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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3
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0answers
36 views

A bounded extension operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
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37 views

Well-posedness in modified H2 space

Can I modify the $H^2$ space such that: $$\tilde{H}^2 := \left[ u(\Omega): ||u||^2_{L_2(\Omega)} + ||\nabla u||^2_{L_2(\Omega)} + ||\Delta u||_{L_2(\Omega)}^2 < \infty \right]$$ and then use the ...
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48 views

Trace inequality normal derivative

For $v(\Omega) \in W^1_2$ and $\Omega \in C^1$ we have a trace inequality: $$\Vert v \Vert _{L_2(\partial \Omega)} \leq C_\Omega \Vert v \Vert _{W_2^1},$$ which can be found in many places in the ...
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45 views

Inclusion $H^1_0(\Omega)\cap C^k(\Omega)\subset C^0(\overline\Omega)$?

Let $\Omega\subset {\bf R}^d$ be a bounded domain with a Lipschitz boundary. Assume that a function $u$ is $C^k$ inside $\Omega$ and that $u$ also belongs to $H^1_0(\Omega)$. Can one conclude that $u$ ...
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1answer
124 views

Is there any nontrivial characterization of weakly differentiable functions?

When $f\in L_\text{loc}^1$, it's distributional derivative can be defined as $D_{f'}\in\mathfrak{D}'$, such that $D_{f'}(\varphi)=-\int f\varphi'$ for all $\varphi\in\mathfrak{D}$, where $\mathfrak{D}$...
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Understanding weak formulation of a linear elliptic pde [closed]

hey I'm working on weak formulation on an elliptic pde and I have these questions: Is there any difference between: $\Delta u=f$ in $\Omega$ and $\Delta u=f$ almost everywhere in $\Omega$ when $\...
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36 views

Domains in $\mathbb{R}^n$ for which Hajlasz-Sobolev spaces and Sobolev Spaces are the same

I'm reading Heinonen's book on metric measure spaces. He writes that for general domains $\Omega \subset \mathbb{R}^n$, $M^{1,p}(\Omega) \subset W^{1,p}(\Omega)$ where the former are Hajlasz-Sobolev ...
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1answer
59 views

Poincaré-type Inequality

In Lieb's paper "On the lowest eigenvalue of the Laplacian for the intersection of two domains" one finds the following remark: Let $u\in L_{loc}^p(\mathbb{R}^N)$ with $\nabla u \in L^{p}$ and $\|\...
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1answer
67 views

A time dependent variational problem coming from a second order linear PDE

Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$. Consider the problem of finding $u:\Omega\times[0,T]\to\...
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1answer
387 views

A characterization of constant functions

In How to recognize constant functions. Connections with Sobolev spaces (Russian Math Surveys 57 (2002); MSN), H. Brezis recalls the following fact: Let $\Omega\subset{\mathbb R}^N$ be connected ...
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66 views

What is the closure of this set in $H^1(\mathbb{R}^2)$?

I'm not sure that if this is a difficult question or not. I asked it on MSE and it hasn't been answered and so I thought I might ask it here: What is (or how can we describe) the closure in $H^1(\...
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1answer
56 views

Compact embedding of space of signed Radon measures into Sobolev space $W^{-1,q}$ from Evans paper; Does it work in one space dimension?

Background: I work on a PDE problem where I have some approximating sequence of measure-valued functions and I need to compactly embed it into some negative Sobolev space $W^{-m,q}$ on the bounded ...
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2answers
102 views

inequality involving the fractional Sobolev space

Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality $$|u(x)...
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1answer
62 views

Density property for Sobolev spaces

My question is as follows: is the space $ C_c^{\infty}(\mathbb{R}^3 \setminus \mathcal{C}) $ dense in $ H^1( \mathbb{R}^3) $ where $ \mathcal{C} $ is the circle $ \{(x,y,z) \in \mathbb{R}^3 \mid x^2 +...
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1answer
191 views

What is the motivation of the $L^p$ differentiability?

I was reading some papers and come up with the next definition : A function is differentiable in the $L^p$ sense at $x$ if there exists a real number $f'_p(x)$ such that $$\bigg(\frac{1}{h}∫_{-h}^...
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52 views

Analogous $H^1$-space for pseudo inner products

Perhaps this is a naive question but I could not find anything related to this. Imagine we are on a bounded and regular open subset $\Omega$ of $\mathbb{R}^3_1$, i.e, $\mathbb{R}^4$ is considered ...
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30 views

Covering number for the unit ball in a reproducing kernel Hilbert space

I am looking for a reference for an upper bound on the covering number for the unit ball $\{ f \in \mathcal{H}: ||f||_{\mathcal{H}} || \leq 1\} $, where $\mathcal{H}$ is a reproducing kernel Hilbert ...
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76 views

Compatibility between the source and the boundary condition for an Helmholtz-type equation

Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
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Is $\Delta \phi$ monotone operator on $H^1(\mathbb{R}^d)$ for monotone $\phi$

Let $H^1(\mathbb{R}^d)$ be the usual Sobolev space and let $\phi: \mathbb{R} \to \mathbb{R}$ be a non decreasing Lipschitz function with $\phi(0)=0$. Is the operator $\Delta \phi $ on $H^1(\mathbb{R}^...
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52 views

Stability and capacity, error in the book of Adams-Hedberg?

I am struggling to understand the proof from the book of Adams and Hedberg, "Function spaces and potential theory". It seems to me that there is a serious flaw, and moreover, the statement is ...
2
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1answer
69 views

Approximating functions in $H^1_0(U) \cap H^2(U)$ via $H^1$ norm and $L^2$ projection

Let $U$ be a bounded domain in the Euclidean space with sufficiently smooth boundary. Let $\{f_i\}$ be a orthonormal basis of $H^1_0(U)$ satisfying $-\Delta f_i = \lambda_i f_i$ where $\lambda_i \leq \...
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1answer
134 views

Limit case of Sobolev space in $1$-D

This might look too an elementary question, but I am confined and is not able to find a textbook which answers the following question. I have a function $f:{\mathbb R}\rightarrow{\mathbb R}$, such ...
7
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1answer
69 views

Sobolev topology on essentially compactly supported Sobolev-“functions”

The locally convex space of essentially compactly-supported $p$-integrable "functions" $\operatorname{L}_{\mathrm{comp}}^p(\mathbb{R}^d,\mathbb{R})$ is defined as the set $$ \bigcup_{n \in \mathbb{N}} ...
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72 views

Sobolev convergence of Fourier series

Consider $f\in H^{\sigma}(S^1)=W^{\sigma, 2}$ (the usual Sobolev space on the circle) and let $S_Nf$ be its truncated Fourier series $S_Nf = \sum_{|n|\leq N} \hat{f}(n)e^{2\pi i n x}$. I am looking ...
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47 views

Regularity theory for parabolic PDEs in fractional Sobolev spaces

I am trying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya and the existence ...
1
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1answer
98 views

Understanding a family of Sobolev-type inequalities

I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following: Denote the following inequality as $S_{r,s}^{\theta}$: $\...
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37 views

Does the function with the αth-weak partial derivative has the βth-weak partial derivative with β≤α?

The definition of weak derivative in the book Partial Differential Equations by Evans is stated as follows: Suppose $u,v \in L_{loc}^1(U)$, and $\alpha$ is a multiindex. We say that $v$ is the $\...
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52 views

Boundedness of $\chi_{\{f_n=0\}}$ in the BV norm

Let $f_n \in H^2(\Omega) \cap C^0(\bar \Omega)$ be a sequence of functions that are uniformly bounded in $H^2(\Omega) \cap C^0(\bar \Omega)$ on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ ...
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46 views

Embedding of anisotropic Besov Spaces in spaces of continuous/differentiable functions

I am interested in anisotropic Besov spaces. In particular I am interested in the embedding of such anisotropic Besov spaces into possibly anisotropic spaces of differentiable functions. $\newcommand{...
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88 views

Uniform bound on $\lVert \chi_{\{u_n=0\}}\rVert_{W^{s,p}(\Omega)}$ for a bounded sequence $u_n$ in $H^1_0(\Omega)$?

Suppose I have a sequence $u_n \to u$ in $H^1_0(\Omega)$ on a smooth and bounded domain. For some $p>1$ and $s \in (0,\frac 12)$, is it possible to estimate the norm of the characteristic function ...
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64 views

If $u_n \to u$ in $H^1_0(\Omega)$, does $\chi_{\{u_n = 0\}} \to g$ for some $g$ in some space, for a subsequence?

Let $\Omega$ be a bounded and smooth domain. Suppose we have $u_n \to u$ in $H^1_0(\Omega)$. We know that for a subsequence, $\chi_{\{u_n = 0\}} \rightharpoonup f$ to some $f$, weak-* in $L^\infty(\...
4
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1answer
101 views

Interpolation for Sobolev spaces

How one can identify the following (complex) interpolation space $$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$ where $\Omega$ is a regular domaine. After research, it seems that ...
2
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0answers
56 views

Proving that $(f,g)$ are Cauchy data for the Schrödinger equation iff $(f,g)$ satisfies an equation

I have to prove that if $f\in H^{1/2}(\partial\Omega)$ then $(f,g)$ are Cauchy data for the Schrödinger equation if and only if $$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
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1answer
90 views

Fractional Sobolev norm of characteristic function of an interval?

Is there an explicit expression giving a fractional Sobolev norm of the characteristic function of some interval $I=[a,b)$? I believe it is true that $\chi_{I} \in W^{s,1}(\mathbb{R})$ for $s < \...
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39 views

(1, 2) stability and Hausdorff dimension

Recall that a set $E \subset \mathbb{R}^n$ is called (1,2) stable if it satisfies the following: $$ W_0^{1,2}(E) = W_0^{1,2}(E^0), $$ where $E^0$ is the interior of $E$ and $W_0^{1,2}(E)$ is defined ...
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58 views

Logarithmic Sobolev growth of time-space-periodic Schrödinger solutions

Consider the following Schrödinger equation $$i\partial_t \psi (t,x) + \Delta \psi - V(t,x) \psi = 0 \, ,$$ where $x\in \mathbb{T}^d$ and $V(t,\cdot)$ is real, smooth, and periodic (with a diophantine ...
2
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1answer
66 views

Joining Hölder continuous functions on Whitney covering

Let $u$ be a bounded function and let a closed set $E$ be given. The compliment of $E$ can be covered with a Whitney type covering $B_i$ such that the following are satisfied: 1) $E^c \subset \...
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47 views

Approximation of a Sobolev map with fixed singular values by smooth maps with the same singular values

Let $0<\sigma_1<\sigma_2$, and let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Let $f \in W^{1,\infty}(D,\mathbb{R}^2)$, and suppose that the singular values of $df$ are a.e. equal to ...
9
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226 views

Second order differentiability of convex functions

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is ...
5
votes
1answer
130 views

Stability of fractional Sobolev spaces under diffeomorphisms

Let $H^s_p(\mathbb{R}^n)$ be the fractional Sobolev space of fractional order $s\in \mathbb{R}$, for $1<p<\infty$, and let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Assume that the ...
2
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1answer
49 views

Weak lower semicontinuity of functional with two arguments

Let $\Omega$ be a bounded domain (smooth if necessary) and let $J:H^1(\Omega) \times H^1_0(\Omega) \to \mathbb{R}$ be defined by $$J(u,v) = \int_\Omega f(u)|\nabla v|^2$$ where $f\colon \mathbb{R} \...
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0answers
34 views

Estimates on density for Stokes equation

Consider a bounded, smooth domain $\Omega\subset \mathbb{R}^3$ and in there the Stokes equations $\nabla p(\rho)-\Delta u=\rho f\\ \operatorname{div}(\rho u)=0\\ u\restriction_{\partial \Omega}=0$ ...
2
votes
1answer
42 views

Sobolev spaces complement of Hausdorff codimension 2, restriction theorem

Let $X$ be an open domain in $R^n$. Let $E$ be a subspace of $X$ with Hausdorff dimension $m$. Fix $k$ and $p$. What are the optimal assumptions on $m$ and $n$ so that the trivial map $W^k_p(X) \to W^...
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1answer
106 views

Orlicz-Sobolev Spaces

let $A$ an N-function and suppose that $$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty $$ we denote by $\widehat{A}$ an N-function equal to A near infinity and $\widehat{A}$ ...
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0answers
43 views

On a interpolation inequality for the Schrödinger unitary group (NLS)

I'm trying to understand scattering for the classical nonlinear Schrödinger equation and for that i'm studying a scattering criterion on Tao's paper. At Lema 3.1 he states that $$\left\|e^{it\Delta}f\...
5
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0answers
76 views

Extension Operator for $W^{1,\infty}(U,X)$

I am reading through some lectures on Sobolev spaces and the vector-valued (or Banach space valued) version of them. At this moment I am very interested in extension operators for the vector-valued ...
2
votes
1answer
127 views

Non-convergence to a Gaussian

Let $f_n: \mathbb R^2 \rightarrow \mathbb R$ be a family of probability distributions with the property that they vanish on the diagonal $f_n(x,x)=0.$ I would like to know: Can we show that a ...
4
votes
1answer
193 views

Approximate constant function

Let $f:[0,1]^2 \rightarrow \mathbb C$ be an $H^1$ function with the property that $f(x,x)=0$ and $\Vert f \Vert_{L^2[0,1]}=1.$ Does there exist a constant $c>0$ such that any such function ...
1
vote
1answer
114 views

Fourier transform for $H^2(\mathbb{R}^N)$, $N\geq 5$

How i can prove that if $u\in H^2(\mathbb{R}^N)$ then $u\in \mathcal{F}(L^{p^*}(\mathbb{R}^N))$, where $1/p+1/{p^*}=1,$ $2\leq p<2N/(N-4)$?
4
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1answer
96 views

Adjoint of the multiplication operator on a Sobolev space

Let $f\colon\mathbb{R}^n\rightarrow\mathbb{C}$ be a bounded function with a bounded first derivative. Then the multiplication operator $H^1(\mathbb{R}^n)\ni x\mapsto A_f x:=fx\in H^1(\mathbb{R}^n)$ is ...

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