# A control problem related to the parabolic dominative $p$-Laplace equation

## TL;DR

This paper demonstrates that the value functions of a specific time-dependent control problem converge uniformly to the viscosity solution of a parabolic dominative p-Laplace equation, linking control theory with nonlinear PDEs.

## Contribution

It establishes a novel connection between a control problem and the parabolic dominative p-Laplace equation, showing convergence of value functions to the PDE's viscosity solution.

## Key findings

- Uniform convergence of value functions to the viscosity solution.
- Characterization of the PDE as a limit of control problems.
- Extension of control methods to nonlinear parabolic PDEs.

## Abstract

We show that value functions of a certain time-dependent control problem in $\Omega\times (0,T)$, with a continuous payoff $F$ on the parabolic boundary, converge uniformly to the viscosity solution of the parabolic dominative $p$-Laplace equation $$2(n+p)u_t=\Delta u+(p-2)\lambda_n(D^2 u),$$ with the boundary data $F$. Here $2\leq p< \infty$, and $\lambda_n(D^2 u)$ is the largest eigenvalue of the Hessian $D^2 u$.