Finsler structure for variable exponent Wasserstein space and gradient flows
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 -Laplacian operator \begin{equation*}\label{equation variable q(x)} \frac{\partial ρ(t,x)}{\partial t}=div_x\left(ρ(t,x)|\nabla_x G^{'}(ρ(t,x))|^{q(x)-2}\nabla_x G^{'}(ρ(t,x)) \right) .\end{equation*} 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 constant, we recover some known results existing in the literature for the -Laplacian operator.
26 pages Submitted for publication in EJDE