Regularity of weak solutions to obstacle problems for nondiagonal quasilinear degenerate elliptic systems

Abstract

Abstract Let X={X1,…,Xm} $X=\{X_{1} ,\ldots ,X_{m} \}$ be a system of smooth real vector fields satisfying Hörmander’s rank condition. We consider the interior regularity of weak solutions to an obstacle problem associated with the nonhomogeneous nondiagonal quasilinear degenerate elliptic system Xα∗(Aijαβ(x,u)Xβuj)=Bi(x,u,Xu)+Xα∗giα(x,u,Xu). $$X_{\alpha }^{\ast } \bigl( {A_{ij}^{\alpha \beta } (x,u)X_{\beta }u ^{j}} \bigr)= B_{i}(x,u,Xu)+X_{\alpha }^{\ast }g_{i}^{\alpha }(x,u,Xu). $$ After proving the higher integrability and a Campanato type estimate for the weak solutions to the obstacle problem for the homogeneous nondiagonal quasilinear degenerate elliptic system, the interior Morrey regularity and Hölder continuity of weak solutions to the obstacle problem for the nonhomogeneous system are obtained.

Keywords

• Nondiagonal quasilinear degenerate elliptic system
• Obstacle problem
• Morrey regularity
• Hölder continuity