  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.
