## Bernoulli

• Bernoulli
• Volume 16, Number 1 (2010), 258-273.

### Viscosity solutions for systems of parabolic variational inequalities

#### Abstract

In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator: $\cases{ \dfrac{\partial u}{\partial t}(t,x )+\mathcal{L}_{t}u(t,x )+f (t,x,u (t,x )) \in\partial\varphi (u (t,x)), & \quad t\in[ 0,T) ,x\in\mathbb{R}^{d},\cr u( T,x ) =h(x), & \quad x\in\mathbb{R}^{d}, }$ where $\partial\varphi$ is the subdifferential operator of the proper convex lower semicontinuous function $\varphi:\mathbb{R}^{k}\rightarrow (-\infty,+\infty]$ and $\mathcal{L}_{t}$ is a second differential operator given by $\mathcal{L}_{t}v_{i}(x)=\frac{1}{2}\operatorname{Tr}% [\sigma(t,x)\sigma^{\ast}(t,x)\mathrm{D}^{2}v_{i}(x) ]+ \langle b(t,x),\nabla v_{i}(x) \rangle$, $i\in\overline{1,k}$.

We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution $u: [ 0,T ] \times\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$ of the above parabolic variational inequality.

#### Article information

Source
Bernoulli, Volume 16, Number 1 (2010), 258-273.

Dates
First available in Project Euclid: 12 February 2010

https://projecteuclid.org/euclid.bj/1265984711

Digital Object Identifier
doi:10.3150/09-BEJ204

Mathematical Reviews number (MathSciNet)
MR2648757

Zentralblatt MATH identifier
05815971

#### Citation

Maticiuc, Lucian; Pardoux, Etienne; Răşcanu, Aurel; Zălinescu, Adrian. Viscosity solutions for systems of parabolic variational inequalities. Bernoulli 16 (2010), no. 1, 258--273. doi:10.3150/09-BEJ204. https://projecteuclid.org/euclid.bj/1265984711

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