Variational problems with two phases and their free boundaries

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.