Geodesic
Symmetry and asymmetry in a multi-phase overdetermined problem
Lorenzo Cavallina1 orcid
1Tohoku University, Sendai, Japan
Interfaces and free boundaries, Tome 26 (2024) no. 3, pp. 473-488

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DOI

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.
DOI : 10.4171/ifb/512
Classification : 35N25, 35J15, 46G05, 47J07
Mots-clés : multi-phase problem, transmission problem, overdetermined problem, free boundary problem, shape derivatives, implicit function theorem

Lorenzo Cavallina  1

1 Tohoku University, Sendai, Japan 
Lorenzo Cavallina. Symmetry and asymmetry in a multi-phase overdetermined problem. Interfaces and free boundaries, Tome 26 (2024) no. 3, pp. 473-488. doi: 10.4171/ifb/512
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     title = {Symmetry and asymmetry in a multi-phase overdetermined problem},
     journal = {Interfaces and free boundaries},
     pages = {473--488},
     year = {2024},
     volume = {26},
     number = {3},
     doi = {10.4171/ifb/512},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/512/}
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