imported>Russell |
imported>Luis |
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| The celebrated "Aleksandrov-Bakelman-Pucci Maximum Principle" (often abbreviated as "ABP Estimate") is a pointwise estimate for weak solutions of elliptic equations. The strength of the ABP estimate is that it is the main tool in the theory of non-divergence elliptic equations which gives a pointwise bound on solutions in terms of a measure theoretic quantity of the equation. It is not only 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"/>, but it also is essential in stochastic homogenization of uniformly elliptic equations in both the second order and integro-differential setting.
| | #REDIRECT[[Alexadroff-Bakelman-Pucci estimates]] |
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| == The classical Aleksandrov-Bakelman-Pucci Theorem ==
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| Let $u$ be a viscosity supersolution of the linear equation:
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| \[ Lu \leq f(x) \;\; x \in B_1\]
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| \[ u \geq 0 \;\; x \in \partial B_1\]
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| \[ Lu:=a_{ij}(x) u_{ij}(x)\]
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| The coefficients $a_{ij}(x)$ are only assumed to be measurable functions such that for positive constants $\lambda<\Lambda$ we have
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| \[ \lambda |\xi|^2 \leq a_{ij}(x) \xi_i\xi_j \leq \Lambda |\xi|^2 \;\;\forall \; \xi \in \mathbb{R}^n \]
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| Moreover, the function $f$ is assumed to be continuous. Then, the ABP Theorem says that
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| \[ \sup \limits_{B_1}\; |u_-|^n \leq C_{n,\lambda,\Lambda} \int_{\{ u=\Gamma_u \} } f_+^n dx \]
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| Where $\Gamma_u$ is the "convex envelope" of $u$: it is the largest non-positive convex function in $B_2$ that lies above $u$ 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"/>.
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| == ABP-type estimates for integro-differential equations ==
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| The setting for integro-differential equations is similar, what changes are the operators: let $u$ be a viscosity supersolution of the equation
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| \[ Lu \leq f(x) \;\; x \in B_1\]
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| \[ u \geq 0 \;\; x \in B_1^c\]
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| \[ Lu:= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{a(x,y)}{|y|^{n+\sigma}} dy\]
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| Here $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$ and $\sigma \in (0,2)$. The function $a(x,y)$ is only assumed to be measurable and such that for some $\Lambda\geq\lambda>0$ we have
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| \[ \lambda \leq a(x,y) \leq \Lambda \;\;\forall\;x,y\in \R^n\]
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| As in the second order ABP, the function $f$ is assumed to be continuous. Then, Caffarelli and Silvestre proved <ref name="CS"/> there is an estimate
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| \[ \inf \limits_{B_1} \; |u_-|^n \leq C_{n,\lambda,\Lambda,\sigma}\sum \limits_{i=1}^m ( \sup \limits_{Q_i^*} |f|^n) |Q^*_i| \]
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| For some finite collection of non-overlapping cubes $\{Q_i \}_{i=1}^m$ that cover the set $\{ u=\Gamma_u\}$, each cube having non-zero intersection with this set. Moreover, all the cubes have diameters $d_i \lesssim 2^{-\frac{1}{2-\sigma}}$. As before, $\Gamma_u$ denotes the "convex envelope" of $u$ in $B_2$.
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| Furthermore, although the sharp constant may depend on $\sigma$, it is uniformly bounded for all $\sigma$ bounded away from zero. In particular, as $\sigma \to 2^-$ the above constant does not blow up, and since the diameter of the cubes goes to zero as $\sigma$ approaches $2$, one can check that this last estimate implies the second order ABP in the limit.
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| So far, only very little is known about an ABP result which can capture the more refined measure theoretic information of the right hand side of the equation for such integro-differential $L$ as above. The only known result applies to a very restricted family of $L$ which are indexed by degenerate elliptic matrices:
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| \[
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| L(u,x):= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{y^TA(x)y}{|y|^{n+\sigma+2}} dy,
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| \]
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| where $\text{Tr}(A(x))\geq \lambda$. For these operators, Guillen and Schwab proved that
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| \[
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| \sup_{B_1}|u_-|\leq \frac{C(n,\sigma)}{\lambda}\lVert f^+\rVert^{(2-\sigma)/2}_{L^\infty(\mathcal C)}\Vert f^+\rVert^{\sigma/2}_{L^n(\mathcal C)},
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| \]
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| where $\mathcal C$ is the contact set between $u$ and a $\sigma$-order replacement for the convex envelope.
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| == References ==
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| {{reflist|refs=
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| <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>
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| <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>
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| }}
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