Sharp Interface Limits for Diffuse Interface Models with Contact Angle

We consider the sharp interface limit for the Allen-Cahn equation and some variants in a bounded smooth domain in the case of boundary contact. The Allen-Cahn equation is a diffuse interface model since (after a short generation time) solutions typically develop so-called diffuse interfaces, where the solution stays smooth but experiences steep gradients. Moreover, the equation contains a small parameter $\varepsilon>0$ that corresponds to the thickness of the diffuse interfaces. The limit $\varepsilon\rightarrow 0$ is called \enquote{sharp interface limit} because - at least heuristically - the solutions should converge to step functions with the jump set evolving in time according to some sharp interface problem. We show the rigorous sharp interface limit, i.e.~that solutions to the diffuse interface and the sharp interface model are related rigorously. The results are local in time and applicable as long as a smooth solution to the limit problem exists. We consider the following cases: \begin{itemize} \item Convergence of the Allen-Cahn equation with Neumann boundary condition to the mean curvature flow with $90^\circ$-contact angle in any dimension $N\geq 2$. \item Convergence of the vector-valued Allen-Cahn equation involving different choices for the potential and with Neumann boundary condition to the mean curvature flow with $90^\circ$-contact angle in any dimension $N\geq 2$, but without the triple junction situation. For this case we expect that a similar strategy works. We give some comments in this direction. \item Convergence of an Allen-Cahn equation with a non-linear Robin boundary condition to the mean curvature flow with an $\alpha$-contact angle in 2D for $\alpha$ close to $90^\circ$. \end{itemize} For the convergence proofs we use the method of de Mottoni, Schatzman \cite{deMS}, i.e.~we \begin{enumerate} \item Rigorously construct an approximate solution for the diffuse interface model with asymptotic expansions. \item Estimate the difference of the exact and approximate solution to the diffuse interface model with a spectral estimate for a linear operator associated to the model. \end{enumerate} The major novelty in the thesis is the consideration of boundary contact for the diffuse interfaces within the method of \cite{deMS}. Therefore we construct suitable curvilinear coordinates. Based on the latter we rigorously set up the asymptotic expansions. In this process new parameter-dependent elliptic problems on the half space in $\mathbb{R}^2$ appear. For the $90^\circ$-case these problems are solved with a splitting method in exponentially weighted Sobolev spaces. The latter seems not possible for angles $\alpha\neq 90^\circ$ and we use the Implicit Function Theorem with respect to $\alpha$ in this case. Moreover, for the spectral estimate for the Allen-Cahn operator in every case (which is obtained by linearization at the approximate solution) we use a new idea: we construct an approximate first eigenfunction using asymptotic expansions. Then we split the space of $H^1$-functions over the domain into a \enquote{small} explicit space formally approximating the first eigenfunctions and the complementing space. Finally, we analyze the associated bilinear form on every part