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...
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| Published in | Interfaces and free boundaries Vol. 26; no. 3; pp. 473 - 488 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
European Mathematical Society Publishing House
01.09.2024
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1463-9963 1463-9971 1463-9971 |
| DOI | 10.4171/ifb/512 |
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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. |
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| AbstractList | 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. 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. 2020 Mathematics Subject Classification. Primary 35N25; Secondary 35J15, 46G05, 47J07. Keywords. Multi-phase problem, transmission problem, overdetermined problem, free boundary problem, shape derivatives, implicit function theorem. |
| Audience | Academic |
| Author | Cavallina, Lorenzo |
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| Snippet | 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... |
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| SubjectTerms | Boundary value problems Mathematical research Symmetry Tests, problems and exercises Torsion |
| Title | Symmetry and asymmetry in a multi-phase overdetermined problem |
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