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Some properties of monotone rearrangement with applications to elliptic equations in cylinders

✍ Scribed by H. Berestycki; T. Lachand-Robert

Publisher
John Wiley and Sons
Year
2004
Tongue
English
Weight
236 KB
Volume
266
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

This paper is concerned with properties and applications of monotone rearrangement as defined in [18]. Unlike the Schwarz and Steiner symmetrization, in the monotone rearrangement framework, the inequality for the Dirichlet integral holds for all functions in H^1^ (not only H^1^~0~). Further, we state and prove a necessary and sufficient condition for equality to hold in this inequality.

These properties allow us to give several applications of monotone rearrangement to semilinear elliptic equations. We study such problems in cylinders with general boundary conditions (Neumann, mixed and Robin type). For these problems, we establish the existence of solutions which are monotone in one direction. (Β© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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