A note on Campanato's $L^p$-regularity with continuous coefficients

In this note we consider local weak solutions of elliptic equations in variational form with data in $L^p$. We refine the classical approach due to Campanato and Stampacchia and we prove the $L^p$-regularity for the solutions assuming the coefficients merely continuous. This result shows that it is possible to prove the same sharp $L^p$-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 $2m$.