Alexandrov-Bakelmann-Pucci maximum principle

02-04

ABP estimate is the most basic estimate in fully nonliear elliptic equation.

The ABP maximum principle states (roughly) that, if

$a^{ij} partial _i partial _j u geq f, in Omega subset mathbb{R}^n (a^{ij} geq C Id >0)$

, then (assuming sufficient regularity of the coefficients),

$sup _{Omega} u leq sup _{partial Omega} u + C (int _{Omega} vert f vert^n )^{1/n} .......... (*)$

I will give an intuitive explanation of the proof of $(*)$ .

Usually, in order to prove maximum principles, the key idea is to use that at a local max the second derivative is negative-definite, then choose a good basis and get some identity of 1-order drivative and inequality for 2-orders. This process is used in such like the proof of the Hopf lemma, and some inter gradient estimate, consider some flexiable function like $e^{Au}$ or sometihng else anyway.

But in the proof of ABP we need more geometric intution and more trick.

First we do a rescaling:

if $a^{ij} partial _i partial _j u geq 1, in B_1 subset mathbb{R}^n (a^{ij} geq C Id >0),u|_{partial B_1}geq 0.$

then:

$|inf_{B_1}u| leq C |A|^{1/n} ....(**)$

And then We explain what is the contact set. It is the subset $Gamma^{+}$ of $Omega$ such that $u$ agree with it convex envelop. i.e. $Gamma^{+}={x|u=convex evolap of u at x}$ . The geometric meaning is it has at least one lower support plane. So what is $Gamma^{+}$ it is just the set that $u$ is very low on it. Or in another way of view you consider $-u$ as a lot of mountains then $Gamma^+$ is the place near the tops of which mountain can see every thing.

Then we look at every point in $Gamma^{+}$ , then determination of the hessian matrix $det(u_{ij})$ at this point have a control due to the PDE $a^{ij} partial _i partial _j u geq 1,$ and the uniformly elliptic property.

The determination of hessian matrix could be view as a determination of Jacobe matrix of the map $(u_1,...,u_n)to (e_1,...,e_n)$ .and by Area formula we have:

$int_{Phi(Omega)}f(Φ^{?1}(y)) dy =int_{Omega}f(x)|JΦ(x)| dx,$

It is easy to see for a constant $c$ , $B_{c|sup_{Omega}|u||}(0)subset Phi(Omega)$ (Base on the PDE on every point, the geometric intution is just the function $u$ could not be very narrow cone at every point). So we have , take $f=chi_{Gamma^{+}}$ ,

$|B_{csup_{Omega}|u|}(0)|^{n}leq int_{Gamma^{+}}chi_{Gamma_{+}}(x)|J_{Phi}(x)|dx$

so we have:

$|B_{csup_{Omega}|u|}(0)|lesssim |{Gamma_{+}}|^{frac{1}{n}}...(***)$

and the classical matrix inequality for every positive definite matrix $A$ we have

: $det(AB)leq (frac{tr(AB)}{n})^n...(****)$ .

combine $(***),(****)$ ,we have:

$sup_{Omega}|u|lesssim ||frac{a^{ij}u_{ij}}{D^*}||_{L^n({Gamma^+})}$ .