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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Variational problems with two phases and their free boundaries
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by Hans Wilhelm Alt, Luis A. Caffarelli and Avner Friedman PDF
Trans. Amer. Math. Soc. 282 (1984), 431-461 Request permission

Abstract:

The problem of minimizing $\int {[\nabla \upsilon {|^2}} + {q^2}(x){\lambda ^2}(\upsilon )]dx$ 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 $|\nabla u{|^2}$ has a jump \[ {q^2}(x)\left ( {\lambda _1^2 - \lambda _2^2} \right )\] across the free boundary $\{ u \ne 0\}$. Regularity and various properties are established for the minimizer $u$ and for the free boundary.
References
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 282 (1984), 431-461
  • MSC: Primary 49A29; Secondary 35J85
  • DOI: https://doi.org/10.1090/S0002-9947-1984-0732100-6
  • MathSciNet review: 732100