Advances in Differential Equations

On Serrin's symmetry result in nonsmooth domains and its applications

Juraj Földes

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Abstract

The paper investigates overdetermined fully nonlinear problems on nonsmooth domains. Under natural regularity assumptions on solutions it is shown that overdetermined problems on reflectionally symmetric, bounded domains can have positive solutions only if the domain is a ball. These results are extensions of results of Serrin, who proved this statement for smooth solutions on smooth domains. The results for overdetermined problems are applied to a study of Dirichlet problems, specifically to the question when a nonnegative solution is positive or zero everywhere. As a consequence, an extension of symmetry results of Gidas--Ni--Nirenberg to nonnegative solutions is obtained.

Article information

Source
Adv. Differential Equations Volume 18, Number 5/6 (2013), 523-548.

Dates
First available in Project Euclid: 14 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1363266256

Mathematical Reviews number (MathSciNet)
MR3086464

Zentralblatt MATH identifier
1272.35018

Subjects
Primary: 35N25: Overdetermined boundary value problems 35B06: Symmetries, invariants, etc. 35J25: Boundary value problems for second-order elliptic equations 35J60: Nonlinear elliptic equations 35B09: Positive solutions

Citation

Földes, Juraj. On Serrin's symmetry result in nonsmooth domains and its applications. Adv. Differential Equations 18 (2013), no. 5/6, 523--548. https://projecteuclid.org/euclid.ade/1363266256.


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