2.3 Hölder continuity of weak solutions (PART 3)

We look into our first regularity result for this video series: that under a continuity assumption on the leading coefficients and slightly stronger integrability conditions on the other terms, we show that weak solutions to elliptic PDEs in divergence form are Hölder continuous in the interior.

The proof of this result is divided into three parts:

Part 1: We start the proof by writing out the calculations/details involving the “perturbation” argument, that is, we freeze the leading coefficients and compare weak solutions to A-harmonic functions.

Part 2: We discuss how to obtain estimates for our weak solution. The techniques used here are standard and appear in the proofs of a priori estimates of similar equations. We talk about how these calculations are “natural” in some sense and try to motivate them. We also explain why we needed stronger integrability conditions on some of the terms in the equation.

Part 3: We finish the proof of the result. We discuss the use of a bootstrap argument and how it is used to improve estimates through iteration.

The proof is taken from “Elliptic Partial Differential Equations, Second Edition” by Qing Han and Fanghua Lin, published by the American Mathematical Society and the Courant Institute of Mathematical Sciences.

NOTE: I am planning to upload my notes for these lectures soon in my website!

[shengzhang]

Really cool ! Look forward to your update.