This research proposes a new methodological framework for deriving asymptotic expansions of solutions to k-Hessian equations using nonlocal operator techniques. The approach introduces nonlocal regularization operators that approximate the k-Hessian operator while preserving key structural properties, enabling rigorous error estimates and convergence proofs.
Key findings
A novel connection between nonlocal operator theory and asymptotic analysis of fully nonlinear equations.
Construction of explicit nonlocal approximations to k-Hessian operators with controlled convergence properties.
Systematic asymptotic expansion methods for singularly perturbed nonlinear elliptic PDEs.
Rigorous error estimates and convergence proofs for the proposed approximations.
Limitations & open questions
The framework's application to concrete singular perturbation problems is yet to be fully explored.