Publication 15-CNA-015

Regularity In Time for Weak Solutions of a Continuum Model for Epitaxial Growth with Elasticity on Vicinal Surfaces

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu

Giovanni Leoni
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
giovanni@andrew.cmu.edu

Xin Yang Lu
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
xinyang@andrew.cmu.edu

Abstract: The evolution equation derived by Xiang (SIAM J. Appl. Math. 63:241-258, 2002) to describe vicinal surfaces in heteroepitaxial growth is $$ht = - \left[H(h_x) + (h^{-1}_2 + h_x)H_{xx}\right]_{xx}, {(1)}$$ where $h$ denotes the surface height of the film, and $H$ is the Hilbert transform. Existence of solutions was obtained by Dal Maso, Fonseca and Leoni (Arch. Rational Mech. Anal. 212: 1037-1064, 2014). The regularity in time was left unresolved. The aim of this paper is to prove existence, uniqueness, and Lipschitz regularity in time for weak solutions, under suitable assumptions on the initial datum.

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