Global Schauder Estimates for the $$\mathbf {p}$$-Laplace System

An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for thep-Laplace equation and system, with the right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder,$$\text {BMO}$$BMOand$${{\,\mathrm{VMO}\,}}$$VMOspaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when$$p=2$$p=2, and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.