## Differential and Integral Equations

- Differential Integral Equations
- Volume 27, Number 3/4 (2014), 269-288.

### Tug-of-War games and parabolic problems with spatial and time dependence

Leandro M. Del Pezzo and Julio D. Rossi

#### Abstract

In this paper, we use probabilistic arguments (Tug-of-War games) to obtain the existence of viscosity solutions to a parabolic problem of the form $$ \begin{cases} K_{(x,t)}(D u )u_t (x,t)= \tfrac12 \langle D^2 u J_{(x,t)}(D u ),J_{(x,t)}(D u) (x,t)\rangle & \mbox{in } \Omega_T,\\ u(x,t)=F(x) & \mbox{on }\Gamma, \end{cases} $$ where $\Omega_T=\Omega\times(0,T]$ and $\Gamma$ is its parabolic boundary. This problem can be viewed as a version with spatial and time dependence of the evolution problem given by the infinity Laplacian, $$ u_t (x,t)= \langle D^2 u (x,t) \frac{D u}{|Du|} (x,t),\, \frac{D u}{|Du|} (x,t)\rangle\, . $$

#### Article information

**Source**

Differential Integral Equations, Volume 27, Number 3/4 (2014), 269-288.

**Dates**

First available in Project Euclid: 30 January 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1391091366

**Mathematical Reviews number (MathSciNet)**

MR3161604

**Zentralblatt MATH identifier**

1324.35065

**Subjects**

Primary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35J25: Boundary value problems for second-order elliptic equations 35J69

#### Citation

Del Pezzo, Leandro M.; Rossi, Julio D. Tug-of-War games and parabolic problems with spatial and time dependence. Differential Integral Equations 27 (2014), no. 3/4, 269--288. https://projecteuclid.org/euclid.die/1391091366