Quasiconcave Solutions to Elliptic Problems in Convex Rings

被引：50

作者：

Bianchini, Chiara ^{[1
]}

Longinetti, Marco ^{[2
]}

Salani, Paolo ^{[1
]}

机构：

[1] Univ Florence, Dipartimento Matemat U Dini, I-50134 Florence, Italy

[2] Univ Florence, Dipartimento Ingn Agr & Forestale, I-50139 Florence, Italy

来源：

INDIANA UNIVERSITY MATHEMATICS JOURNAL | 2009年 / 58卷 / 04期

关键词：

convexity; level sets; elliptic equations; quasiconcave envelope; Minkowski addition; PARTIAL-DIFFERENTIAL EQUATIONS; VISCOSITY SOLUTIONS; LEVEL SETS;

D O I：

10.1512/iumj.2009.58.3539

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We investigate the convexity of level sets of solutions to general elliptic equations in a convex ring Q. In particular, if u is a classical solution which has constant (distinct) values on the two connected components of partial derivative Omega, we consider its quasi-concave envelope u* (i.e., the function whose superlevel sets are the convex envelopes of those of u) and we find suitable assumptions which force u* to be a subsolution of the equation. If a comparison principle holds, this yields u = u* and then u is quasi-concave.

引用

页码：1565 / 1589

页数：25

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