ALEX 2018 Workshop: Abstracts

Sharp interface limit for the Allen–Cahn equation with a contact angle

Helmut Abels and Maximilian Moser

University of Regensburg, Faculty of Mathematics (Germany)

We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary conditions in a two-dimensional domain $\mathrm{\Omega }$, in the situation where an interface has developed and intersects $\partial \mathrm{\Omega }$. Here a parameter $\epsilon >0$ in the equation, which is related to the thickness of the diffuse interface, is sent to zero. The limit problem is given by mean curvature flow with a $\pi /2$-contact angle condition and convergence using strong norms is shown for small times. Here we assume that a smooth solution to this limit problem exists on $[0,T]$ for some $T>0$ and that it can be parametrized suitably. The strategy is as in Chen, Hilhorst, Logak [3] and Abels, Liu [1]: With asymptotic expansions we construct an approximate solution ${(u_A^{\epsilon })}_{\epsilon \in (0,{\epsilon }_0]}$ for the Allen-Cahn equation and estimate the difference of the exact and approximate solution with the aid of a spectral estimate for the (around $u_A^{\epsilon }(.,t)$) linearized Allen-Cahn operator $-\mathrm{\Delta }+\frac{1}{{\epsilon }^2}{f}^{\mathrm{\prime \prime }}(u_A^{\epsilon }(.,t))$ for $t\in [0,T]$. Here the main new difficulty lies in the contact points. Therefore a suitable curvilinear coordinate system based on work of Vogel [4] is constructed. For the asymptotic expansion and the proof of the spectral estimate also ideas from Alikakos, Chen, Fusco [2] are used.

Acknowledgments: This research and the second author was supported financially by the DFG-RTG 1692 ”Curvature, Cycles, and Cohomology”. The support is gratefully acknowledged.