2017/04/05 by Harold R. Parks, Parks, Harold R., Jon T. Pitts +1
Computer Science · Mathematics · #49Q05 #49Q15 #49Q20 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1704.01658
openalex publication_date 2017/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let N≥3 be an integer and B be a smooth, compact, oriented, (N-2)-dimensional boundary in \Bbb RN. In 1960, H. Federer and W. Fleming proved that there is an (N-1)-dimensional integral current spanning surface of least area. The proof was by compactness methods and non-constructive. In 1970 H. Federer proved the definitive regularity result for such a codimension one minimizing surface. Thus it is a question of long standing whether there is a numerical algorithm that will closely approximate the area minimizing surface. The principal result of this paper is an algorithm that solves this problem. Specifically, given a neighborhood U around B in \Bbb RN and a tolerance ε>0, we prove that one can explicitly compute in finite time an (N-1)-dimensional integral current T with the following approximation requirements: (1) spt(∂ T)⊂ U. (2) B and ∂ T are within distance ε in the Hausdorff distance. (3) B and ∂ T are within distance ε in the flat norm distance. (4) \mathbb M(T)