Elliptic Problems Involving Phase Boundaries Satisfying a Curvature Condition |
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Authors: | CAGINALP G; FIFE P C |
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Institution: |
Mathematics Department, University of Pittsburgh Pittsburgh, PA 15260
Mathematics Department, University of Arizona Tucson, AZ 85721
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Abstract: | Two theorems related to equilibrium free-boundary problems arepresented. One arises as a time-independent solution to thephase-field equations. The other is the relevant time-independentproblem for the Stefan model, modified for the surface tensioneffect. It also serves as a preliminary result for the phase-fieldformulation. Under appropriate conditions, we prove that, givenan appropriate positive constant and a smooth function u: R;,where is an annular domain in R2, there exists a curve suchthat u(x)=K(x) for all x , where K is the curvature.Using this result, we prove the existence of solutions to O=2+ ?(3) + 2u that have a transition layer behaviour (from=1 to =+1) for small and make the transition on thecurve . This proves there exist solutions to the phase fieldmodel that satisfy a Gibbs-Thompson relation. |
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