Topological Methods in Nonlinear Analysis

Degree and Sobolev spaces

Haïm Brezis, Yanyan Li, Petru Mironescu, and Louis Nirenberg

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Let $u$ belong (for example) to $W^{1,n+1}(S^n\times \Lambda, S^n)_{\lambda\in\Lambda}$ where $\Lambda$ is a connected open set in ${\mathbb R}^k$. For a.e. $\lambda\in\Lambda$ the map $x\mapsto u(x,\lambda)$ is continuous from $S^n$ into $S^n$ and therefore its (Brouwer) degree is well defined. We prove that this degree is independent of $\lambda$ a.e. in $\Lambda$. This result is extended to a more general setting, as well to fractional Sobolev spaces $W^{s,p}$ with $sp\geq n+1$.

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Topol. Methods Nonlinear Anal., Volume 13, Number 2 (1999), 181-190.

First available in Project Euclid: 29 September 2016

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Brezis, Haïm; Li, Yanyan; Mironescu, Petru; Nirenberg, Louis. Degree and Sobolev spaces. Topol. Methods Nonlinear Anal. 13 (1999), no. 2, 181--190.

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