The Poisson problem consists in finding an immersed surface
Σ⊂Rm 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)=∫Σ∣IIΣ∣gΣ2dvolΣ
(IIΣ is the second fundamental form of Σ),
which is variationally equivalent to the elastic energy, in the case of
boundary data of class C1,1 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 C1,α up to
the boundary for some 0<α<1, and whose Gauss map extends to a map of
class C0,α up to the boundary.Comment: 65 pages, 3 figure