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: {\Delta u = 0}. 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 {u} be a {C^2} function in the unit ball {B_1} of {\mathbb{R}^n}. If {\Delta u = 0} and {S} is a sphere contained in {B_1} and centered at {x_0}, then {u(x_0)} equals the average of {u} on {S}

It is not hard to prove with some calculus, one basically looks at the function `Average of {u} on the sphere of radius {r} centered at {x_0}‘={f(r)} and shows that {f'(r)=0}, and since by continuity {f(0)=u(x_0)}, the theorem follows. To show {f'(r)=0} one sees (by say, a change of variables) that {\frac{d}{dr} \frac{1}{|S|}\int_{S_r}u(x)d\sigma =\frac{1}{|S}\int_{S_r}u_nd\sigma} and this last integral is zero thanks to Stokes’ theorem and the fact that {\Delta u = 0}. 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 {u} in {B_1} we have the inequality

\displaystyle  u(x) \leq 2^nu(0) \;\;\; \mbox{ for all } x \in B_{1/2}

Proof. Let {x \in B_{1/2}}, then the ball of radius {1/4} centered at {x} (call it {B}) is completely contained in {B_1}, thus by the mean value property

u(x)=\int_B u(y)dy

but B is also contained in B_{1/2} and since u is nonnegative we have \frac{1}{|B|}\int_B u(y)dy \leq \frac{2^n}{|B_{1/2}|} \int_{B_{1/2}}u(y)dy=2^nu(0), 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.

(Note: this post was made using Luca Trevisan’s Latex to WordPress program, which is very useful although I am still getting used to using it. It allows you to prepare your post in a latex editor and then translate it into HTML code which WordPress can read, I strongly recommend it)

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3 Responses to “Reviewing the regularity theory of elliptics PDEs via the Laplace equation. Part I.”


  1. 1 K January 12, 2010 at 7:52 pm

    Hey Nestor, would you define a few things for me?

    1. f(r)
    2. u_n that appears in the second paragraph of the first proof
    and S_r, I assume means the sphere of radius r centered at x.

    Thanks

  2. 2 Nestor January 13, 2010 at 9:52 pm

    Hi K,

    1) f(r) is the average of the the function u on the sphere of radius r, in formulas

    f(r)= \frac{1}{|S_r|}\int_{S_r}u(x')dx'

    here dx' stands for the surface measure of the sphere.

    2) u_n is the outer normal derivative of u on S_r, which is indeed the sphere of radius r.

    Hope that clears it up!

  3. 3 K January 17, 2010 at 3:03 pm

    Oh, I’m sorry. I just noticed the tick mark (quotations) defining the f. Before, I read it as x_0 = f(r), which made no sense. Thanks!


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