Aleksandrov-Bakelman-Pucci estimates

The celebrated "Alexandroff-Bakelman-Pucci Maximum Principle" (often abbreviated often 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:

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

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 convex function in $B_2$ such that it lies below $u$ 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