Nonlinear mobility continuity equations and generalized displacement convexity

J.A. Carrillo
ICREA (Institució Catalana de Recerca i Estudis Avançats
and
Departament de Matemàtiques
Universitat Autònoma de Barcelona
Bellaterra, Spain
carrillo@mat.uab.es

S. Lisini
Dipartimento di Matematica
``F. Casorati''
Universitdegli Studi di Pavia
Pavia, Italy
stefano.lisini@unipv.it

G. Savaré
Dipartimento di Mateiatica ``F. Casorati''
Università degli Studi di Pavia
Pavia, Italy
giuseppe.savare@unipv.it

D. Slepčev
Carnegie Mellon University
Department of Mathematical Sciences
Pittsburgh, PA
slepcev@math.cmu.edu

Abstract: We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a non-rigorous argument indicating that they are not displacement semiconvex.

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