2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60726The 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 $\ge 2$)16 pagesAnalysis of PDEsClassical Analysis and ODEsDifferential Geometrytext