Hessian estimates for linear and nonlinear second-order boundary value problems

This project investigates boundary regularity for elliptic PDEs with second-order Venttsel-type boundary conditions. The first part concerns the (σ_k, B_k)-Yamabe problem, a fully nonlinear generalization of the classical boundary Yamabe problem. Unlike the classical (semilinear, uniformly elliptic) case, the (σ_k, B_k) problem couples a fully nonlinear, non-uniformly elliptic interior equation with a nonlinear, second-order boundary operator; general solvability remains largely open. Building on our recent Liouville-type theorem, this proposal tackles the next major obstacle --- boundary C^2 estimates --- beginning with the model (σ_2, B_2) setting in the umbilic case. The main analytic difficulty is that the boundary operator includes second-order terms and is bothtangentially non-uniformly elliptic and non-uniformly oblique.Motivated by this, the second part develops linear theory for Venttsel boundary value problems. While existing results are mostly "Dirichlet-like", relying on uniform tangential ellipticity, we investigate the complementary "oblique-like" regime, where quantitative obliqueness drives the estimates even when tangential ellipticity may degenerate. Our goal is to obtain sharp C^{2,α} and W^{2,p} estimates on rough domains, extending the "boundary relaxation" phenomenon established for pure oblique derivative problems in my earlier work. Together, these advances aim to provide a more complete picture of second-order boundary operators and the linear tools needed for subsequent nonlinear analysis in conformal geometry.