Finsler structure for variable exponent Wasserstein space and gradient flows

2019/12/28 by Aboubacar Marcos, Marcos, Aboubacar, Ambroise Soglo +1
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Advanced Differential Geometry Research

paper · pdf · doi:10.48550/arxiv.1912.12450

Abstract

In this paper, we propose a variational approach based on optimal transportation to study the existence and unicity of solution for a class of parabolic equations involving q(x)-Laplacian operator (∂ ρ(t,x))/(∂ t)=div_x(ρ(t,x)|∇_x G^'(ρ(t,x))|^q(x)-2∇_x G^'(ρ(t,x)) ) . The variational approach requires the setting of new tools such as appropiate distance on the probability space and an introduction of a Finsler metric in this space. The class of parabolic equations is derived as the flow of a gradient with respect the Finsler structure. For q(x)≡ q constant, we recover some known results existing in the literature for the q-Laplacian operator.

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