Hölder continuity of solutions of elliptic systems

Abstract

The purpose of this note is to observe that a variant of the method of Morrey, as exposed in [4] and [5], can be used to show that weak solutions of a certain class of elliptic systems of quasilinear equations of arbitrary order of the form

$$\mathop {\sum\limits_{\left| \alpha \right| \leqslant m} {( - 1)^{\left| \alpha \right|} D^\alpha F_{\alpha ,v} (x,u,Du, \ldots ,D^m u) = 0,v = 1,2 \ldots ,N} }\limits_{u = (u_1 ,u_2 , \ldots ,u_{N)} } $$

are Hölder continuous, thus partially extending results of Ladyženskaja-Ural'ceva [3] and Serrin [8] to higher order equations. A full extension is not possible. With suitable assumptions the Hölder continuity holds out to the boundary.