Reviewing the regularity theory of elliptics PDEs via the Laplace equation [Part I]
There is a tedious, simple but hopefully fruitful exercise I always wanted to do. It is to review all the different proofs of the Harnack inequality and regularity of solutions to elliptic equations that I know, but only for the Laplace equation. First, because it is a good way to really get your hands on some of the ideas of several deep theorems (like those of De Giorgi-Nash-Moser and Krylov-Safonov) in the simplest possible setting. Second, because looking at all the different proofs it is possible to trace the evolution of analysis and PDEs through the last century (and a bit before that) and appreciate the level maturity reached in several fields: potential theory, singular integrals, calculus of variations, fully non linear elliptic PDE and free boundary problems. The `simple’ and `elementary’ Laplace equation lies at the intersection of all these fields, so every new breakthrough reflected on our understanding of this equation, each new proof emphasizing a different approach or point of view. Each of the proofs that I will discuss are based on one of the following:
- The mean value property (the proof you learn in your typical complex variables or introductory PDE course).
- The Poisson Kernel for the ball (the proof from potential theory).
- The Calderón-Zygmund theorem (ok not exactly a `Harnack inequality’, but it should be on this list anyway) which uses the machinery of singular integrals.
- The De Giorgi-Nash-Moser theorem, which follows the variational point of view and it is best suited for quasilinear equations or equations in divergence form.
- The Aleksandrov-Bakelman-Pucci estimate and the Krylov-Safonov’s `Harnack’s inequality’, which follows the comparison principle point of view and it is best suited for fully non linear equations or equations in non-divergence form.
So I am going to review each theorem and its proof but only for Laplace’s equation: . To start off easy, I am going to do first the proof via the mean value property.
First proof: mean value property
The mean value property says basically this
Let be a function in the unit ball of . If and is a sphere contained in and centered at , then equals the average of on
It is not hard to prove with some calculus, one basically looks at the function `Average of on the sphere of radius centered at ‘= and shows that , and since by continuity , the theorem follows. To show one sees (by say, a change of variables) that and this last integral is zero thanks to Stokes’ theorem and the fact that . Moreove, integrating the result with respect to the radius of the sphere one gets the same statement where instead of average over a sphere we have an average over a ball.
With this, one may prove easily Harnack’s inequality for harmonic functions, which I will state formally for the first time
Theorem 1 For any nonnegative harmonic function in we have the inequality
Proof. Let , then the ball of radius centered at (call it ) is completely contained in , thus by the mean value property
but is also contained in and since is nonnegative we have , again by the mean value property. This finishes the proof.
That is for today, in the next post I will explain some of the consequences of this theorem and maybe move on to the proof with potential theory methods.
Posted on August 25, 2011, in education and tagged Elliptic operator, Fermat's Last Theorem, Laplace transform, Math, Number Theory, Partial differential equation, Poisson Kernel, Stokes' theorem. Bookmark the permalink. 1 Comment.