On the best constants in Sobolev inequalities on the solid torus in the limit case p=1

被引：6

作者：

Labropoulos, Nikos ^{[1
]}

Radulescu, Vicentiu D. ^{[2
,3
]}

机构：

[1] Univ Patras, Dept Math, GR-26110 Patras, Greece

[2] King Abdulaziz Univ, Fac Sci, Dept Math, POB 80203, Jeddah 21589, Saudi Arabia

[3] Romanian Acad, Inst Math Simion Stoilow, POB 1-764, Bucharest 014700, Romania

来源：

ADVANCES IN NONLINEAR ANALYSIS | 2016年 / 5卷 / 03期

关键词：

Sobolev inequalities; best constants; limit case; manifolds with boundary; symmetries; solid torus; RIEMANNIAN-MANIFOLDS; TRACE INEQUALITIES; SYMMETRIES; BOUNDARY; SPACES; MAP;

D O I：

10.1515/anona-2015-0125

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper, we analyze the problem of determining the best constants for the Sobolev inequalities in the limiting case vvherep = 1. Firstly, the special case of the solid torus is studied, whenever it is proved that the solid torus is an extremal domain with respect to the second best constant and totally optimal with respect to the best constants in the trace Sobolev inequality. Secondly, in the spirit of Andreu, Mazon and Rossi [3], a Neumann problem involving the 1-Laplace operator in the solid torus is solved. Finally, the existence of both best constants in the case of a manifold with boundary is studied, when they exist. Further examples are provided where they do not exist. The impact of symmetries which appear in the manifold is also discussed.

引用

页码：261 / 291

页数：31

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