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A Resolution of the Poisson Problem for Elastic Plates
- Source :
- Arch. Rational. Mech. Anal. 236 (2020) 1593-1676
- Publication Year :
- 2018
-
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.<br />Comment: 65 pages, 3 figures
Details
- Database :
- arXiv
- Journal :
- Arch. Rational. Mech. Anal. 236 (2020) 1593-1676
- Publication Type :
- Report
- Accession number :
- edsarx.1807.09373
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s00205-020-01499-2