  This work puts forth a new optimal design formulation for planar elastic
membranes. The goal is to minimize the membrane's compliance through choosing
the material distribution described by a positive Radon measure. The
deformation of the membrane itself is governed by the convexified F\"{o}ppl's
model. The uniqueness of this model lies in the convexity of its variational
formulation despite the inherent nonlinearity of the strain-displacement
relation. It makes it possible to rewrite the optimization problem as a pair of
mutually dual convex variational problems. In the primal problem a linear
functional is maximized with respect to displacement functions while enforcing
that point-wisely the strain lies in an unbounded closed convex set. The dual
problem consists in finding equilibrated stresses that are to minimize a convex
integral functional of linear growth defined on the space of Radon measures.
The pair of problems is analysed: existence and regularity results are
provided, together with the system of optimality criteria. To demonstrate the
computational potential of the pair, a finite element scheme is developed
around it. Upon reformulation to a conic-quadratic & semi-definite programming
problem, the method is employed to produce numerical simulations for several
load case scenarios.
