Higher interior regularity for quasilinear subelliptic systems (Q1357534)

The authors study the problem of interior regularity for weak solutions of quasilinear systems of the form \[ \sum_{i,j=1}^m X_j^* (a^{ij}(x,u(x)) X_i u^\alpha (x)) = f^\alpha (x,u(x),X u(x)), \quad 1 \leq \alpha \leq N, \] where \( X_1, \dots ,X_m\) are \(C^\infty\) vector fields satisfying Hörmander's condition, the matrix \(a_{ij}\) is symmetric and positive definite and the known term has quadratic growth in the gradient \(X u(x).\) In particular, if the coefficients \(a_{ij}\) and the known term \(f^\alpha\) are smooth functions, they prove that every continuous weak solution is also \(C^\infty\) in \(\Omega.\) In the proofs Morrey space tools are used.