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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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Approximation on $H^1_0(B)$ and cut-off functions

Let $u \in H^1_0(B)$, where $B$ is the unit ball in $\mathbb{R}^N$. Given $\epsilon > 0$, I need to show there exists a function $\chi_\epsilon \in C^\infty_0(\mathbb{R}^N)$ such that $$ \| u - \...
Lucas Linhares's user avatar
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Continuity of the set of critical points of a Morse function when the function varies

Let me first give the result I am looking for, then what I found up to now and some related questions. Definition/Notation. Denote $Y_k(f)$ the set of critical values $x$ of $f$ such that the index of ...
Chev's user avatar
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Radial functions in $H^1_0(B)$ and a Strauss type inequality

Let $B$ be the unit ball in $\mathbb{R}^N$. Denote by $\|\cdot\|$ the usual norm in $H^1_0(B)$. In (2) of the paper is said that $$ |u(x)| \leq \frac{\|u\|}{|x|^{(N-2)/2}}, $$ for any radial function $...
Lucas Linhares's user avatar
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1 answer
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Dense subspace of space of radial functions in $H^1_0(\Omega)$

Consider $\Omega\subset \mathbb{R}^N$ a bounded domain. By definition, the space of radial functions in $H^1_0(\Omega)$ is $$ H^1_{0, \text{rad}}(\Omega) = \{u \in H^1_0(\Omega) : u = u \circ R, \...
Lucas Linhares's user avatar
1 vote
0 answers
60 views

Multivariate polynomial approximation

Let $f$ be a function on $[-1,1]^d$ with some smoothness property, for example, it is in the Sobolev space $W^{k,p}$. Let $P_n$ is a space of polynomials with degree $n$. My question is what is the ...
Iris's user avatar
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$H^{s,\infty}$ via Triebel Lizorkin spaces

I know that $F^s_{p,2} = H^{s,p}$ for $p \in (1, \infty)$, but what about $H^{s,1}$ and $H^{s,\infty}$? I know one extension for $F^0_{\infty,2}$ which should be BMO. But how do I get the $H^{s,\infty}...
Kilian Koch's user avatar
2 votes
1 answer
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Examples showing Rellich-Kondrakov theorem fails for domains with non-Lipschitz boundary?

The Rellich-Kondrakov theorem requires the domain to have Lipschitz boundary, but after searching in many places, I have not found an example to show that this is necessary. Can anyone share an ...
Ma Joad's user avatar
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61 views

For any $p, q \in [1,\infty]$ and $s \in (0,\infty)$, can we find some $f \in L^q - W^{s,p}$?

Sobolev inequalities show us when we can embed a Sobolev space into another. However, I wonder if these inclusions are always proper. More specifically, let $\Omega \subset \mathbb{R}^n$ be a bounded ...
Isaac's user avatar
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4 votes
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Multivariate polynomial approximation of functions in Sobolev space

I found a result of the estimation error of polynomial approximation in page 6 of https://scg.ece.ucsb.edu/publications/theses/ARajagopal_2019_Thesis.pdf The statement is for $f \in W^{k, p}\left([-1,...
Iris's user avatar
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How can i show that the last equality in the text is true?

Suppose that $v$ is critical point of $$ f(u)=\frac{1}{2} \int_D|\nabla u|^2-\frac{1}{p+1} \int_D|u|^{p+1}, \quad u \in H_{0, \text{rad}}^1(D),\quad D(r, d)=\left\{z \in R^N: r^2<|z|^2<(r+d)^2\...
Pádua's user avatar
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3 votes
1 answer
140 views

Notation for weak derivatives

I remember that, as a student, I felt a bit uncomfortable because I had to use the same notation (say $f'$, $D^\alpha f$, $\frac{\partial f}{\partial x^j}$, $\nabla \cdot f$ etc...) for classical and ...
Alessandro Della Corte's user avatar
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Dual of homogeneous Triebel-Lizorkin

Let $ p, q \in (1,\infty)$ and consider the homogeneous Triebel- Lizorkin space $\dot{F}^{s}_{p,q}$ to be the space of all tempered distributions (modulo polynomials) with $$ [f]^{p}_{\dot{F}^{s}_{p,q}...
User091099's user avatar
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Regularity and decay of Fourier-like series on a manifold

Let $D$ be a first-order self-adjoint elliptic differential operator acting on sections of a vector bundle $S$ over a closed manifold $M$. Then it is well-known that the various eigenspaces $E_\lambda$...
geometricK's user avatar
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Trace theorem for $L^2([0,1]; H^k(S^2))$

Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer. Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
Laithy's user avatar
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Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?

The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...
vmist's user avatar
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Image of trace operator on $W^{2,1}(\mathbb{R}^2)$

It is known that for a domain $\Omega\subset \mathbb{R}^2$ with $C^1$ boundary $\partial\Omega$, that the trace operator is bounded and surjective from $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$. For ...
vmist's user avatar
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3 answers
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Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?

If $\Omega$ is an open Lipschitz domain with bounded boundary then their exists a continuous operator $E: W^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$ where $s\in (0,1)$ such that $Eu=u$ on $\...
Perelman's user avatar
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1 vote
1 answer
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Bounding supremum norm in terms of gradient L2-norm using a Poincare-like inequality

Suppose $f$ is a Lipschitz continuous real-valued function over a bounded domain $\Omega \subset \mathbb{R}^d$ with smooth boundary, and let $\overline{f} := \frac{1}{|\Omega|}\int_\Omega f(x) dx$. Is ...
Greg O.'s user avatar
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1 answer
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On the weak derivative of $|u|^{(p-2)/2}u$

Let $u$ be a function such that $|u|^{(p-2)/2}u$ is in $H^1_0(G)$, $G$ is open and $p>2$. How can I show that $$ D(|u|^{(p-2)/2}u)=p/2|u|^{(p-2)/2}D(u) \label{1}\tag{1} $$ or how can I show that, ...
Perelman's user avatar
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3 votes
1 answer
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Does the union of fractional Sobolev spaces fills $L^p$?

Let $\Omega\subset \Bbb R^d$ be any open set. Recall that for $s\in (0,1)$, the fractional Sobolev space $W^{s,p}(\Omega)$ is the collection of function in $L^p(\Omega$ such that \begin{align*} \iint_{...
Guy Fsone's user avatar
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1 answer
51 views

$L^\infty$ estimate for elliptic PDE with mixed boundary conditions

Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint. Consider the problem $$\Delta u = f \quad\text{in $\...
BBB's user avatar
  • 31
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1 answer
76 views

Uniform convergence of differential quotients in $L^1$

I know that the question arose already in other contexts. However, I think this question might be different. If I have $f\in W^{1,1}$ then it is obvious that $\frac{f(x+t)-f(x)}{t}$ converges ...
Mario Vasilija's user avatar
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0 answers
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Extension operator maps which fractional Sobolev space to $W^{p,s}(R)$

Let us assume we have the following extension operator: $$ \operatorname{ext}_R^\sigma u= \begin{cases} u(x) & \text{if }x \in (0,T)\\ u(0) & \text{if }x \in(0,T)^c. \end{cases} $$ We ...
Fractional analysics's user avatar
2 votes
2 answers
216 views

$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?

Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has $$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
Iosif Pinelis's user avatar
2 votes
2 answers
180 views

$L^p$ domination of mixed partial derivatives by the unmixed ones?

Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has $$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
Iosif Pinelis's user avatar
2 votes
0 answers
50 views

Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE

Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of the ...
BBB's user avatar
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0 answers
208 views

Gauss transformation in fractional Sobolev space

Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that $$ \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
Muniain's user avatar
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Is it possible to bound the L2 norm of the gradient of a divergent by the L2 norm of the Lapacian?

Is it possible to show for $u:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ that $$\||\nabla(\nabla\cdot u)|\|_2^2\leq C\||\Delta u|\|_2^2?$$ Here $\||f|\|_2$ is the norm in $(L^2(\Omega))^3$ and ...
Alberto Leandro's user avatar
2 votes
0 answers
69 views

Any solution of an evolution problem tends to a steady state in $L^2$?

I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\...
Bogdan's user avatar
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0 answers
37 views

Finding an element of Gelfand triple with a designated time derivative

Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that \begin{equation} V \subset H \subset V' \end{equation} where $V'$ is the dual of $V$ and the inclusions are ...
Isaac's user avatar
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9 votes
1 answer
612 views

Prove J.L. Lions’s Lemma without using Fourier transform

When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states Let $\Omega \subset \mathbb R^n$ be a ...
Zhang Yuhan's user avatar
3 votes
0 answers
74 views

Absolute continuity of $t \to \lVert u(t) \rVert^2_{H}$ and Gelfand triple : are they equivalent?

Let $V$ be a separable Banach space and $H$ be a separable Hilbert space such that \begin{equation} V \subset H \subset V' \end{equation} and the inclusion maps are continuous with dense images. Here $...
Isaac's user avatar
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3 votes
1 answer
206 views

$L^{\infty}$ estimate for heat equation with $L^2$ initial data

Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem: $$\begin{...
Bogdan's user avatar
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2 votes
0 answers
153 views

Does integration by parts formula hold in $H^1(0,T,L^2(\Omega))$?

Let $\Omega$ be an open set from $\mathbb{R}^N$. How can we prove that if $u,v\in H^1(0,T,L^2(\Omega))$ (in Bochner sense) then $(u\cdot v)'\in L^2(0,T,L^1(\Omega))$ with $(uv)'=u'v+v'u$ and the ...
Bogdan's user avatar
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1 vote
1 answer
114 views

Can functions with "big" discontinuities be in $H^1$?

How can I prove that the function: $$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...
Bogdan's user avatar
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4 votes
1 answer
133 views

Embeddings of the maximal domain for the Laplacian

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function: $$D = \left\{ f \in L^2(\...
MeS's user avatar
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3 votes
2 answers
242 views

Can every $L^p$ function be written as the weak derivative of a Sobolev function?

Let $\mathbb B^n$ be the open unit ball in $\mathbb R^n$, and $g: \mathbb B^n \to \mathbb R^n$ a measurable function with $|g| \in L^p (\mathbb B^n)$. Does there exist some function $f$ in the Sobolev ...
Nate River's user avatar
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1 vote
0 answers
79 views

On Talenti's proof of optimal constant in Sobolev inequality

I'm reading the paper by Giorgio Talenti on the best constant for the Sobolev inequality. The main theorem states that for $u:\mathbb{R}^m\rightarrow \mathbb{R}$ sufficiently smooth (eg. Lipschitz) ...
user519428's user avatar
3 votes
1 answer
216 views

Weighted Lebesgue space with exponential weights: smoothing effect and properties

I am researching whether there are weighted Lebesgue spaces of the type $$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$ ...
Ilovemath's user avatar
  • 585
0 votes
0 answers
47 views

Constants in the entropy number of the Sobolev space

For a Sobolev space with $W^s(\Omega)$, where $\Omega\subset R^d$ is a compact space with smooth boundary, we know that the entropy number satisfies $e(\delta, W^s(\Omega, 1),\|\cdot\|_{L_\infty})\leq ...
NullOfMatrix's user avatar
1 vote
0 answers
72 views

$T$ trace, then $Tg(u)=g(T(u))$ for all $u$ on $W^{1,p}$

The trace operator $T$ is defined for bounded domain $U$ with $C^1$ boundaries as the linear, continuous operator $T: W^{1,p}(U) \rightarrow L^p(\partial U)$ such that $$ Tu=u\;\text{ on }\partial U $$...
Furkan's user avatar
  • 11
2 votes
1 answer
92 views

Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function

Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-...
alia's user avatar
  • 23
6 votes
0 answers
115 views

A function whose derivatives belong to $BMO(\mathbb{R}^n)$

I am reading the paper "Bounded Mean Oscillation and Sobolev Spaces" by Robert s. Strichartz, Indiana University Mathematics Journal , 1980, Vol. 29, pp. 539-558. In this paper he defines ...
Gio67's user avatar
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2 votes
0 answers
75 views

How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$

Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...
SAKLY's user avatar
  • 63
0 votes
0 answers
45 views

Fractional Bochner spaces $H^s((0,\infty); V)$

I'm looking for "intrinsic" definitions of fractional Bochner spaces such as $H^s((0,\infty); V)$ for a Banach space $V$ and exponent $s \in (0,1)$. I have seen these spaces being defined as ...
Blah000's user avatar
  • 29
2 votes
0 answers
66 views

Derivative of a functional involving integral and level set

Let $\Omega$ be a bounded smooth domain. For $u\colon \Omega \to \mathbb{R}$, define the functional $$F(u) = \int_{\{u=0\}}g(x) \; \mathrm{d}x$$ where eg. $u \in H^2(\Omega) \cap C^0(\bar\Omega)$ and ...
Blah000's user avatar
  • 29
2 votes
0 answers
145 views

$H^s$-mild solution for Navier–Stokes : why do we restrict attention to the function spaces "without Fourier zero mode"? (Related to Terence Tao blog)

This question has been triggered by the Definition 32 and Remark 33 in the blog of Terence Tao. There, every function space is restricted to ones without the Fourier zeroth mode. And the Remark 33 ...
Isaac's user avatar
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0 votes
0 answers
50 views

Weighted version of the Gagliardo Nirenberg inequality

I'm searching for a weighted Gagliardo-Nirenberg inequality similar as in https://arxiv.org/pdf/1307.1363.pdf where the weight is a power of the last component. Is there an inequality of the form $$ \...
user99432's user avatar
  • 173
5 votes
1 answer
270 views

If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)

I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask. I repeat the question for the sake of completeness: Let $f(x,t) ...
Isaac's user avatar
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4 votes
1 answer
525 views

$f\in C(B_1)\cap W^{1,2}(B_1\setminus \{f=0\})$ implies $f\in W^{1,2}(B_1)$?

In a paper I am writing I need to show that a certain real-valued function $f\in C^0(B_1)$ belongs to the Sobolev space $W^{1,2}(B_1)$ ($B_1$ is the unit ball). So far I have been able to show that ...
No-one's user avatar
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