Partial regularity of solutions of nonlinear quasimonotone systems
Vector-valued weak solutions of the nonlinear system \(\text{div}\;A(x,u,Du)+B(x,u,Du)=0\) are studied, assuming strict quasimonotonicity on \(A\) and some natural polynomial growth, as well as Hölder continuity of \(A\) in the first and second and differentiability in the third variable. Moreover, it is assumed that \(A(x,u,P)\cdot P\geq F(x,P)\) for some continuous function \(F\) which is strictly quasiconvex at zero. Under such conditions, it is proved that weak solutions have Hölder continuous derivatives almost everywhere, with optimal Hölder exponent. The proof is an indirect blow-up argument, using higher integrability.
scientific article; zbMATH DE number 2053122