Vol. 3, No. 4, 2010

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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
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