Advances in Differential Equations
- Adv. Differential Equations
- Volume 18, Number 5/6 (2013), 523-548.
On Serrin's symmetry result in nonsmooth domains and its applications
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.
Adv. Differential Equations Volume 18, Number 5/6 (2013), 523-548.
First available in Project Euclid: 14 March 2013
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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