Plane-like minimal surfaces in periodic media with inclusions

textThe mathematical areas of minimal surfaces and homogenization of PDE have been subjects of research for many decades. In this work we consider the particular case of minimal surfaces in heterogeneous media. We prove theoretical results and develop numerical algorithms that characterize the behavior of these surfaces. We also show that this work turns out to be related to the theory of homogenization of Hamilton-Jacobi equations. In this work we think of Rn as a lattice of points with integer coordinates, where each cube of edge length 1 has an internal inclusion (we can think of an inclusion as a hole or as a part of the domain containing another material). All inclusions are compact and periodic. Within this framework we measure the area of a surface of codimension one by neglecting the parts that are inside the inclusions, and measuring the outside parts in the standard way. We say that the surface is a minimal surface if any compact perturbation of it increases the area (in this degenerate metric). In this work we prove the existence of minimal surfaces that always stay at a bounded distance (universal) from a given hyperplane. While we know that the surface is smooth outside the inclusions, in this work we prove a result concerning the behaviour of the minimal surfaces at the boundary: that the intersection between the inclusions and the surface locally looks like two perpendicular hyperplanes. Within this degenerate metric, the smallest distance between two points is no longer a line. We analyze the behavior of this distance when the edge length of the cube goes to zero. In particular, we want to find, for the case n = 2 and the inclusions being closed balls, the effective norm in the homogenized limit. The effective norm depends on the radius of the inclusions, and our results suggest that as the radius gets smaller the behavior of the effective norm changes, though it is always polygonal with more and more sides, until it becomes a circle in the limit. We implement an algorithm to compute our weighted minimal surfaces. We extend the Bence-Merriman-Osher algorithm to the case of heterogeneous domains. We implement the algorithm in 2 and 3 dimensions using adaptive finite element methods.Mathematic