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The best Sobolev trace constant as limit of the usual Sobolev constant for small strips near the boundary

Published online by Cambridge University Press:  14 July 2008

José M. Arrieta
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain (arrieta@mat.ucm.es; arober@mat.ucm.es)
Aníbal Rodríguez-Bernal
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain (arrieta@mat.ucm.es; arober@mat.ucm.es)
J. D. Rossi
Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina (jrossi@dm.uba.ar) Present address: IMDEA-Matemáticas, C-IX, Universidad Autónoma de Madrid, Campus Cantoblanco, Madrid, Spain
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Abstract

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In this paper we prove that the best constant in the Sobolev trace embedding $H^1(\varOmega)\hookrightarrow L^q(\partial\varOmega)$ in a bounded smooth domain can be obtained as the limit as $\varepsilon\to0$ of the best constant of the usual Sobolev embedding $H^1(\varOmega) \hookrightarrow L^q(\omega_\varepsilon,\mathrm{d} x/\varepsilon)$, where $\omega_\varepsilon=\{x\in\varOmega:\mathrm{dist}(x,\partial\varOmega)<\varepsilon\}$ is a small neighbourhood of the boundary. We also analyse symmetry properties of extremals of the latter embedding when $\varOmega$ is a ball.

Type
Research Article
Copyright
2008 Royal Society of Edinburgh