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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Variational problems with two phases and their free boundaries


Authors: Hans Wilhelm Alt, Luis A. Caffarelli and Avner Friedman
Journal: Trans. Amer. Math. Soc. 282 (1984), 431-461
MSC: Primary 49A29; Secondary 35J85
MathSciNet review: 732100
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Abstract: The problem of minimizing $ \int {[\nabla \upsilon {\vert^2}} + {q^2}(x){\lambda ^2}(\upsilon )]dx$ in an appropriate class of functions $ \upsilon $ is considered. Here $ q(x) \ne 0$ and $ {\lambda ^2}(\upsilon ) = \lambda _1^2$if $ \upsilon < 0, = \lambda _2^2$ if $ \upsilon > 0$. Any minimizer $ u$ is harmonic in $ \{ u \ne 0\} $ and $ \vert\nabla u{\vert^2}$ has a jump

$\displaystyle {q^2}(x)\left( {\lambda _1^2 - \lambda _2^2} \right)$

across the free boundary $ \{ u \ne 0\} $. Regularity and various properties are established for the minimizer $ u$ and for the free boundary.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0732100-6
PII: S 0002-9947(1984)0732100-6
Article copyright: © Copyright 1984 American Mathematical Society