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 Hajlasz Dec 29 '18 at 2:40

$\begingroup$ @PiotrHajlasz Thank you for the heads up. Done. $\endgroup$ – Hiro Dec 29 '18 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.


$\begingroup$ 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), 247302. N.S.Trudinger
On Harnack type inequalities and their applications to quasilinear elliptic equations.", Comm. Pure. Appl. Math., 20, (1967), 721747. 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. $\endgroup$ – JCM Jan 7 at 18:31