What is the heuristic idea of the proof of the boundary Harnack inequality presented in the appendix of Caffarelli's 1998 lectures on the obstacle problem (page 38 here)?
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1$\begingroup$ You should expand the question. People are not supposed to search for Caffarelli's lectures. You need to provide a statement of the Harnack inequality and give clear references. $\endgroup$– Piotr HajlaszCommented Dec 29, 2018 at 2:40
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$\begingroup$ @PiotrHajlasz Thank you for the heads up. Done. $\endgroup$– user124345Commented Dec 29, 2018 at 10:54
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An illustration of the proof of an associated Harnack inequality for elliptic pdi is given in the book "The Maximum Principle" by Pucci and Serrin (see Chapter 7). Hopefully that helps. I can pull up paper references within if that is more helpful.
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$\begingroup$ Yes, further references would be helpful. Thanks. $\endgroup$– user124345Commented Jan 3, 2019 at 14:12
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$\begingroup$ Instead of the book, you can see the following 2 references for similar results: J. Serrin,
Local behavior of solutions of quasilinear elliptic equations", Acta Math., 111, (1964), 247-302. N.S.TrudingerOn Harnack type inequalities and their applications to quasilinear elliptic equations.", Comm. Pure. Appl. Math., 20, (1967), 721-747. Alternatively, a somewhat more contemporary discussion of related Harnack inequalities appears in A.I.Nazarov and N.N.Ural'tseva foud at arxiv.org/abs/1011.1888. Hopefully that helps and sorry about the way the comment is rendered. $\endgroup$– JCMCommented Jan 7, 2019 at 18:31