Aleksandrov-Bakelman-Pucci estimates

The celebrated "Alexandroff-Bakelman-Pucci Maximum Principle" (often abbreviated as "ABP Estimate") is a pointwise estimate for weak solutions of elliptic equations. It is the backbone of the regularity theory of fully nonlinear second order elliptic equations and more recently for Fully nonlinear integro-differential equations .

The classical Alexsandroff-Bakelman-Pucci Theorem

Let $u$ be a viscosity supersolution of the linear equation:

\[ Lu \leq f(x) \;\; x \in B_1\] \[ u \geq 0 \;\; x \in \partial B_1\] \[ Lu:=a_{ij}(x) u_{ij}(x)\]

The coefficients $a_{ij}(x)$ are only assumed to be measurable functions such that for positive constants $\lambda<\Lambda$ we have

\[ \lambda |\xi|^2 \leq a_{ij}(x) \xi_i\xi_j \leq \Lambda |\xi|^2 \;\;\forall \; \xi \in \mathbb{R}^n \]

Moreover, the function $f$ is assumed to be continuous. Then, the ABP Theorem says that

\[ \sup \limits_{B_1}\; |u_-|^n \leq C_{n,\lambda,\Lambda} \int_{\{ u=\Gamma_u \} } f_+^n dx \]

Where $\Gamma_u$ is the "convex envelope" of $u$: it is the largest non-positive convex function in $B_2$ that lies above $u$ in $B_1$. The fact that the integration on the right hand side takes place only on the set where $u$ agrees with its convex envelope is an important feature of the estimate and it is not to be overlooked .

ABP-type estimates for integro-differential equations

The setting for integro-differential equations is similar, what changes are the operators: let $u$ be a viscosity supersolution of the equation

\[ Lu \leq f(x) \;\; x \in B_1\] \[ u \geq 0 \;\; x \in B_1^c\] \[ Lu:= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{a(x,y)}{|y|^{n+\sigma}} dy\]

Here $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$ and $\sigma \in (0,2)$. The function $a(x,y)$ is only assumed to be measurable and such that for some $\Lambda\geq\lambda>0$ we have

\[ \lambda \leq a(x,y) \leq \Lambda \;\;\forall\;x,y\in \R^n\]

As in the second order ABP, the function $f$ is assumed to be continuous. Then, Caffarelli and Silvestre proved there is an estimate

\[ \inf \limits_{B_1} \; |u_-|^n \leq C_{n,\lambda,\Lambda,\sigma}\sum \limits_{i=1}^m ( \sup \limits_{Q_i^*} |f|^n) |Q^*_i| \]

For some finite collection of non-overlapping cubes $\{Q_i \}_{i=1}^m$ that cover the set $\{ u=\Gamma_u\}$, each cube having non-zero intersection with this set. Moreover, all the cubes have diameters $d_i \lesssim 2^{-\frac{1}{2-\sigma}}$. As before, $\Gamma_u$ denotes the "convex envelope" of $u$ in $B_2$.

Furthermore, although the sharp constant may depend on $\sigma$, it is uniformly bounded for all $\sigma$ bounded away from zero. In particular, as $\sigma \to 2^-$ the above constant does not blow up, and since the diameter of the cubes goes to zero as $\sigma$ approaches $2$, one can check that this last estimate implies the second order ABP in the limit.