Regularity for weak solutions to nondiagonal quasilinear degenerate elliptic systems

Abstract

The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to Hörmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We first prove L p (p ≥ 2) estimates for gradients of weak solutions by using a priori estimates and a known reverse Hölder inequality, and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems. Then we get higher Morrey and Campanato estimates for gradients of weak solutions to original systems and Hölder estimates for weak solutions.Regularity for solutions to elliptic systems in Euclidean spaces has been studied by many authors and a lot of important conclusions were got. Campanato in [2] obtained gradient estimates for weak solutions to linear elliptic system with discontinuous coefficients.For related articles, we quote [1,14] and the references therein. Huang in [18] derived Morrey estimates for uniformly elliptic systems by applying Campanato's technique. Zheng and Feng in [28] established Hölder estimates for weak solutions to quasilinear elliptic systems by reverse Hölder inequality and Dirichlet growth theorem, where the coefficients belong to L ∞ ∩ V MO, and low terms satisfy controlled

Higher integrability for weak solutions to a degenerate parabolic system with singular coefficients

“…Based on Hömander's fundamental work [11], there has been tremendous work on degenerate PDEs arising from non-commuting vector fields; see, for example, [12][13][14][15][16][17][18][19][20][21]. Di Fazio and Fanciullo in [14] obtained gradient estimates for weak solutions to linear diagonal elliptic systems with bounded VMO coefficients.…”

“…Di Fazio and Fanciullo in [14] obtained gradient estimates for weak solutions to linear diagonal elliptic systems with bounded VMO coefficients. Dong and Niu [17] got the higher L p estimates for the gradient of weak solutions to nondiagonal quasilinear degenerate elliptic systems. In [16,18], Dong and her collaborators studied Morrey and Hölder regularity for weak solutions to diagonal and nondiagonal parabolic systems with bounded VMO coefficients.…”

Regularity of weak solutions to obstacle problems for nondiagonal quasilinear degenerate elliptic systems

“…The study of interior regularity for degenerate elliptic equations and systems has attracted much attention (see, e.g., [16][17][18][19], etc.). Di Fazio and Fanciullo in [16] pointed out that the local gradient estimates in [6] still hold true for the diagonal degenerate elliptic systems.…”

Interior regularity of obstacle problems for nonlinear subelliptic systems with VMO coefficients

“…In [15] and [16] Di Fazio and Fanciullo proved that the local gradient estimates in [9] still hold true for the subelliptic systems structured on Hömander’s vector fields. Dong and Niu [14] established the Morrey and Campanato regularity for weak solutions to the nondiagonal subelliptic systems. The direct methods were mainly used to prove the desired results in the papers mentioned above.…”