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Calderón-Zygmund estimates and non-uniformly elliptic operators.(English)Zbl 1479.35158

Summary: We consider a class of non-uniformly nonlinear elliptic equations whose model is given by \[- \operatorname{div}(| D u |^{p - 2} D u + a(x) | D u |^{q - 2} D u) = - \operatorname{div}(| F |^{p - 2} F + a(x) | F |^{q - 2} F)\] where \(p < q\) and \(a(x) \geq 0\), and establish the related nonlinear Calderón-Zygmund theory. In particular, we provide sharp conditions under which the natural, and optimal, Calderón-Zygmund-type result \[(| F |^p + a(x) | F |^q) \in L_{\operatorname{loc}}^\gamma \Longrightarrow(| D u |^p + a(x) | D u |^q) \in L_{\operatorname{loc}}^\gamma\] holds for every \(\gamma \geq 1\).
These problems naturally emerge as Euler-Lagrange equations of some variational integrals introduced and studied by Marcellini and Zhikov in the framework of Homogenisation and Lavrentiev phenomenon.

MSC:

35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35D30 Weak solutions to PDEs
35J20 Variational methods for second-order elliptic equations
35J70 Degenerate elliptic equations
35J92 Quasilinear elliptic equations with \(p\)-Laplacian

Cite

References:

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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.
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