Elliptic Problems Involving Phase Boundaries Satisfying a Curvature Condition

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Abstract
Two theorems related to equilibrium free-boundary problems are presented. One arises as a time-independent solution to the phase-field equations. The other is the relevant time-independent problem for the Stefan model, modified for the surface tension effect. It also serves as a preliminary result for the phase-field formulation. Under appropriate conditions, we prove that, given an appropriate positive constant σ and a smooth function u: Ω→R;, where Ω is an annular domain in R 2 , there exists a curve Γ such that u ( x )=—σK( x ) for all x ε Γ, where K is the curvature. Using this result, we prove the existence of solutions φ to O=ξ 2 Δφ + ½(φ—φ 3 ) + 2uξ that have a transition layer behaviour (from φ=—1 to φ=+1) for small ξ and make the transition on the curve Γ. This proves there exist solutions to the phase field model that satisfy a Gibbs-Thompson relation.

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