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arXiv · 1912.12450

Finsler structure for variable exponent Wasserstein space and gradient flows

Aboubacar Marcos·Ambroise Soglo
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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 \begin{equation*}\label{equation variable q(x)} \frac{\partial \rho(t,x)}{\partial t}=div_x\left(\rho(t,x)|\nabla_x G^{'}(\rho(t,x))|^{q(x)-2}\nabla_x G^{'}(\rho(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 $q(x)\equiv q$ constant, we recover some known results existing in the literature for the $q$-Laplacian operator.

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Aboubacar Marcos, Ambroise Soglo. 2019-12-28. Finsler structure for variable exponent Wasserstein space and gradient flows. https://arxiv.org/abs/1912.12450

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