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.

Filter by
Sorted by
Tagged with
3 votes
1 answer
114 views

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_{...
9 votes
1 answer
2k views

Chain rule for distributional derivative

Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$). Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
0 votes
1 answer
40 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 $\...
2 votes
1 answer
213 views

Periodic solution for linear parabolic equation - existence, regularity

I am interested in proving the existence and regularity of solution to the following problem: $$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)...
1 vote
1 answer
282 views

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)...
1 vote
1 answer
222 views

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 ...
0 votes
1 answer
53 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 ...
5 votes
1 answer
982 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 $\mathfrak{D}$...
2 votes
2 answers
192 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+\|...
1 vote
0 answers
44 views

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 ...
2 votes
2 answers
178 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^...
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 ...
0 votes
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}...
4 votes
1 answer
338 views

Regularity of solutions to $-\Delta u = \operatorname{div} F$, $F\in L^1$

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary. What are the regularity results for solutions to $$ -\Delta u= \operatorname{div} F, \qquad F\in L^1(\Omega,\mathbb{R}^n)? $$...
1 vote
0 answers
106 views

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 ...
15 votes
1 answer
2k views

Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain

Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space $$X = \left\{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{...
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{\...
0 votes
0 answers
35 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 ...
9 votes
1 answer
595 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 ...
3 votes
1 answer
183 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{...
3 votes
0 answers
73 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 $...
4 votes
1 answer
353 views

Contractivity of Neumann Laplacian

I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian. In ...
2 votes
0 answers
152 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 ...
3 votes
1 answer
199 views

Parabolic Sobolev inequality in Sobolev mixed norm spaces

Assume $p,q\in (1,\infty)$, $r\in [p,\infty)$, $s\in [q,\infty)$ and $$ 1<\frac{d}{p}+\frac{2}{q}=1+\frac{d}{r}+\frac{2}{s}. $$ Let $u\in C_c^\infty((0,1)\times B_1)$, where $B_1=\{x\in \mathbb{R}^...
1 vote
1 answer
113 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 ...
4 votes
1 answer
125 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(\...
3 votes
2 answers
231 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 ...
4 votes
1 answer
235 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) ...
1 vote
0 answers
70 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) ...
3 votes
1 answer
198 views

Weighted Lebesgue space with exponential weights: smoothing effect and properties

I am researching whether there are weighted Lebesgue spaces of the type $$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$ ...
0 votes
1 answer
457 views

Orlicz–Sobolev spaces

Let $A$ be 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{...
0 votes
0 answers
45 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 ...
1 vote
0 answers
71 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 $$ Tu=u\;\text{ on }\partial U $$...
2 votes
1 answer
90 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-...
1 vote
1 answer
819 views

Does weak-* convergence in $W^{1,\infty}$ imply weak-* convergence in $L^\infty$?

Let $\Omega \subset \mathbb{R}^n$ be open and bounded. What does weak-* convergence for a sequence of functions $\{f_k\}_{k \in \mathbb{N}}$ in $W^{1,\infty}(\Omega)$ mean? It seems to me that there ...
6 votes
1 answer
996 views

Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t =...
5 votes
0 answers
105 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 ...
2 votes
0 answers
73 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 ...
0 votes
0 answers
44 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 ...
2 votes
0 answers
61 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 ...
8 votes
1 answer
485 views

A fractional weighted Poincaré inequality

Does there exists a constant $C>0$ such that $$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$ for all $u\in C^{\infty}_0((-1,1))$?
2 votes
0 answers
138 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 ...
4 votes
1 answer
315 views

Equivalent norms of fractional Sobolev spaces on bounded Lipschitz domain

Let $s>0$, $1<p<\infty$ and let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. Set $H^{s,p}(\Omega)=\{u\in L^p(\Omega):\exists\tilde u\in L^p(\mathbb R^n),\tilde u|_{\Omega}=u,(I-\...
0 votes
0 answers
45 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 $$ \...
4 votes
1 answer
491 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 ...
1 vote
0 answers
178 views

Is this a well known space? Perhaps homogeneous Sobolev-like space?

The homogeneous Sobolev space $\dot H^s(\mathbb{R}^n) $ is often defined as the closure of $\mathcal{S}(\mathbb{R}^n)$ under the norm $$ || |\omega|^s \widehat{f} ||_{L^2(\mathbb{R}^d)} =\int_{\...
8 votes
1 answer
1k views

Noncompactness of the Sobolev embedding in the critical exponent case

Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$. It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the ...
2 votes
0 answers
62 views

Localized estimate for divergence free vector field

Suppose $\Omega \subset \mathbb{R}^3$ is a simply connected Lipchitz domain. For a divergence free field $w\in [L^2(\Omega)]^d$, it is well known that there exists a vector field $v\in [W^{1,2}(\Omega)...
6 votes
1 answer
264 views

Regularity of the Jacobian of a $W^{2,n}$ Sobolev mapping

Given a mapping in the Sobolev space $f\in W^{2,n}_{\rm loc}(\mathbb{R}^n,\mathbb{R}^n)$ I would like to know what is the Sobolev regularity of the Jacobian $J_f=\operatorname{det} Df$. It is well ...
1 vote
0 answers
85 views

Dependence of Sobolev embedding theorem constant on smoothness

Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that $$ \|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...

1
2 3 4 5
21