Sensitivity of the Compliance and of the Wasserstein Distance with Respect to a Varying Source

We show that the compliance functional in elasticity is differentiable with respect to horizontal variations of the load term, when the latter is given by a possibly concentrated measure; moreover, we provide an integral representation formula for the derivative as a linear functional of the deformation vector field. The result holds true as well for the p-compliance in the scalar case of conductivity. Then we study the limit problem as p→ + ∞, which corresponds to differentiate the Wasserstein distance in optimal mass transportation with respect to horizontal perturbations of the two marginals. Also in this case, we obtain an existence result for the derivative, and we show that it is found by solving a minimization problem over the family of all optimal transport plans. When the latter contains only one element, we prove that the derivative of the p-compliance converges to the derivative of the Wasserstein distance in the limit as p→ + ∞.