×

zbMATH — the first resource for mathematics

Harnack’s inequality and Hölder continuity for weak solutions of degenerate quasilinear equations with rough coefficients. (English) Zbl 1323.35045
Summary: We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form \[ \operatorname{div}(A(x, u, \nabla u)) = B(x, u, \nabla u) \quad \text{ for }x \in \Omega \] as considered in our paper [Differ. Integral Equ. 25, No. 1–2, 143–200 (2012; Zbl 1249.35117)]. There we proved only local boundedness of weak solutions. Here, we derive a version of Harnack’s inequality as well as local Hölder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin [Acta Math. 111, 247–302 (1964; Zbl 0128.09101)] and N. S. Trudinger [Commun. Pure Appl. Math. 20, 721–747 (1967; Zbl 0153.42703)] for quasilinear equations, as well as ones for subelliptic linear equations obtained by E. T. Sawyer and R. L. Wheeden [Mem. Am. Math. Soc. 847, 157 p. (2006; Zbl 1096.35031); Trans. Am. Math. Soc. 362, No. 4, 1869–1906 (2010; Zbl 1191.35085)].

MSC:
35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35D30 Weak solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chua, Seng-Kee; Rodney, S.; Wheeden, R. L., A compact embedding theorem for generalized Sobolev spaces, Pacific J. Math., 265, 17-57, (2013) · Zbl 1360.46015
[2] Chua, Seng-Kee; Wheeden, R. L., Self-improving properties of inequalities of Poincaré type on measure spaces and applications, J. Funct. Anal., 255, 2977-3007, (2008) · Zbl 1172.46020
[3] Fabes, E. B.; Kenig, C. E.; Serapioni, R. P., The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7, 77-116, (1982) · Zbl 0498.35042
[4] C. Fefferman, D.H. Phong, Subelliptic eigenvalue problems, in: Conference in Honor of A. Zygmund, Wadsworth Math. Series, 1981. · Zbl 0503.35071
[5] Franchi, B.; Lu, G.; Wheeden, R. L., Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations, Potential Anal., 4, 4, 361-375, (1995) · Zbl 0841.46018
[6] Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order, (1998), Springer Verlag · Zbl 0691.35001
[7] Hörmander, Hypoelliptic second order differential equations, Acta Math., 119, 147-171, (1967) · Zbl 0156.10701
[8] Jerison, D., The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J., 53, 503-523, (1986) · Zbl 0614.35066
[9] Korobenko, L.; Maldonado, D.; Rios, C., From Sobolev inequality to doubling, Proc. Amer. Math. Soc., (2015), in press · Zbl 1325.35055
[10] Lu, G., Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications, Rev. Mat. Iberoamericana, 8, 367-439, (1992) · Zbl 0804.35015
[11] Monticelli, D. D.; Rodney, S.; Wheeden, R. L., Boundedness of weak solutions of degenerate quasilinear equations with rough coefficients, J. Differ. Integral Equ., 25, 143-200, (2012) · Zbl 1249.35117
[12] Nagel, A.; Stein, E. M.; Wainger, S., Balls and metrics defined by vector fields I: basic properties, Acta Math., 155, 103-147, (1985) · Zbl 0578.32044
[13] Sánchez-Calle, A., Fundamental solutions and geometry of the sums of squares of vector fields, Invent. Math., 78, 143-160, (1984) · Zbl 0582.58004
[14] Sawyer, E. T.; Wheeden, R. L., Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math., 114, 813-874, (1992) · Zbl 0783.42011
[15] Sawyer, E. T.; Wheeden, R. L., Hölder continuity of weak solutions to subelliptic equations with rough coefficients, Mem. Amer. Math. Soc., 847, (2006) · Zbl 1096.35031
[16] Sawyer, E. T.; Wheeden, R. L., Degenerate Sobolev spaces and regularity of subelliptic equations, Trans. Amer. Math. Soc., 362, 1869-1906, (2010) · Zbl 1191.35085
[17] Serrin, J., Local behavior of solutions of quasi-linear equations, Acta Math., 111, 247-302, (1964) · Zbl 0128.09101
[18] Trudinger, N., On Harnack type inequalities and their application to quasilinear equations, Comm. Pure Appl. Math., 20, 721-747, (1967) · Zbl 0153.42703
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.