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Harnack inequalities for quasi-minima of variational integrals. (English) Zbl 0565.35012

The main result is that the Harnack inequality can be proved directly for functions in the De Giorgi classes. It is well known that weak solutions of quasilinear elliptic equations, in divergence form, under appropriate structure conditions, belong to De Giorgi classes. Therefore this work provides alternate proofs of the Harnack inequalities for those weak solutions. These results also imply that very non-negative Q-minimum (in the terminology of Giaquinta and Giusti) satisfies a Harnack inequality. The main tools are modifications of the De Giorgi estimates for elliptic equations and a fundamental covering lemma due to Krylov and Safonov.
Reviewer: S.M.Lenhart

MSC:

35B45 A priori estimates in context of PDEs
35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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References:

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