A priori estimates for infinitely degenerate quasilinear equations. (English) Zbl 1224.35149

Summary: We prove a priori bounds for derivatives of solutions \(w\) of a class of quasilinear equations of the form \(\operatorname {div}\mathcal {A}(x,w)\nabla w+\vec \gamma (x,w)\cdot \nabla w+f(x,w)=0\), where \(x=(x_{1},\dots ,x_{n})\), and where \(f\), \(\vec \gamma =(\gamma ^{i})_{1\leq i\leq n}\) and \(\mathcal {A}=(a_{ij})_{1\leq i,j\leq n}\) are \(\mathcal {C}^\infty \). The rank of the square symmetric matrix \(\mathcal {A}\) is allowed to degenerate, as all but one eigenvalue of \(\mathcal {A}\) are permitted to vanish to infinite order. We estimate derivatives of \(w\) of arbitrarily high order in terms of just \(w\) and its first derivatives. These estimates will be applied in a subsequent work to establish existence, uniqueness and regularity of weak solutions of the Dirichlet problem.


35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs