# 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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### Operator norm of linear functional $\varphi \mapsto \int_\Omega f\varphi$ with respect to different norms

Let $\Omega \subseteq \mathbb{R}^n$ be open. For some $f \in L^2(\Omega)$ consider the continuous linear functional $$T \colon C^\infty_c(\Omega) \to \mathbb{R}, \qquad T(\varphi) := \int_\Omega f \...

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### A chain rule for weak time derivatives in Bochner space

Let $u \in W(0,T)$ where
$$W(0,T) := \{ v \in L^2(0,T;H^1_0(\Omega)) : v' \in L^2(0,T;H^{-1}(\Omega))\}$$
where $v'$ refers to the weak temporal derivative and $\Omega$ is a bounded and smooth domain.
...

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### Extending Hölder functions

I originally asked this question on MathStackExchange some time ago, but it seems that MathOverflow would be more appropriate. Essentially, I would like to find references for extension theorems for (...

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### Existence of a smooth extension

In the three dimensional Euclidean space $\mathbb R^3$ let us define the hypersurface
$$ S= \{(x,y,z) \in \mathbb R^3\,:\, z^2= x^2+y^2\}.$$
Suppose that $f \in C^{\infty}(S)$. Does there exist $u\in ...

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### Question about calculation in Schwartz space

While reading a paper Hengang Li and Weiping Yan - Asymptotic stability and instability of explicit self-similar waves for a class of nonlinear shallow water equations, I experienced that my ...

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### Can we interpret fractional Sobolev spaces in terms of fractional derivatives?

Let $1 \leq p < \infty$, $0<s<1$, and $\Omega \subseteq R^n$ be a domain. The Banach space $W^{s,p}(\Omega)$ is defined as
$$W^{s,p}(\Omega) := \left\{ f \in L^p(\Omega) \colon \int_{\Omega \...

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### Finite dimensionality of a subspace

Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds:
$$ \...

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### Compact embedding of homogeneous weighted Sobolev spaces

Let $n\geq 2$ and let $\Omega$ be the open unit ball with the origin removed. For each $\delta>0$ and each $u\in C^{\infty}(\Omega)$ let us define
$$ \|u\|^2_{L'^1_\delta(\Omega)}= \int_{\Omega} |x|...

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### Inequality for a weighted bilinear form in Fourier variables

Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$.
Consider the ...

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### Sobolev trace inequality with exterior domains

Let $x_1\in \mathbb{R}^n$, $n\geq 3$, $\Omega=\mathbb{R}^n\backslash B_1(x_1)$, define $D_{\Omega}$ by taking the closure of $C_{c}^{\infty}(\overline{\Omega})$ under the norm
\begin{align*}
\|u\|_{D_{...

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### Nonlocal elliptic problem - what is its associated energy?

It is well known that for any smooth domain $\Omega\subset\mathbb{R}^N$ the energy functional (the one for which the Euler-Lagrange equation is our b.v.p.) associated to the following local problem:
$$...

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### Can Sobolev space be characterized by spectral decomposition?

Consider a homogeneous Carnot group $\mathbb{G}$ with step $r$. Let $X_1,\cdots,X_m$ be the first layer of its Lie algebra. Denote by $\mathcal{L}=-\sum_{i=1}^m X_i^2$ the sub-Laplacian on $\mathbb{G}$...

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### Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?

The following startles me. Let $\Omega \subseteq \mathbb R^n$ and write $W^{\sigma,1}(\Omega)$ for the fractional Sobolev space with norm
$$|u|_{W^{\sigma,1}(\Omega)} := \iint \frac{|u(x) - u(y)|}{|x-...

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### Sobolev inequality with holes

Classical Sobolev inequality says, $n\geq 3$, we have
\begin{equation}
\left(\int_{\mathbb{R}^n}|u|^{2 n /(n-2)}\right)^{(n-2) /(2 n) } \leq C(n)\left(\int_{\mathbb{R}^n}|\nabla u|^{2}\right)^...

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### The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$

Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...

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### A question about Gauss-Green formula - a weaker assumption

The question I have in mind is the following: how can we prove that for any $v\in H^1(\Omega)$ and for any $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ the Gauss-Green identity takes place
$$\...

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### Varifold convergence of images of Sobolev maps

Suppose I have a sequence of maps $\{f_k:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^{n+1}\}$ such that:
$f_k\rightharpoonup f_*$ weakly in $W^{1,p}(\Omega,\mathbb{R}^{n+1})$,
The images $\Sigma_k:...

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### Domain where the fractional Laplacian operator is a closed operator

Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$
Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\...

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### Bound for the product of Sobolev functions in $W^{s,1}$

I would like to bound the product of two functions $f$, $g$ in the space $W^{s,1}$.
$$ \lVert fg\rVert_{W^{s,1}}\leq c\lVert f \rVert \lVert g \rVert. $$
It seems reasonable to want to use Hölder's ...

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### Elliptic regularity and compact embedding in a weighted Sobolev space

Let $B \subset \mathbb{R}^3$ be the open unit ball centered at $0$ and consider the weight function $w(x)=|x|^2$. Suppose $u \in C_0^\infty(B)$ and consider weighted Sobolev $H^2_w(B)$ norm
$$\|u\|_{H^...

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### First-order interpolation inequalities with weights by L.Caffarelli, R.Kohn and L.Nirenberg

L. Caffarelli, R. Kohn and L. Nirenberg showed in this article that, under some conditions, the following weighted interpolation inequality is valid
THEOREM: There exists a positive constant $C$ such ...

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### Zero trace Sobolev space on Carnot group

Let $\mathbb{G}=(\mathbb{R}^{n},\circ)$ be a Lie group on $\mathbb{R}^n$ and $\mathfrak{g}$ be the corresponding Lie algebra of $\mathbb{G}$. Let $X_{1},\ldots,X_{m}$ be the left invariant vector ...

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### Dense properties of weighted Sobolev space define on $\mathbb{R}^n$

Consider the usual Sobolev space $H^1(\mathbb{R}^n)$ and $H^1_0(\mathbb{R}^n)$, where $H^1_0(\mathbb{R}^n)$ is the closure of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm of $H^1(\mathbb{R}^n)$....

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### Derivative in Sobolev space extended by zero

Let $u(x) \in H^1_0$ is a complex function in sobolev space extension by zero.
How to find $J'(u)$ for
$$
J(u)= \int\limits_0^l |u(x)|^2\operatorname{d\!}x\;??
$$
In $L_2$ it's easy:
$$
J'(u) = \left(\...

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### Bounds on dimension of a subspace

Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that:
$$ \| u\|_{...

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### Optimal constant to compare $L^2$ norm of smooth function on $[0, 1]$ to a grid

Suppose that $f \colon [0, 1] \to \mathbb{R}$ is a $C^\infty$ function satisfying the constraints
$$
f(0) = f'(0) = f(1) = f'(1) = 0, \quad \mbox{and} \quad \int_0^1 (f''(y))^2 \, dy \leq 1.
$$
...

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### Compactly contained subset with Sobolev functions

I recently asked the very same question on math.stackexchange but unfortunately nobody answered thus far and I would "need" an answer rather quickly, so first of all sorry for doubly posting ...

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### The Cauchy problem in general relativity, hyperbolic PDEs, and Sobolev spaces on manifolds

(I apologize in advance if this question is ill-posed or not suitable for Math Overflow, I am not yet a research mathematician, just a student.)
Let $(\Sigma,\bar{g})$ be an $n$-dimensional Riemannian ...

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### Higher integrability for Sobolev functions - part 2

This is a follow-up to the question asked in Higher integrability for Sobolev functions
Updated question: Given the very helpful counterexamples and the ideas, I have the following question: Suppose ...

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### Higher integrability for Sobolev functions

Let $u \in W^{1,2}(\mathbb{R}^2)$ be a given function satisfying $$\frac{1}{|B_r|}\int_{B_r(x)} |\nabla u|^2 dy \leq \frac{1}{r^{\delta}}$$ for all $r \leq 1$, $x \in \mathbb{R}^2$ and some fixed $\...

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### Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?

Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...

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### Lebesgue differentiation theorem at boundary points for Sobolev traces

$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...

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### Perhaps an application of Hardy's inequality

Let $f \in H_{0}^{1}(0,1)$ and $\lambda >0$ big enough. Consider $0 <\alpha < 1$ and some $k > 0$. I would like to show the following inequality
$$
\int_{\lambda^{-k}}^{1}|f(x)|^{2}dx \leq ...

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### Sobolev embedding [closed]

I was trying to understand Sobolev embedding, some results about this topic are not clear to me.
My question is the following:
what are the condition on $p_1 , \alpha_1, p_2 $and $\alpha_2$ for
$W^{...

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### Boundedness of integral under certain assumptions

we have $$\displaystyle\int_{E\times(0,T)}f(x,\tau)^{p-1}\partial_\tau g(x,\tau)\,dx\,d\tau$$
with $f\in C(0,T;L^p(E))$ for some domain $E$ in $\mathbb{R}^n$ and $p>1$. As continuous functions on ...

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### Using a maximum principle to deduce regularity

Suppose $\Omega \subset \mathbb{R}$ is an bounded domain and that $u \in C(0,T; H^{2}) \cap L^{2}(0,T; H^{3})$ where $T >0$.
Consider the PDE on $\Omega \times [0,T]$
$$ \partial_{t}u = a_{1}(x,t) \...

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### Functions whose zero extension are in $H^1$

Let $W^{1,p}(\Omega)$ be the classical Sobolev space on an open set $\Omega\subseteq \mathbb{R}^N$. Denote by $W_0^{1,p}(\Omega)$ the closure of $C_c^\infty(\Omega)$ in $W^{1,p}(\Omega)$.
Question.
...

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### Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$

I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...

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### Interpolation between Sobolev spaces

In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by
$$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$
where $D^sf$ is defined by the Fourier transform
$$(D^...

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### Equivalence between two Sobolev norms on manifolds

On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following.
Use pseudo-differential operators on $M$...

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### Sobolev embedding on sphere

Let $S$ be a two-dimensional sphere, $\Delta$ be the Laplace-Beltrami operator on $S$ and $L^p(S)$, $p\geq 1$, be the usual $L^p$ space of real-valued functions on $S$. We also set $\|f\|_{H^\alpha(S)}...

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### The space of Sobolev maps between Riemannian manifolds

Let $\mathcal{M}, \mathcal{N}$ be two Riemannian manifods. Suppose that $\mathcal{N}$ is properly and isometrically embedded in $\mathbb{R}^n$. The space of Sobolev maps between $\mathcal{M}$ and $\...

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### A question of interpolation space on homogeneous Carnot group

Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows:
A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...

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### A ball with slit at the radius is not $W^{1,1}$-extension domain

Recall that: A domain $\Omega\subset \mathbb{R}^d$ is an $W^{1,1}$-extension domain if there exists an operator $E:W^{1,1}(\Omega)\to W^{1,1}(\mathbb{R^d})$ and a constant $c= c(d,\Omega)>0$ such ...

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### Research in analysis of PDEs

I am currently figuring out what topic to work on for my undergraduate thesis and was able to narrow it down to mathematical analysis. As of now, I have two main options: a thesis working on Sobolev ...

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### Interpolation on Sobolev space on $[0, 1]^d$ over finite meshes

Let $\Omega = [0, 1]^d$ and suppose that $f \colon \Omega \to \mathbb{R}$ lies in order $m > d/2$ Sobolev space; i.e.,
$$
\|f\|_{H^m(\Omega)}^2 = \sum_{|\alpha| \leq m} \|D^\alpha f\|_{L^2(\Omega)}^...

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### Error bounds for Sobolev space norm approximation on a finite grid

Suppose that $f : [0, 1] \to \mathbb{R}$ is an element of the order-$k$ Sobolev space,
\begin{multline}
f^{(k - 1)}~\text{is absolutely continuous},\quad \|f\|_{W^k}^2 := \int_0^1 f^{(k)}(x)^2 \, dx &...

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### Extension for fractional Sobolev spaces, s>0

In their paper, Fractional Sobolev extension and imbedding, the author describes all extension domains for $s \in (0,1)$ -- meaning spaces functions in which are not required to have weak derivatives. ...

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### A question about approximation in Gaussian Sobolev space

Let $\gamma_d$ be the standard Gaussian measure in $\mathbb R^d$, and let $W^{1,2}(\mathbb R^d,\gamma_d)$ be the Gaussian Sobolev space on $\mathbb R^d$. Fix $f_0 \in W^{1,2}(\mathbb R^d,\gamma_d)$.
...

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### Fourier characterization of weighted Sobolev space $W^{1,2}(\mathbb R^n, \gamma_n)$

For integers $n \ge 1$ and $m \ge 0$, the Sobolev space $W^{m,2}(\mathbb R^n)$ is characterized by
$$
f \in W^{m,2}(\mathbb R^n) \text{ iff } \tilde f_m \in L^2(\mathbb R^n),
\label{1}\tag{1}
$$
where ...