No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem nonlinear coefficient in the principal part that is countably $PC^2$ (continuous and $C^2$ apart from many points). We prove control-to-state operator continuously differentiable even though non-smooth. enables us to establish "no-gap" necessary sufficient terms of an abstract curvature functional, i. e., which condition only differs one fact inequality strict. A equivalent could be useful error estimates in, e.g., finite element discretizations also provided.