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A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in    $${\mathbb R^n}$$  R n are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the  p -Laplace system, our conclusions broaden the range of the admissible values of the exponent  p  previously known.

A pointwise differential inequality and second-order regularity for nonlinear elliptic systems