# Aleksandrov-Bakelman-Pucci estimates

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(Created page with "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 a...") |
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- | \[ \sup \limits_{B_1} u^n \leq C_{n,\lambda,\Lambda} \int_{u=\Gamma_u} f_+^n dx \] | + | \[ \sup \;\limits_{B_1} u^n \leq C_{n,\lambda,\Lambda} \int_{u=\Gamma_u} f_+^n dx \] |

== ABP-type estimates for integro-differential equations == | == ABP-type estimates for integro-differential equations == |

## Revision as of 23:20, 3 June 2011

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 a fundamental result which is the backbone of the regularity theory of fully nonlinear second order elliptic equations (reference Caffarelli-Cabré) and more recently for Fully nonlinear integro-differential equations (reference Caffarelli-Silvestre).

## The classical Alexsandroff-Bakelman-Pucci Theorem

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

\[ a_ij(x) u_ij(x) \leq f(x) \;\; x \in B_1\] \[ u \leq 0 \;\; x \in \partial B_1\]

where the coefficients $a_ij(x)$ are only assumed to be measurable functions such that for positive constants $\lambda<\Lambda$ we have

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

Then,

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