One-sided Mullins-Sekerka flow does not preserve convexity

Uwe F. Mayer

Abstract: The Mullins-Sekerka model is a nonlocal evolution model for hypersurfaces, which arises as a singular limit for the Cahn-Hilliard equation. Assuming the existence of sufficiently smooth solutions we will show that the one-sided Mullins-Sekerka flow does not preserve convexity. The main tool is the strong maximum principle for elliptic second order differential equations.

Key words: Mullins-Sekerka flow, Hele-Shaw flow, Cahn-Hilliard equation, free boundary problem, convexity, curvature, Laplace operator.

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