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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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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$L^\infty$ 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
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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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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
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2 answers
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$L^p$ 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
179 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
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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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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
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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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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
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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
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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 $...
Isaac's user avatar
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$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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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
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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
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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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2 answers
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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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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
198 views

Weighted Lebesgue space with exponential weights: smoothing effect and properties

I am researching whether there are weighted Lebesgue spaces of the type $$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$ ...
Ilovemath's user avatar
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0 answers
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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
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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 $$ Tu=u\;\text{ on }\partial U $$...
Furkan's user avatar
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1 answer
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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
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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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0 answers
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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
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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
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2 votes
0 answers
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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
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2 votes
0 answers
139 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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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
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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 $f(x,t) ...
Isaac's user avatar
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4 votes
1 answer
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$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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1 vote
0 answers
179 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_{\...
Dan1618's user avatar
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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 $u\in C^{\infty}_0((-1,1))$?
Ali's user avatar
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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)}...
user515999's user avatar
2 votes
1 answer
108 views

Equivalent characterization of weak derivative in Bochner space

Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff $$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
Mandelbrot's user avatar
0 votes
0 answers
59 views

Friedrich's second inequality for functions with zero average

Friedrich's second inequality (or Maxwell Estimates or Gaffney’s inequality in the literature) is referred as follows: for all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n} \cdot \...
Ryan Li's user avatar
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2 votes
0 answers
165 views

finite dimensionality of a subspace of a Banach space

Let $H$ be the space of measurable functions on $(0,1)$ such that $$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$ Let $C>0$ be a constant. Suppose that $W \...
Ali's user avatar
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2 votes
0 answers
40 views

Blow up for certain classes of distributions

Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that $$u|_{\mathbb D\setminus \{0\}} \in C^{\...
Ali's user avatar
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2 votes
1 answer
148 views

A compact embedding claim

Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms $$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$ Let $H_2$ be the weighted Sobolev space with the ...
Ali's user avatar
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5 votes
1 answer
235 views

Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?

Let $f \in W^{1, 1} (\mathbb R^d)$. For every $\varepsilon > 0$, we consider the local maximal function $M_\varepsilon f: \mathbb R^d \to \mathbb R$, defined by $$M f_{\varepsilon} (x) = \sup_{r \...
Nate River's user avatar
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2 votes
1 answer
351 views

On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$

Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$. I have found in several research ...
Akundu's user avatar
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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)...
Ryan Li's user avatar
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4 votes
1 answer
339 views

Norm inequality for the inclusion $L^2(\partial \Omega)\hookrightarrow H^{-1/2}(\partial \Omega)$

Let $\Omega \subset \mathbb{R}^3$ be a lipschitz domain. We then have the trace operator $\tau : H^1(\Omega) \to L^2(\partial \Omega)$ and can define the space $H^{1/2}(\partial \Omega) := \tau(H^1(\...
Mandelbrot's user avatar
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0 answers
117 views

About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"

I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1. We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
Elio Li's user avatar
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1 vote
1 answer
113 views

Sobolev-type estimate for irrational winding on a torus

Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
user197284's user avatar
2 votes
0 answers
128 views

Self-adjointness of fractional laplacian

Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
B.Hueber's user avatar
  • 987
3 votes
1 answer
188 views

Blowup of Sobolev norms in approximating a non-absolutely continuous function

Let $f: [0, 1] \to \mathbb R$ be a continuous function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that if $f$ fails to be absolutely ...
Nate River's user avatar
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3 votes
1 answer
494 views

Existence of a curve of finite length on the image of an embedding which is Sobolev

Suppose that we have an embedding $f:\mathbb{R}^2\to\mathbb{R}^3$ which belongs in the Sobolev space $W^{1,p}_{loc}(\mathbb{R^2},\mathbb{R}^3)$ for some $p>2$. Is it true then that for any two ...
Mad Max's user avatar
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5 votes
0 answers
197 views

Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$

My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
Lorenzo Pompili's user avatar
2 votes
0 answers
70 views

Sobolev inequalities in weighted Sobolev spaces

My definition for "weighted Sobolev space" $W^{1,p} _w (\Omega)$ is just a set of functions with the property that they admit distributional derivatives and that $$ \int_\Omega |f|^p (x) w(x)...
tommy1996q's user avatar

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