## Posts Tagged 'Harnack’s inequality'

### Reviewing the regularity theory of elliptics PDEs via the Laplace equation. Part III. (representation formulas)

This is the third of a series of posts dealing with the regularity theory of elliptic equations. My motivation in writing these is outlined in the first post. The previous post is here.

Let us recall Green’s identity, if ${u,v}$ are any functions smooth in ${\bar{\Omega}}$ and ${\Omega}$ is a bounded domain with smooth boundary we have

$\displaystyle \int_\Omega u\Delta v - v \Delta u dx = \int_{\partial \Omega}u\frac{\partial v}{\partial \nu}-v\frac{\partial u}{\partial \nu}dS$

this identity can be obtained with a couple of integration by parts involving the vector fields ${u \nabla v}$ and ${v \nabla u}$.

Lets rewrite the identity as

$\displaystyle \int_\Omega u\Delta vdx = -\int_\Omega v \Delta u dx + \int_{\partial \Omega}u\frac{\partial v}{\partial \nu}-v\frac{\partial u}{\partial \nu}dS$

thus, at least formally, if somehow we could find for every ${x \in \Omega}$ a function ${v_x(y)}$ such that

$\displaystyle \Delta v_x(y)=\delta_x(y)$

$\displaystyle v_x(y) \equiv 0 \mbox{ on } \partial \Omega$

then Green’s identity applied to both ${u}$ and ${v_x}$ in ${\Omega}$ would give us an integral representation formula for harmonic functions

### Reviewing the regularity theory of elliptics PDEs via the Laplace equation. Part II.

This is the second of a series of posts dealing with the regularity theory of elliptic equations. My motivation in writing these is outlined in the first post.

Some consequences of Harnack’s inequality the Mean value property

The mean value property is characteristic of harmonic functions, but the fact that harmonic functions control their pointwise values by their local average is a general fact that is characteristic of elliptic equations (as we will see later, less sharp but more general theorems for nonlinear elliptic equations still have this flavor and are at the very heart of the regularity theory of fully nonlinear elliptic PDEs). Let me mention a few of its consequences, I already talked last time about Harnack’s inequality, as it follows from the mean value theorem, the mean value theorem (at least for harmonic functions) is more fundamental.

### 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)