# Global Schauder estimates for the $p$-Laplace system

## TL;DR

This paper develops a comprehensive regularity theory for solutions to the $p$-Laplace equation and system, establishing optimal global estimates in oscillation-based function spaces, extending classical results even in the linear case.

## Contribution

It introduces a new regularity framework in oscillation spaces for the $p$-Laplace system, enhancing classical Schauder and gradient theories with optimal global estimates.

## Key findings

- Established optimal regularity in oscillation spaces for $p$-Laplace solutions.
- Extended classical Schauder theory to nonlinear and oscillation-based contexts.
- Demonstrated sharpness of results with specific examples.

## Abstract

An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the $p$-Laplace equation and system, with 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 H\"older, $\mathrm{BMO}$ and $\mathrm{VMO}$ spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when $p=2$, 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 H\"older spaces, and complements the Jerison-Kenig gradient theory in Lebesgue spaces with a parallel in the oscillation spaces realm. The sharpness of our results is demonstrated by apropos examples.