A Note on Campanato’s Lp-Regularity with Continuous Coefficients

Abstract

In this note we consider local weak solutions of elliptic equations in variational form with data in Lp. We refine the classical approach due to Campanato and Stampacchia and we prove the Lp-regularity for the solutions assuming the coefficients merely continuous. This result shows that it is possible to prove the same sharp Lp-regularity results that can be proved using classical singular kernel approach also with the variational regularity approach introduced by De Giorgi. This method works for general operators: parabolic, in nonvariational form, of order 2m.