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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Asymptotics of Sobolev embeddings and singular perturbations for the $p$-Laplacian
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by Manuel del Pino and César Flores PDF
Proc. Amer. Math. Soc. 130 (2002), 2931-2939 Request permission

Abstract:

We consider the best constant $S(\Omega _\lambda )$ for the embedding of $W^{1,p} (\Omega _\lambda )$ into $L^q(\Omega _\lambda )$ where $1<p<2$, $p<q< {Np\over N-p}$. Here $\Omega _\lambda = \lambda \Omega$ with $\Omega$ a smooth, bounded domain in $\mathbb {R}^n$ and $\lambda$ a large positive number. It is proven by the validity of the expansion \begin{equation} S( \Omega _\lambda ) = S(\mathbb {R}^n_+) - \lambda ^{-1} \gamma \max _{x\in \partial \Omega } H(x) + o ( \lambda ^{-1} ), \nonumber \end{equation} as $\lambda \to \infty$, where $\gamma$ is a positive constant depending on $p,q$ and $N$. The behavior of associated extremals, which satisfy an equation involving the $p$-Laplacian operator, is also analyzed.
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Additional Information
  • Manuel del Pino
  • Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMR2071 CNRS-UChile), Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile
  • MR Author ID: 56185
  • Email: delpino@dim.uchile.cl
  • César Flores
  • Affiliation: Departamento de Matemáticas, FCFM Universidad de Concepción, Casilla 160-C, Concepción, Chile
  • Email: cflores@dim.uchile.cl
  • Received by editor(s): May 1, 2001
  • Published electronically: April 10, 2002
  • Additional Notes: This work was supported by grants Fondecyt Lineas Complementarias 8000010, DIUC 200.015.015-1.0, ECOS/CONICYT, and FONDAP

  • Dedicated: To the memory of Carlos Cid
  • Communicated by: David S. Tartakoff
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 130 (2002), 2931-2939
  • MSC (2000): Primary 35J20; Secondary 35B40
  • DOI: https://doi.org/10.1090/S0002-9939-02-06535-8
  • MathSciNet review: 1908916