The vector-valued Allen-Cahn equation with Robin boundary conditions converges locally in time to mean curvature flow with contact angle, and the limits solve harmonic heat flow and a minimal pair condition.
BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness
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abstract
We study weak solutions to mean curvature flow satisfying Young's angle condition for general contact angles $\alpha \in (0,\pi)$. First, we construct BV solutions using the Allen-Cahn approximation with boundary contact energy as proposed by Owen and Sternberg. Second, we prove the weak-strong uniqueness and stability for this solution concept. The main ingredient for both results is a relative energy, which can also be interpreted as a tilt excess.
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The vector-valued Allen-Cahn equation with potentials of high-dimensional double-wells under Robin boundary conditions
The vector-valued Allen-Cahn equation with Robin boundary conditions converges locally in time to mean curvature flow with contact angle, and the limits solve harmonic heat flow and a minimal pair condition.