Hokkaido Mathematical Journal

DEPNER Daniel, GARCKE Harald,

Linearized stability analysis of surface diffusion for hypersurfaces with triple lines.

Hokkaido Mathematical Journal, 42 (2013) pp.11-52

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Abstract

The linearized stability of stationary solutions for surface diffusion is studied. We consider three hypersurfaces that lie inside a fixed domain and touch its boundary with a right angle and fulfill a non-flux condition. Additionally they meet at a triple line with prescribed angle conditions and further boundary conditions resulting from the continuity of chemical potentials and a flux balance have to hold at the triple line. We introduce a new specific parametrization with two parameters corresponding to a movement in tangential and normal direction to formulate the geometric evolution law as a system of partial differential equations. For the linearized stability analysis we identify the problem as an H-1-gradient flow, which will be crucial to show self-adjointness of the linearized operator. Finally we study the linearized stability of some examples.

MSC(Primary)35G30
MSC(Secondary)35R35, 35B35, 35K55, 53C44
Uncontrolled Keywordssurface diffusion, partial differential equations on manifolds, linearized stability, gradient flow, triple lines