Some mass transport problems as limits of 𝑝-Laplacian problems

par José M. Mazón (Dept. de Analisis Matematico, Universitat de Valencia, Spain)

vendredi 21 mars 2014, 09:30 → 10:30 Europe/Paris

203 (XLIM)

Following the method introduced by Evans y Gangbo to solve the classical Monge- Kantorovich mass transport problem, in this lecture we present two mass transport prob- lems obtained as limit when 𝑝 → ∞ of the solutions of some problem related with the 𝑝-Laplacian operator. The first one is an optimal matching problem that consists in trans- porting two commodities to a prescribed location, the target set in such a way that they match there and the total cost of the operation, measured in terms of the Euclidean dis- tance that the commodities are transported, is minimized. We show that such a problem has a solution with matching measure concentrated on the boundary of the target set. Furthermore we perform a method to approximate the solution of the problem taking limit as 𝑝 → ∞ in a system of PDE’s of 𝑝−Laplacian type. The second problem consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger or equal than a fixed one to fulfil a demand also larger or equal than a fixed one, with the obligation of paying an extra cost of −𝑔1(𝑥) for extra production of one unit at location 𝑥 and an extra cost of 𝑔2(𝑦) for creating one unit of demand at 𝑦. The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as 𝑝 → ∞ to a double obstacle problem (with obstacles 𝑔1, 𝑔2) for the 𝑝–Laplacian. In fact, under a certain natural constraint on the extra costs (that is equivalent to impose that the total optimal cost is bounded) we prove that this limit gives the extra material and extra demand needed for optimality and a Kantorovich potential for the mass transport problem involved.