Viscosity solutions for systems of parabolic variational inequalities

Details Viscosity solutions for systems of parabolic variational inequalities Viscosity solutions for systems of parabolic variational inequalities

2008-07-28

arxiv.org/abs/0807.4415v2

Bernoulli 2010, Vol. 16, No. 1, 258-273

Mathematics

Dynamical Systems

Published in at http://dx.doi.org/10.3150/09-BEJ204 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/09-BEJ204

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.

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