Natural Second-Order Regularity for Systems in the Case 1 < p ≤ 2 Using the A-Approximation

The chapter delves into the boundary value problem associated with nonlinear elliptic systems, focusing on the case p > 2. It introduces the concept of natural second-order regularity and discusses the challenges in regularity for this case, which cannot be addressed using standard approaches. The chapter then presents a novel theory based on the multiple approximation of the operator, allowing for a unified treatment of all p ∈ (1, ∞). This approach involves defining an A-approximation of the operator and function, and proving relevant properties needed for the analysis. The chapter also discusses the existence, uniqueness, and regularity of solutions for the approximate problem, as well as estimates independent of A. Finally, it addresses the passage to the limit as A → ∞ to obtain a regular solution for the original problem.