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Regularity for quasilinear second-order subelliptic equations. (English) Zbl 0827.35023

Summary: We study the regularity of solutions of the quasilinear equation \(\sum^m_{i,j = 1} A_{ij} (x,u,Xu) X_iX_ju + B(x,u,Xu) = 0\), where \(X = (X_1, \ldots, X_m)\) is a system of real smooth vector fields, \(A_{ij}\), \(B \in C^\infty (\Omega \times \mathbb{R}^{m + 1})\). Assume that \(X\) satisfies the Hörmander condition and \((A_{ij} (x,z, \xi))\) is positive definite. We prove that if \(u \in S^{2, \alpha} (\Omega)\) is a solution of the above equation, then \(u \in C^\infty (\Omega)\).

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35J70 Degenerate elliptic equations
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