Mean curvature motion of graphs with constant contact angle at a free boundary

Freire, Alexandre

doi:10.2140/apde.2010.3.359

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Mean curvature motion of graphs with constant contact angle at a free boundary

Alexandre Freire

Vol. 3 (2010), No. 4, 359–407

DOI: 10.2140/apde.2010.3.359

Abstract

We consider the motion by mean curvature of an $n$ -dimensional graph over a time-dependent domain in $ℝ^{n}$ intersecting $ℝ^{n}$ at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary and derive a continuation criterion based on the second fundamental form. If the initial graph is concave, we show this is preserved and that the solution exists only for finite time. This corresponds to a symmetric version of mean curvature motion of a network of hypersurfaces with triple junctions with constant contact angle at the junctions.

Keywords

mean curvature flow, triple junctions, free boundaries

Mathematical Subject Classification 2000

Primary: 35K55, 53C44

Milestones

Received: 8 December 2008

Revised: 8 October 2009

Accepted: 17 October 2009

Published: 8 September 2010

Authors

Alexandre Freire
Department of Mathematics University of Tennessee Knoxville, TN 37996-1300 United States
http://www.math.utk.edu/~freire

Vol. 3, No. 4, 2010