Computing the area-minimizing surface by the Allen-Cahn equation with the fixed boundary

Abstract

The Allen-Cahn equation is a famous nonlinear reaction-diffusion equation used to study geometric motion and minimal hypersurfaces. This link has been scrutinized to construct minimal surfaces for many years. The shape of soap film is very interesting, and it can stimulate mathematical inspirations since it explains curvatures and equilibrium shapes in nature. There are many interesting ways to create area-minimizing surfaces with the boundaries, called frame boundaries. However, dealing with surface's ends (boundaries) numerically is not easy for constructing surfaces. This paper presents a mathematical formulation and numerical construction of area-minimizing surfaces, also known as minimal surfaces. We use differential geometry knowledge for numerical verification. The proposed numerical scheme involves fixed frame boundary conditions in the Laplacian operator. We treat the Laplacian with the constraint implicitly and explicitly solve the nonlinear free energy term. This approach ensures stable and efficient construction of area-minimizing surfaces with frame boundaries. In the numerical aspect, we suggest the construction of minimal surfaces by illustrating two classical examples, which are Scherk's minimal surface and catenoid. Both examples have the frame boundaries. Scherk's first surface is a doubly periodic, complete and properly embedded one with parallel ends. The catenoid is formed between two coaxial circular rings and is classified mathematically as the only properly embedded minimal surface with two ends and finite curvature. To be specific, we deal with two different frame boundaries, right angle frame and round frame boundaries, via two examples, Scherk's surface and catenoid.

Keywords

• allen-cahn equation
• phase-field equations
• ginzburg-landau energy functional
• finite difference method
• boundedness
• scherk's first surface
• catenoid
• weierstrass–enneper representation