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

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