# 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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### Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$?

Can someone kindly confirm that the Schwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all ...
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### Relation between the norm of Sobolev space $H^1$ and $L^p$ norm for non-increasing radial functions

I am interested to find $$\sup\|u\|_{p}^{p},$$ when $u$ are non-increasing radial functions on the unit ball $B_1$ of $\mathbb{R}^{n}$ such that $$\|u\|_{H1}^2 < r$$ for some $r > 0$. Since $u$ ...
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### Self-ajointness of the Laplacian over a Riemannian manifold with boundary

I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf). Let $(M,g)$ be a Riemannian manifold with boundary; $E\to M$ be an hermitian fiber bundle; $\Delta$ ...
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### Action of fractional Laplacian on Hölder / Besov spaces on Riemannian manifolds

Let $M$ be a compact Riemannian manifold (without boundary) and $\Delta$ be the corresponding (positive) Laplace-Beltrami operator. We also define the operators $I^s = (\mathrm{Id} + \Delta)^{-s}$ for ...
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### Reference for an extension theorem for Neumann boundary data

$\DeclareMathOperator\Tr{Tr}$Let $\Omega \subset \mathbb{R}^d$ be a smooth bounded domain (we denote by $n$ the normal to $\partial\Omega$) and $p\in(1,\infty)$. Do you know where I can find (book or ...
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### Deriving the general interior elliptic estimate from the compactly supported case

This is an exercise (10.3.4 in the third edition) from Nicolaescu's Lectures on the Geometry of Manifolds. Let $L$ be an elliptic differential operator of order $k$ and $1 < p < \infty$. The ...
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### Estimatives for elliptic systems involving the laplacian

Considering the problem \begin{equation} \left\{ \begin{array}[c]{11} \Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\ \Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\ \end{...
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### Stability of Hajłasz-Sobolev class under post-composition

Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev? Assumptions/Setup Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
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### On Sobolev spaces on domains in Riemannian manifolds

There is extensive literature on Sobolev spaces on complete Riemannian manifolds but are there any standard references regarding the definition and properties of Sobolev spaces on domains (possessing ...
I use a lot in the study of pde bounded Lipschitz domains $\Omega\subseteq\mathbb{R}^N$. However I have noticed that there are some major differences in their definitions. I will put here two of them, ...