Xu, Chao-Jiang Regularity for quasilinear second-order subelliptic equations. (English) Zbl 0827.35023 Commun. Pure Appl. Math. 45, No. 1, 77-96 (1992). 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 \(u \in C^\infty (\Omega)\). Cited in 54 Documents MSC: 35B65 Smoothness and regularity of solutions to PDEs 35J70 Degenerate elliptic equations Keywords:Schauder type estimate; Hörmander condition PDFBibTeX XMLCite \textit{C.-J. Xu}, Commun. Pure Appl. Math. 45, No. 1, 77--96 (1992; Zbl 0827.35023) Full Text: DOI References: [1] Bony, Ann. Inst. Fourier 19 pp 227– (1969) · Zbl 0176.09703 · doi:10.5802/aif.319 [2] and , Subelliptic eigenvalue problems, Proceedings of the Conference on Harmonic Analysis, in honor of A. Zygmund, Wadsworth Math. Series, 1981, pp. 590–606. [3] and , Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften 224, Springer-Verlag, Berlin, 1983. · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0 [4] Hörmander, Acta Math. 119 pp 141– (1967) [5] and , Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. [6] Métivier, Comm. P.D.E. 1 pp 467– (1976) [7] Nagel, Acta Math. 155 pp 103– (1985) [8] and , Second Order Equations with Nonnegative Characteristic Form, American Math. Soc., New York, 1973. · doi:10.1007/978-1-4684-8965-1 [9] Rothschild, Acta Math. 137 pp 247– (1977) [10] Sanchez-Calle, Invent. Math. 78 pp 143– (1984) [11] Xu, Ann. Inst. Fourier 37 pp 105– (1987) · Zbl 0609.35023 · doi:10.5802/aif.1088 [12] Xu, Comm. P.D.E. 11 pp 1575– (1986) [13] Xu, Bull. Soc. Math. France 118 pp 147– (1990) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.