Sandbox and Aleksandrov-Bakelman-Pucci estimates: Difference between pages
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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, | |||
= | \[ \sup \limits_{B_1}\; u^n \leq C_{n,\lambda,\Lambda} \int_{u=\Gamma_u} f_+^n dx \] | ||
== | == ABP-type estimates for integro-differential equations == | ||
Revision as of 18:20, 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,
\[ \sup \limits_{B_1}\; u^n \leq C_{n,\lambda,\Lambda} \int_{u=\Gamma_u} f_+^n dx \]