2023/10/22 by Cătălin I. Cârstea, Matti Lassas, Cârstea, Cătălin I. +5 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Numerical methods in inverse problems #Thermoelastic and Magnetoelastic Phenomena
paper · pdf · doi:10.48550/arxiv.2310.14268
openalex publication_date 2023/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we consider an inverse problem of determining a minimal surface embedded in a Riemannian manifold. We show under a topological condition that if Σ is a 2-dimensional embedded minimal surface, then the knowledge of the Dirichlet-to-Neumann map associated to the minimal surface equation determines Σ up to an isometry. Without the topological condition, we show that a conformal factor of a general minimal surface Σ can be recovered. We develop a semiclassical nonlinear calculus for complex geometric optics solutions, which allows an efficient error analysis for multiplication of the correction terms of the solutions. The calculus is independent of the application to the minimal surface equation and we expect it to have applications in various inverse problems for nonlinear equations in dimension 2, in both ℝ2 and geometric settings. Other applications of the results include generalized boundary rigidity problem and the AdS/CFT correspondence in physics.