The problem of minimizing ∫ [∇ υ |² + q²(x)λ ²(υ )]dx in an appropriate class of functions υ is considered. Here q(x) ≠ 0 and λ ²(υ ) = λ ₁²if υ < 0, = λ ₂² if υ > 0. Any minimizer u is harmonic in \ u ≠ 0\ and |∇ u|² has a jump \[ {q^2}(x) ( {λ _1^2 - λ _2^2} )\] across the free boundary \ u ≠ 0\. Regularity and various properties are established for the minimizer u and for the free boundary.
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Alt et al. (1984) studied this question.