Shape optimization solutions via Monge-Kantorovich equation

We consider the optimization problem max{epsilon(mu) : mu nonnegative measure, integral d mu = m}, where epsilon(mu) is the energy associated to mu: epsilon(mu) = inf{1/2 integral \du\(2) d mu - [f, u] : u is an element of D(R-n)}. The datum f is a signed measure with finite total variation and zero average. We show that the optimization problem above admits a solution which is not in L-1(R-n) in general. This solution comes out by solving a suitable Monge-Kantorovich equation.