Viscosity solutions for systems of parabolic variational inequalities

TL;DR

This paper introduces a new notion of viscosity solutions for systems of parabolic variational inequalities involving subdifferential operators, proving their uniqueness and existence through stochastic methods.

Contribution

It defines viscosity solutions for complex PDE systems with subdifferential operators and establishes their uniqueness and existence using stochastic approaches.

Findings

Established a unique notion of viscosity solutions for the system.

Proved the existence of solutions via stochastic methods.

Demonstrated the applicability to systems involving convex subdifferentials.

Abstract

In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:\[\{[c]{l}\dfrac{\partial u}{\partial t}(t,x)+\mathcal{L}_tu(t,x)+f(t,x,u(t,x))\in\partial\phi (u(t,x)),\quad t\in[0,T),x\in\mathbb{R}^d, u(T,x)=h(x),\quad x\in\mathbb{R}^d,\] where $\partial\phi$ is the subdifferential operator of the proper convex lower semicontinuous function $\phi:\mathbb{R}^k\to (-\infty,+\infty]$ and $\mathcal{L}_t$ is a second differential operator given by $\mathcal{L}_tv_i(x)={1/2}\operatorname {Tr}[\sigma(t,x)\sigma^*(t,x)\mathrm{D}^2v_i(x)]+< b(t,x),\nabla v_i(x)>$, $i\in\bar{1,k}$. We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution $u:[0,T]\times\mathbb{R}^d\to\mathbb{R}^k$ of the above parabolic variational inequality.