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

This paper establishes the existence, uniqueness, and global $C^{1,β}$ 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 Itô 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.