A Resolution of the Poisson Problem for Elastic Plates

Abstract

The Poisson problem consists in finding an immersed surface $Σ\subset\mathbb{R}^m$ 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 $E(Σ)=\int_Σ|\operatorname{I\!I}_Σ|^2_{g_Σ}\,\mathrm{d}vol_Σ$ ($\operatorname{I\!I}_Σ$ is the second fundamental form of $Σ$), 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 $C^{1,α}$ up to the boundary for some $0<α<1$, and whose Gauss map extends to a map of class $C^{0,α}$ up to the boundary.