Limits for Monge-Kantorovich mass transport problems
Jesus Garcia Azorero Juan J. Manfredi I. Peral Julio D. Rossi
Communications on Pure & Applied Analysis 2008, 7(4): 853-865 doi: 10.3934/cpaa.2008.7.853
In this paper we study the limit of Monge-Kantorovich mass transfer problems when the involved measures are supported in a small strip near the boundary of a bounded smooth domain, $\Omega$. Given two absolutely continuos measures (with respect to the surface measure) supported on the boundary $\partial \Omega$, by performing a suitable extension of the measures to a strip of width $\varepsilon$ near the boundary of the domain $\Omega$ we consider the mass transfer problem for the extensions. Then we study the limit as $\varepsilon$ goes to zero of the Kantorovich potentials for the extensions and obtain that it coincides with a solution of the original mass transfer problem. Moreover we look for the possible approximations of these problems by solutions to equations involving the $p-$Laplacian for large values of $p$.
keywords: quasilinear elliptic equations Mass transport Neumann boundary conditions.
Bernd Kawohl Juan J. Manfredi
Communications on Pure & Applied Analysis 2015, 14(1): i-i doi: 10.3934/cpaa.2015.14.1i
This special issue Emerging Trends in Nonlinear PDE of this journal was conceived during the Fall 2013 research semester “Evolutionary Problems” held at the Mittag-Leffler-Institute and its colophon conference Quasilinear PDEs and Game Theory held at Uppsala University in early December. The editors of this special issue participated in these activities. Following several conversations with other participants, we solicited manuscripts from participants of the Mittag-Leffler special semester, the Uppsala conference, as well as from colleagues working in closely related fields. Seventeen papers (out of twenty-one) are authors by participants in the research program at the Mittag-Leffler or the conference at Uppsala.
An obstacle problem for Tug-of-War games
Juan J. Manfredi Julio D. Rossi Stephanie J. Somersille
Communications on Pure & Applied Analysis 2015, 14(1): 217-228 doi: 10.3934/cpaa.2015.14.217
We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war.
keywords: Obstacle Problem infinity laplacian. Tug-of-War games

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