Generalized Snell's Law for Weighted Minimal Surface in Heterogeneous Media

Abstract

The weighted minimal surface problem in heterogeneous media is studied in this paper. The solution to the weighted minimal surface problem is continuous but the derivatives have a jump across the interface where the medium property is discontinuous. The jump condition of the derivatives derived in this paper generalized the Snell's law in geometric optics to weighted minimal surfaces of co-dimension one in any dimensional space. A numerical method based on the gradient flow and the maximum principal preserving immersed interface method is developed to solve this nonlinear elliptic interface problem with jump conditions. Numerical computations are presented to verify both the analysis and the numerical algorithm. 1. Introduction. The minimal surface problem, that is, the problem of finding the surface of the least area among all surfaces having fixed boundary data, has been extensively studied. A recent workshop on minimal surfaces presented the latest re- search on minimal surface applications in chemistry and biology (5). Many phenomena that occur in nature relate to this problem which has been a motivation for devel- oping new mathematical theories and techniques to solve the problem analytically and numerically. Minimal surfaces were shown to be important in various chemical micro-structures and their corresponding phase transitions (5). Computer graphics and image analysis use minimal surfaces frequently for boundary detection, and to construct surfaces that are visually appealing (2), (15). Soap films and other mem- branes passing through a fixed boundary provide mechanical examples of minimal surfaces (14). A related concept is the idea of capillary surfaces, which result from surface tension in liquids. These surfaces are closely related to minimal surfaces (6). For a precise mathematical description of the minimal surface problem we refer, for example, to the classical treatises (7) and (16). The minimal surface problem can be described in two different ways, using the parametric or the non-parametric formulation. In the non-parametric setting the candidate surfaces are graphs of functions, while in the parametric setting the surfaces are treated as boundaries of sets (7). The former is usually seen in more physically- based treatments of the problem, whereas the later provides an excellent framework for the mathematical analysis of minimal surfaces. When the medium is homogeneous, the energy density at each point is constant, and therefore the surface energy is equivalent to the surface area. This is the standard minimal surface problem. In this paper, we consider the weighted minimal surface problem in a heterogeneous medium in which the energy density is piecewise smooth. For example this is the case for capillary interfaces in porous media or composite materials. In particular we derive a jump condition for the weighted minimal surface at the interface between two different media. The jump condition can be regarded as a generalized Snell's law which describes the refraction of minimal surfaces instead of light rays in geometric optics.