Second-Order Two-Sided Estimates in Nonlinear Elliptic Problems

Best possible second-order regularity is established for solutions to p-Laplacian type equations with p∈ (1, ∞) and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L²-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all.