Symmetry and asymmetry in a multi-phase overdetermined problem

A celebrated theorem of Serrin asserts that one overdetermined condition on the boundary is enough to obtain radial symmetry in the so-called one-phase overdetermined torsion problem. It is also known that imposing just one overdetermined condition on the boundary is not enough to obtain radial symm...

Full description

Saved in:
Bibliographic Details
Published inInterfaces and free boundaries Vol. 26; no. 3; pp. 473 - 488
Main Author Cavallina, Lorenzo
Format Journal Article
LanguageEnglish
Published European Mathematical Society Publishing House 01.09.2024
Subjects
Boundary value problems
Mathematical research
Symmetry
Tests, problems and exercises
Torsion
Japan
Online AccessGet full text

Cover

More Information
Summary:A celebrated theorem of Serrin asserts that one overdetermined condition on the boundary is enough to obtain radial symmetry in the so-called one-phase overdetermined torsion problem. It is also known that imposing just one overdetermined condition on the boundary is not enough to obtain radial symmetry in the corresponding multi-phase overdetermined problem. In this paper we show that, in order to obtain radial symmetry in the two-phase overdetermined torsion problem, two overdetermined conditions are needed. Moreover, it is noteworthy that this pattern does not extend to multi-phase problems with three or more layers, for which we show the existence of nonradial configurations satisfying countably infinitely many overdetermined conditions on the outer boundary.
ISSN:1463-9963
1463-9971
DOI:10.4171/ifb/512