6 research outputs found
Remarks on Neumann boundary problems involving Jacobians
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
Third order open mapping theorems and applications to the end-point map
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 Parametric Approach to the Willmore Flow
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
A Resolution of the Poisson Problem for Elastic Plates
The Poisson problem consists in finding an immersed surface
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
( is the second fundamental form of ),
which is variationally equivalent to the elastic energy, in the case of
boundary data of class 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 up to
the boundary for some , and whose Gauss map extends to a map of
class up to the boundary.Comment: 65 pages, 3 figure