Variational problems with two phases and their free boundaries

Abstract

The problem of minimizing ∫ [ ∇ υ | 2 + q 2 ( x ) λ 2 ( υ ) ] d x \int {[\nabla \upsilon {|^2}} + {q^2}(x){\lambda ^2}(\upsilon )]dx in an appropriate class of functions υ \upsilon is considered. Here q ( x ) ≠ 0 q(x) \ne 0 and λ 2 ( υ ) = λ 1 2 {\lambda ^2}(\upsilon ) = \lambda _1^2 if υ > 0 , = λ 2 2 \upsilon > 0, = \lambda _2^2 if υ > 0 \upsilon > 0 . Any minimizer u u is harmonic in { u ≠ 0 } \{ u \ne 0\} and | ∇ u | 2 |\nabla u{|^2} has a jump \[ q 2 ( x ) ( λ 1 2 − λ 2 2 ) {q^2}(x)\left ( {\lambda _1^2 - \lambda _2^2} \right ) \] across the free boundary { u ≠ 0 } \{ u \ne 0\} . Regularity and various properties are established for the minimizer u u and for the free boundary.