A Resolution of the Poisson Problem for Elastic Plates

TL;DR

This paper proves the existence of smooth solutions for the elastic energy minimization problem of thin plates with prescribed boundary conditions, using a variational approach involving total curvature energy.

Contribution

It introduces a new variational framework for solving the Poisson problem for elastic plates, establishing existence of immersed disk solutions with controlled regularity.

Findings

Existence of minimizers as immersed disks with finite branch points.

Minimizers are of class C^{1,eta} up to the boundary.

Gauss map extends continuously to the boundary.

Abstract

The Poisson problem consists in finding an immersed surface $\Sigma\subset\mathbb{R}^m$ minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area which constitutes a non-linear model for the equilibrium state of thin, clamped elastic plates originating from the work of S. Germain and S.D. Poisson or the early XIX century. We present a solution to this problem consisting in the minimisation of the total curvature energy $E(\Sigma)=\int_\Sigma |\operatorname{I\!I}_\Sigma|^2_{g_\Sigma}\,\mathrm{d}vol_\Sigma$ ($\operatorname{I\!I}_\Sigma$ is the second fundamental form of $\Sigma$), which is variationally equivalent to the elastic energy, in the case of boundary data of class $C^{1,1}$ and when the boundary curve is simple and closed. The minimum is realised by an immersed disk, possibly with a finite number of branch points in its interior, which is of class $C^{1,\alpha}$ up to the boundary for some $0<\alpha<1$, and whose Gauss map extends to a map of class $C^{0,\alpha}$ up to the boundary.