∞}(M, UTM)$). ...
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Extreme case of K-interpolation

Suppose $X_0$ and $X_1$ are Banach spaces living in a larger Banach space $X$. The $K$-functional is defined for each $f\in X_0+X_1$ and $t>0$ as $$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
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5 votes
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$L^p$ estimates for linear parabolic pdes

Let $u$ solve the linear parabolic equation $$ u_t - \Delta u = f \text{ on } \Omega \times (0,T) $$ with initial condition $u(0)=u_0$ and homogeneous Dirichlet boundary condition on $\partial \Omega ...
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2 votes
0 answers
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Would you help me to find this expression?

I'm studying a paper, which I'll include a small piece here. And I'm struggling to calculate $$C_n\|u_{m,n}\|^{\left(\frac{2*}{2}\right)^k\frac{2*-q}{(r_k)^k}}_{L^{2*}(\Omega)}$$ Where $\Omega$ is an ...
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Compact embedding of anisotropic Sobolev space

I am wondering if the embedding from $W^{2,1}_p(\Omega \times [0,T])$ to $C^{\alpha,\alpha/2}(\Omega \times [0,T])$ is compact, for some suitable domain, $p$ and $\alpha$. I have found some results. I ...
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4 votes
1 answer
316 views

Best constant for Poincaré inequality on spheres

I am interested in the following Poincaré-type inequality, $$ \int_{S(r)} \lvert u-\bar{u}\rvert^2 d\sigma \leq C(N) \int_{S(r)} |u_{\theta}|^2 d\sigma$$ where $\bar{u} = \frac{1}{\lvert S(r)\rvert}\...
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1 vote
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Target space of Green's operator on $L^p$-differential forms on closed manifolds

Let $M$ be a closed (i.e., compact without boundary) smooth oriented Riemannian manifold endowed with a regular atlas in the sense of C. Scott [1], i.e., with a finite atlas $\mathcal{A}$ so that for ...
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6 votes
0 answers
156 views

Sobolev embedding theorems on manifolds

I had asked the following question on math.stackexchange but did not get any response: I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional ...
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$H^s$ norm of dispersive semigroup

The Bourgain space is $X^{s,b} := X^{s,b}(\mathbb R \times \mathbb{T}^3)$ is the completion of $C^\infty (\mathbb R; H^s(\mathbb{T}^3))$ under the norm $$\| u\|_{X^{s,b}}:= \|e^{- i t \triangle} u(t,x)...
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1 answer
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Intersection of the kernel with the interpolation space

$\DeclareMathOperator\Ker{Ker}$Given two Banach spaces $X$ and $Y$ with a continuous inclusion $X\subset Y$, and another couple $X’ \subset Y’$ with the same properties. Take $f : Y \longrightarrow ...
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Deriving the general interior elliptic estimate from the compactly supported case

This is an exercise (10.3.4 in the third edition) from Nicolaescu's Lectures on the Geometry of Manifolds. Let $L$ be an elliptic differential operator of order $k$ and $1 < p < \infty$. The ...
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Estimatives for elliptic systems involving the laplacian

Considering the problem \begin{equation} \left\{ \begin{array}[c]{11} \Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\ \Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\ \end{...
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1 vote
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Stability of Hajłasz-Sobolev class under post-composition

Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev? Assumptions/Setup Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
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5 votes
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On Sobolev spaces on domains in Riemannian manifolds

There is extensive literature on Sobolev spaces on complete Riemannian manifolds but are there any standard references regarding the definition and properties of Sobolev spaces on domains (possessing ...
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1 answer
126 views

Lipschitz domains ambiguous definitions

I use a lot in the study of pde bounded Lipschitz domains $\Omega\subseteq\mathbb{R}^N$. However I have noticed that there are some major differences in their definitions. I will put here two of them, ...
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6 votes
1 answer
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Eigenvalues and eigenfunctions of the Laplace operator on entire plane

According to the answers in the the following questions: How to prove the spectrum of the Laplace operator? and What is spectrum for Laplacian in $\mathbb{R}^n$ , the spectrum of the Laplace operator $...
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Inequality on the dual space of $H^s$

Does there exist a theorem that allow us to say that, if we have an estimate on the Sobolev space $H^s\,,\, s\geq 0$ then we can deduce an estimate on the dual space $H^{-s}$ ? For instance, assume ...
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11 votes
0 answers
163 views

Factorization of metric space-valued maps through vector-valued Sobolev spaces

Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that $$ \int_{x\in X}\,d(y_0,f(x)...
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1 vote
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Compute surface Sobolev norm using local coordinate

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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3 votes
3 answers
159 views

Non convex optimization problem in $W_0^{1,2}$

Let $0< \alpha \ll 1$. I'm trying to minimize $\int_0^\pi |f'|^2 dx$ over the functions $f \in W_0^{1,2}([0,\pi])$ (or at least find "good" lower bound in terms of $\alpha$) such that ...
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0 answers
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Reference for smoothness of Nemytskii operator on fractional Sobolev spaces

Let $\varphi:\mathbb{R}\to\mathbb{R}$ be smooth and bounded (together with all of its derivatives). Define the operator $$ \big(N_\varphi x\big)(t)=\varphi\big(x(t)\big) $$ for $x\in H^s(T^d)$, the ...
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2 votes
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Interpolation of Sobolev/Besov spaces in the limiting case q = ∞

I'm interested in the interpolation space ($1\le p_0,p_1\le\infty$, $0<\theta<1$) $$ X=(L_{p_0}(0;1),W^1_{p_1}(0,1))_{\theta,q}\quad\text{with}\quad q=\infty\ \ \text{and}\ \ p_0\ne p_1 . $$ It ...
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2 votes
1 answer
146 views

First derivative of cut off function

I am working on proving the following: Let $\rho(x)= \frac{2}{2+x^2}$, $\theta >1$ (assumed integer here) and $B \subset H^1_{ul}$,(uniformly local Sobolev space), be any subset which is bounded in ...
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  • 388
2 votes
0 answers
57 views

Best constant for Sobolev-type inequality

I am currently reading a paper from Del Pino/Dolbeault about optimal constants of GNS inequalities (http://capde.cmm.uchile.cl/files/2015/06/pino2002.pdf). The author wants to prove the following GNS ...
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1 vote
0 answers
131 views

Sobolev interpolation inequality for relatively compact subdomains

I was looking at Nicolaescu's Lectures on the Geometry of Manifolds (3rd edition). In Theorem 10.2.29 he presents (without proof) the following inequality: For $m \geq 1, p \geq 1, 0 < r \leq R$ ...
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