Mathematical Sciences: New Directions in the Phase Field Approach to Free Boundary Problems

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With this award the principal investigator will continue his analytical studies of free boundary problems using asymptotic methods and he will initiate attempts to construct efficient numerical schemes for the resolution of interfacial phenomena. Overall he plans to develop a unified theory of free boundary problems that arise in phase transitions based upon the Phase Field Model that he himself has helped to popularize. The principal investigator will consider progressively realistic models of alloys and contact with external containers. Many technologically important natural phenomena involve the behavior of different media along the interfaces or surfaces of contact between them. With this award the principal investigator will continue his important work on developing approximate solutions of models of phase transitions using asymptotic methods and he will begin developing numerical schemes in order to resolve increasingly complicated interfacial phenomena. This combined asymptotic-numerical approach should pay big dividends toward understanding the physics of transition layers and free boundary problems.