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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)].

35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35D30 Weak solutions to PDEs
Full Text: DOI
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