Global Schauder Estimates for the p-Laplace System

Dominic Breit, Andrea Cianchi, Lars Diening, Sebastian Schwarzacher

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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 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, BMO and 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 Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.
Original languageEnglish
Pages (from-to)201-255
Number of pages55
JournalArchive for Rational Mechanics and Analysis
Volume243
Issue number1
Early online date24 Nov 2021
DOIs
Publication statusPublished - Jan 2022

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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