$C^{(0,\alpha)}-$REGULARITY OF THE SOLUTIONS OF DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS IN DIVERGENCE FORM

Abstract

The author considers second order divergence form operators ${\cal L}$with coefficients in $VMO$ and proves that the solutionof ${\cal L}u = div \, f,$ with $f$ in the Morrey class $L^{p,\lambda},$have derivatives in $L^{p,\lambda}$ locally andglobally. Moreover the solution is H\"older if $p>n-\lambda,$ generalizing aclassical result for $\lambda =0.$ The method, well known as Korn's trick,is based on perturbations of fundamental solutions of constant coefficientsequation.