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We establish a minimal positive existence time of the parametric Willmore flow for any smooth initial data (smooth immersion of a closed oriented surface). The minimal existence time is a function exclusively of geometric data which in particular are all well defined for general weak lipschitz $W^{2,2}$ immersions. This fact combined with the conservation law formulation of the equation given by the first author in 2008 opens the possibility for defining the Willmore flow for weak lipschitz $W^{2,2}$ initial data.

We introduce a parametric framework for the study of Willmore gradient flows which enables to consider a general class of weak, energy-level solutions and opens the possibility to study energy quantization and finite-time singularities. We restrict in this first work to a small-energy regime and prove that, for small-energy weak immersions, the Cauchy problem in this class admits a unique solution., Comment: Revised version to appear in Adv. Math

This paper is devoted to a third order study of the end-point map in sub-Riemannian geometry. We first prove third order open mapping results for maps from a Banach space into a finite dimensional manifold. In a second step, we compute the third order term in the Taylor expansion of the end-point map and we specialize the abstract theory to the study of length-minimality of sub-Riemannian strictly singular curves. We conclude with the third order analysis of a specific strictly singular extremal that is not length-minimizing.

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., Comment: 65 pages, 3 figures

In this short note we explore the validity of Wente-type estimates for Neumann boundary problems involving Jacobians. We show in particular that such estimates do not in general hold under the same hypotheses on the data for Dirichlet boundary problems.