Radial and Non-Radial Solution Structures for Quasilinear Hamilton--Jacobi--Bellman Equations in Bounded Settings

TL;DR

This paper proves existence, uniqueness, and regularity of solutions for a class of quasilinear Hamilton-Jacobi-Bellman equations on bounded convex domains, connecting stochastic control with elliptic PDE analysis and demonstrating applications in various fields.

Contribution

It introduces a constructive existence proof using a weighted linear monotone iteration scheme and bridges stochastic control theory with PDE regularity analysis for sub-quadratic growth sources.

Findings

Established existence and regularity of solutions.

Developed a constructive iterative solution method.

Applied the theory to practical problems in production planning and image restoration.

Abstract

This paper establishes the existence, uniqueness, and global $C^{1,\beta}$ regularity of positive classical solutions to a class of quasilinear Hamilton--Jacobi--Bellman (HJB) equations with Dirichlet boundary conditions on bounded convex domains. The core technical contribution is a constructive existence proof based on a weighted linear monotone iteration scheme. This scheme's stability and convergence are rigorously established through the construction of adaptive sub- and super-solutions leveraging the torsion function of the domain. Additionally, we provide a complete probabilistic derivation of the quasilinear PDE from the framework of controlled It\^{o} diffusions, formally bridging the gap between stochastic optimal control theory and elliptic regularity analysis. Our results extend beyond the classical quadratic cost regime to the wider class of sub-quadratic growth source terms. Finally, we demonstrate the utility of this theoretical framework through high-precision numerical implementations in two distinct fields: stochastic production planning and nonlinear contrast enhancement in image restoration.