Aleksandrov-Bakelman-Pucci estimates

The celebrated "Aleksandrov-Bakelman-Pucci Maximum Principle" (often abbreviated as "ABP Estimate") is a pointwise estimate for weak solutions of elliptic equations. The strength of the ABP estimate is that it is the main tool in the theory of non-divergence elliptic equations which gives a pointwise bound on solutions in terms of a measure theoretic quantity of the equation. It is not only the backbone of the regularity theory of fully nonlinear second order elliptic equations and more recently for Fully nonlinear integro-differential equations , but it also is essential in stochastic homogenization of uniformly elliptic equations in both the second order and integro-differential setting.

The classical Aleksandrov-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.

So far, only very little is known about an ABP result which can capture the more refined measure theoretic information of the right hand side of the equation for such integro-differential $L$ as above. The only known result applies to a very restricted family of $L$ which are indexed by degenerate elliptic matrices:

\[ L(u,x):= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{y^TA(x)y}{|y|^{n+\sigma+2}} dy, \] where $\text{Tr}(A(x))\geq \lambda$. For these operators, Guillen and Schwab proved that \[ \sup_{B_1}|u_-|\leq \frac{C(n,\sigma)}{\lambda}\lVert f^+\rVert^{(2-\sigma)/2}_{L^\infty(\mathcal C)}\Vert f^+\rVert^{\sigma/2}_{L^n(\mathcal C)}, \] where $\mathcal C$ is the contact set between $u$ and a $\sigma$-order replacement for the convex envelope.