Mathématiques/Mathematics Shape optimization solutions via Monge-Kantorovich equation

Abstract We consider the optimization problem max ∫ e(μ) : μ nonnegative measure, ∫ dμ = m ∫, where eS(μ) is the energy associated to μ : e(μ) = inf/2 ¦ Du¦ 2 dμ − 〈f,u: u e 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.