2023/08/03 by Timothy J. Healey, Healey, Timothy J., Gokul G. Nair +1
Computer Science · Mathematics · #35D30 #74B20 #74G65 #74K15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2308.02070
openalex publication_date 2023/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a model for nonlinearly elastic membranes undergoing finite deformations while confined to a regular frictionless surface in ℝ3. This is a physically correct model of the analogy sometimes given to motivate harmonic maps between manifolds. The proposed energy density function is convex in the strain pair comprising the deformation gradient and the local area ratio. If the target surface is a plane, the problem reduces to 2-dimensional, polyconvex nonlinear elasticity addressed by J.M. Ball. On the other hand, the energy density is not rank-one convex for unconstrained deformations into ℝ3. We show that the problem admits an energy-minimizing configuration when constrained to lie on the given surface. For a class of Dirichlet problems, we demonstrate that the minimizing deformation is a homeomorphism onto its image on the given surface and establish the weak Eulerian form of the equilibrium equations.