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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.

587 questions
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Question on Sobolev spaces in domains with boundary

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Define the Sobolev norm on $C^\infty(\bar \Omega)$ $$||u||_{W^{1,2}}:=\sqrt{\int_\Omega (|\nabla u|^2+u^2)dx}.$$ ...
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Let $\mathbb{D}^n$ be the closed unit disk, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be harmonic; suppose that $n \ge 2$, and that $\det df >0$ a.e. on $\mathbb{D}^n$. Are there harmonic maps $\... 0answers 77 views Discrete Sobolev embedding It is true in one dimension that$H^1$is continuously embedded in$L^{\infty}.$Now, consider a compact interval$[0,1]$with a partition$I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$and ... 0answers 198 views Regularity result for the boundary value problem for the heat equation Let$\Omega$be an open bounded subset of$\mathbb R^N$. Let$u_0 \in L^\infty(\Omega)$and$f \in L^\infty((0,T)\times\Omega).$Consider the following boundary value problem for the heat equation: ... 0answers 150 views Is polar decomposition of a smooth map Sobolev? Motivation: Let$\mathbb{D}^2$be the closed unit disk. I am studying the "elastic energy" functional$E(f)=\int_{\mathbb{D}^2} \text{dist}^2(df,\text{SO}_2)$, where$f \in C^{\infty}(\mathbb{D}^2,\...
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The setting-a "linear algebra" fact: Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \... 0answers 84 views Relationship between$p$-capacity and Riesz$s$-capacity of a set What is the relationship between the definitions of$s$-capacity (page 13 here) and$p\$-capacity (here) of a set? Are they equivalent? If not, what inequalities hold? What is the difference (in ...

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山西福彩快乐十分钟

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