Aleksandrov-Bakelman-Pucci estimates: Difference between revisions
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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 | 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 <ref name="CC"/> and more recently for [[Fully nonlinear integro-differential equations]] <ref name="CS"/>. | ||
== The classical Alexsandroff-Bakelman-Pucci Theorem == | == The classical Alexsandroff-Bakelman-Pucci Theorem == | ||
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\[ u \leq 0 \;\; x \in \partial B_1\] | \[ u \leq 0 \;\; x \in \partial B_1\] | ||
The coefficients $a_{ij}(x)$ are only assumed to be measurable functions such that for positive constants $\lambda<\Lambda$ we have | |||
\[ \lambda |\xi|^2 \leq a_{ij}(x) \xi_i\xi_j \leq \Lambda |\xi|^2 \;\;\forall \xi \in \mathbb{R}^n \] | \[ \lambda |\xi|^2 \leq a_{ij}(x) \xi_i\xi_j \leq \Lambda |\xi|^2 \;\;\forall \xi \in \mathbb{R}^n \] | ||
Then, | Moreover, the function $f$ is assumed to be continuous. Then, the ABP Theorem says that | ||
\[ \sup \limits_{B_1}\; u^n \leq C_{n,\lambda,\Lambda} \int_{\{ u=\Gamma_u \} } f_+^n dx \] | \[ \sup \limits_{B_1}\; u^n \leq C_{n,\lambda,\Lambda} \int_{\{ u=\Gamma_u \} } f_+^n dx \] | ||
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 <ref name="CC"/> | |||
== ABP-type estimates for integro-differential equations == | == ABP-type estimates for integro-differential equations == | ||
== References == | |||
{{reflist|refs= | |||
<ref name="CC">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Cabré | first2=Xavier | title=Fully nonlinear elliptic equations | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=American Mathematical Society Colloquium Publications | isbn=978-0-8218-0437-7 | year=1995 | volume=43}}</ref> | |||
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref> | |||
}} | |||
Revision as of 12:31, 4 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 the backbone of the regularity theory of fully nonlinear second order elliptic equations [1] and more recently for Fully nonlinear integro-differential equations [2].
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 [1]
ABP-type estimates for integro-differential equations
References
- ↑ 1.0 1.1 Caffarelli, Luis A.; Cabré, Xavier (1995), Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, 43, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0437-7
- ↑ Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations", Communications on Pure and Applied Mathematics 62 (5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20274