Calculus of Variations and Geometric Measure Theory

J. A. Carrillo - S. Lisini - G. Savaré - D. Slepčev

Nonlinear mobility continuity equations and generalized displacement convexity

created by lisini on 24 Jan 2009
modified on 31 Jul 2010


Published Paper

Inserted: 24 jan 2009
Last Updated: 31 jul 2010

Journal: J. Funct. Anal.
Volume: 258
Pages: 1273-1309
Year: 2010


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.

Keywords: displacement convexity, generalized Wasserstein distance, Gradient flows, nonlinear mobility, diffusion equation