Systems of fully nonlinear parabolic obstacle problems with Neumann boundary conditions

TL;DR

This paper establishes the existence and uniqueness of viscosity solutions for systems of fully nonlinear parabolic PDEs with interconnected obstacles and Neumann boundary conditions, motivated by optimal switching problems.

Contribution

It adapts viscosity solution techniques to interconnected obstacle systems with Neumann conditions and constructs explicit bounds for Perron's method.

Findings

Proved existence and uniqueness of solutions

Developed a method for interconnected obstacle systems

Applied to optimal switching problems

Abstract

We prove the existence of a unique viscosity solution to certain systems of fully nonlinear parabolic partial differential equations with interconnected obstacles in the setting of Neumann boundary conditions. The method of proof builds on the classical viscosity solution technique adapted to the setting of interconnected obstacles and construction of explicit viscosity sub- and supersolutions as bounds for Perron's method. Our motivation stems from so called optimal switching problems on bounded domains.