Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients.

*(English)*Zbl 1228.35260The paper deals with local and global estimates for gradient of solutions to a nonhomogeneous quasilinear equation on certain Lebesgue spaces. The equation under consideration is of weak type and has measurable coefficients \(A_{ij}\). The matrix \(A=A_{ij}(x)\) is symmetric and satisfies the ellipticity condition.

The author obtains a weighted regularity estimate assuming that, in the bounded domain under study, the equation is of class \(C^1\) and that each of its components belongs to the Sarason class VMO. The approach could be easily modified to cover the case where the domain \(\Omega\) where the functions under consideration belong, is Lipschitz with small Lipschitz constant and \(A\) has small bounded mean oscillation coefficients (i.e., they are in the John-Nirenberg class BMO).

The approach makes use of the Fefferman-Stein sharp maximal function and \(C^{1,\alpha}\) regularity estimates obtained earlier for homogeneous \(p\)-Laplace type equations. In particular, a weighted version of Fefferman-Stein inequality for a local dyadic sharp maximal function is obtained, which enables the author to apply some available results to the weighted case.

The author obtains a weighted regularity estimate assuming that, in the bounded domain under study, the equation is of class \(C^1\) and that each of its components belongs to the Sarason class VMO. The approach could be easily modified to cover the case where the domain \(\Omega\) where the functions under consideration belong, is Lipschitz with small Lipschitz constant and \(A\) has small bounded mean oscillation coefficients (i.e., they are in the John-Nirenberg class BMO).

The approach makes use of the Fefferman-Stein sharp maximal function and \(C^{1,\alpha}\) regularity estimates obtained earlier for homogeneous \(p\)-Laplace type equations. In particular, a weighted version of Fefferman-Stein inequality for a local dyadic sharp maximal function is obtained, which enables the author to apply some available results to the weighted case.

Reviewer: Maria Alessandra Ragusa (Catania)

##### MSC:

35R05 | PDEs with low regular coefficients and/or low regular data |

35J15 | Second-order elliptic equations |

42B25 | Maximal functions, Littlewood-Paley theory |

35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |

42B37 | Harmonic analysis and PDEs |