Some inequalities related to isoperimetric inequalities with partial free boundary

Details Some inequalities related to isoperimetric inequalities with partial free boundary Some inequalities related to isoperimetric inequalities with partial free boundary

2001-01-09

arxiv.org/abs/math/0101078v2

Mathematics

Analysis of PDEs

16 pages

Scientific paper

The main purpose of this paper is to prove a sharp Sobolev inequality in an exterior of a convex bounded domain. There are two ingredients in the proof: One is the observation of some new isoperimetric inequalities with partial free boundary, and the other is an integral inequality (due to Duff [9]) for any nonnegative function under Schwarz equimeasurable rearrangement. These ingredients also allow us to establish some Moser-Trudinger type inequalities, and obtain some estimates on the principal frequency of a membrane with partial free boundary, which extend early results of Nehari [15] and Bandle [5] for two dimensional domains to the one for any dimensional domains (dimension $\ge 2$)

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