# Tagged Questions

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

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### Rellich-Kondrachov compacteness theorem for the Euclidean space with Gaussian measure

Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$\gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}.$$ Let $d\gamma_n$ denote the following measure ...
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### Equivalent descriptions of Sobolev spaces on compact manifolds

While reading through a set of lectures on the Laplacian on manifolds, I encountered two descriptions of Sobolev spaces. The first one, valid only for compact manifolds (because it needs to globalize ...
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### Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
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### The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E$, where $\mathcal E$ denotes the ...
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### Prove difference quotient converges to weak derivative in $L^p$

I am trying to solve the following exercise: Let $U$ be an open set in $\mathbb{R}^n$ and let $V$ be a compact subset with Lipschitz boundary. Assume that $f$ is in $L^p(U)$ with $1<p<\infty$ ...
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### Regularity of weak solution for $u_t - \Delta u = f$ with $u(0) = u_0 \in L^1(\Omega)$

Let $\Omega$ be a bounded domain, and consider the equation $$u_t - \Delta u = f$$ $$u(0) = u_0 \in L^1(\Omega)$$ with Neumann BCs (or Dirichlet if convenient) where $f$ is smooth. Using energy/...
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### if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
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### What is a good way to prove the weak form of the Sobolev embedding theorem?

From Richard Melrose, From microlocal to global analysis, Chapter 1, Problem 11. Suppose $u\in L^{2}(\mathbb{R}^{n})$ and that $$D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n})$$ ...
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### Sobolev estimation of second derivative against Laplacian and higher terms

Given $u \in H^2(\Omega)$ (and $\Omega \subseteq \mathbb{R}^n$ with appropriate properties) is there a way to estimate the norm of the second derivative $\Vert D^2 u\Vert_{L^2(\Omega)}^2$ against the ...
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### $W_0^{1,\:p}(\Lambda)$ is dense in $L^2(\Lambda)$
Let $d\in\mathbb N$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Lambda\subseteq\mathbb R^d$ be open with $\lambda(\Lambda)<\infty$ $p\ge 2$ $W^1(\Lambda)$ denote the set of weakly ...