The ABP maximum principle states (roughly) that, if $a^{ij} \partial _i \partial _j u \geq f$, over a domain $\Omega$ in $\mathbb{R}^n$ (where $a^{ij} \geq C Id >0$), then (assuming sufficient regularity of the coefficients), $\sup _{\Omega} u \leq \sup _{\partial \Omega} u + C (\int _{\Omega} \vert f \vert^n )^{1/n}$. Its proof seems to be quite unconventional (the final inequality is proved by an inequality of the measures of certain sets constructed using the normal mapping). Is there an intuitive explanation of the proof? (Perhaps using some probabilistic ideas or something?) I mean, usually, in order to prove maximum principles, the key idea is to use that at a local max the second derivative is negative-definite. But, this one seems to use some strange ideas..

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    $\begingroup$ Your differential inequality is missing the function $u$, and also the hypothesis of ellipticity. $\endgroup$ – YangMills Sep 4 '12 at 1:33

In my view, the key point the ABP estimate is that it ties pointwise information (the PDE) to information in measure (the contact set). This is crucial to regularity theory for non-divergence equations; it enables the proof of a Harnack inequality (due to Krylov and Safonov), and $C^{1,\alpha}$ regularity for viscosity solutions of fully nonlinear elliptic equations. For divergence equations, whose solutions are already defined by integrals, it is a bit easier to obtain regularity.

Note that the technique of looking at a local maximum, we only use the equation at a single point. In the proof of ABP that I know, we use the equation at many points. Here's the theorem I'd like to discuss:

If $a^{ij}(x)u_{ij} \leq 1$ in $B_1$ (uniformly elliptic)and $u|_{\partial B_1} \geq 0$ then $|inf_{B_1}u| \leq C |A|^{1/n}$ where $A$ is the set where $u$ agrees with its convex envelope. (The contact set).

Another way to describe $A$ is the collection of points in $B_1$ for which the graph of $u$ can be touched below by a supporting hyperplane. It is reasonable to believe that, given a supersolution, it doesn't have "cone-like" behavior, i.e. few points in $A$, roughly because the Hessian would be too large and contradict the equation at these points. This observation is made rigorous by the Area Formula; all the planes of slope $c|inf_{B_1}u|$ touch $u$ by below somewhere in $B_1$, so the derivative mapping $Du(B_1)$ contains a ball of radius $c|\inf_{B_1}u|$. The Jacobian determinant of this mapping is $detD^2u$, which tells us locally "how much we curved", i.e. the measure of the set of slopes of supporting hyperplanes nearby. The PDE tells us that at each of these points in $A$, $det D^2u$ is not too large, completing the proof.

Prof. Ovidiu Savin uses a similar technique in his paper "Small Perturbation Solutions of Elliptic Equations" to prove an ABP-type estimate; in this version, we slide paraboloids up from below the solution until they touch and show that the set of contact points has measure controlled below by the set of vertices of these paraboloids. This gives information in measure that can be exploited to prove a Harnack inequality and $C^{\alpha}$ regularity in the setting of the paper.

I hope this helps!

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