Viscosity solutions for systems of parabolic variational inequalities

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

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