Symmetry and asymmetry in a multi-phase overdetermined problem

  • Lorenzo Cavallina

    Tohoku University, Sendai, Japan
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Abstract

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.

Cite this article

Lorenzo Cavallina, Symmetry and asymmetry in a multi-phase overdetermined problem. Interfaces Free Bound. 26 (2024), no. 3, pp. 473–488

DOI 10.4171/IFB/512