# Aleksandrov-Bakelman-Pucci estimates

(Difference between revisions)
 Revision as of 23:20, 3 June 2011 (view source)Nestor (Talk | contribs) (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...")← Older edit Revision as of 23:20, 3 June 2011 (view source)Nestor (Talk | contribs) Newer edit → Line 14: Line 14: Then, Then, - $\sup \limits_{B_1} u^n \leq C_{n,\lambda,\Lambda} \int_{u=\Gamma_u} f_+^n dx$ + $\sup \;\limits_{B_1} u^n \leq C_{n,\lambda,\Lambda} \int_{u=\Gamma_u} f_+^n dx$ == 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,

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