Aleksandrov-Bakelman-Pucci estimates - Revision history

imported>Russell at 01:33, 12 March 2012

2012-03-12T01:33:30Z

← Older revision		Revision as of 20:33, 11 March 2012
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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,		L(u,x):= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{y^TA(x)y}{\|y\|^{n+\sigma+2}} dy,
	\]		\]
	where $\text{Tr}(A(x))\geq \lambda$. For these operators, Guillen and Schwab proved that		where $\text{Tr}(A(x))\geq \lambda$. For these operators, Guillen and Schwab <ref name="GS"/> proved that
	\[		\[
	\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)},		\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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	<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="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>		<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>
			<ref name="GS">{{Citation \| last1=Guillen \| first1=N. \| last2=Schwab \| first2=R. \| title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations \| year=2010 \| journal=Arxiv preprint arXiv:1101.0279}}</ref>
	}}		}}

imported>Russell: moved Alexadroff-Bakelman-Pucci estimates to Aleksandrov-Bakelman-Pucci estimates: typo in the original title of the page

2012-03-12T01:23:56Z

moved Alexadroff-Bakelman-Pucci estimates to Aleksandrov-Bakelman-Pucci estimates: typo in the original title of the page

← Older revision	Revision as of 20:23, 11 March 2012
(No difference)

imported>Russell at 01:22, 12 March 2012

2012-03-12T01:22:35Z

← Older revision		Revision as of 20:22, 11 March 2012
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	The celebrated "~~Alexandroff~~-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.		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.

	== The classical ~~Alexsandroff~~-Bakelman-Pucci Theorem ==		== The classical Aleksandrov-Bakelman-Pucci Theorem ==

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

imported>Russell at 22:00, 10 March 2012

2012-03-10T22:00:00Z

← Older revision		Revision as of 17:00, 10 March 2012
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	The celebrated "Alexandroff-Bakelman-Pucci Maximum Principle" (often abbreviated as "ABP Estimate") is a pointwise estimate for weak solutions of elliptic equations. 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.		The celebrated "Alexandroff-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.

	== The classical Alexsandroff-Bakelman-Pucci Theorem ==		== The classical Alexsandroff-Bakelman-Pucci Theorem ==
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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.		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.

			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:

			\[
			L(u,x):= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{y^TA(x)y}{\|y\|^{n+\sigma+2}} dy,
			\]
			where $\text{Tr}(A(x))\geq \lambda$. For these operators, Guillen and Schwab proved that
			\[
			\sup_{B_1}\|u_-\|\leq \frac{C(n,\sigma)}{\lambda}\lVert f^+\rVert^{(2-\sigma)/2}_{L^\infty(\mathcal C)}\Vert f^+\rVert^{\sigma/2}_{L^n(\mathcal C)},
			\]
			where $\mathcal C$ is the contact set between $u$ and a $\sigma$-order replacement for the convex envelope.

	== References ==		== References ==

imported>Russell at 21:41, 10 March 2012

2012-03-10T21:41:32Z

← Older revision		Revision as of 16:41, 10 March 2012
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	The celebrated "Alexandroff-Bakelman-Pucci Maximum Principle" (often abbreviated 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 celebrated "Alexandroff-Bakelman-Pucci Maximum Principle" (often abbreviated as "ABP Estimate") is a pointwise estimate for weak solutions of elliptic equations. 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.

	== The classical Alexsandroff-Bakelman-Pucci Theorem ==		== The classical Alexsandroff-Bakelman-Pucci Theorem ==

imported>Nestor: /* ABP-type estimates for integro-differential equations */

2011-06-06T17:41:30Z

ABP-type estimates for integro-differential equations

← Older revision		Revision as of 12:41, 6 June 2011
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	\[ Lu \leq f(x) \;\; x \in B_1\]		\[ Lu \leq f(x) \;\; x \in B_1\]
	\[ u \~~leq~~ 0 \;\; x \in B_1^c\]		\[ u \geq 0 \;\; x \in B_1^c\]
	\[ Lu:= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{a(x,y)}{\|y\|^{n+\sigma}} dy\]		\[ Lu:= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{a(x,y)}{\|y\|^{n+\sigma}} dy\]

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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		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

	\[ \~~sup~~ \limits_{B_1} \; u^n \leq C_{n,\lambda,\Lambda,\sigma}\sum \limits_{i=1}^m ( \sup \limits_{Q_i^} \|f\|^n) \|Q^_i\| \]		\[ \inf \limits_{B_1} \; \|u_-\|^n \leq C_{n,\lambda,\Lambda,\sigma}\sum \limits_{i=1}^m ( \sup \limits_{Q_i^} \|f\|^n) \|Q^_i\| \]

	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 "~~concave~~ envelope" of $u$ in $B_2$.		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 $B_2$.

	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.		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.

imported>Nestor: /* The classical Alexsandroff-Bakelman-Pucci Theorem */

2011-06-06T17:40:43Z

The classical Alexsandroff-Bakelman-Pucci Theorem

← Older revision		Revision as of 12:40, 6 June 2011
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	\[ Lu \leq f(x) \;\; x \in B_1\]		\[ Lu \leq f(x) \;\; x \in B_1\]
	\[ u \~~leq~~ 0 \;\; x \in \partial B_1\]		\[ u \geq 0 \;\; x \in \partial B_1\]
	\[ Lu:=a_{ij}(x) u_{ij}(x)\]		\[ Lu:=a_{ij}(x) u_{ij}(x)\]

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	Moreover, the function $f$ is assumed to be continuous. Then, the ABP Theorem says that		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 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"/>.		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"/>.

imported>Nestor: /* The classical Alexsandroff-Bakelman-Pucci Theorem */

2011-06-06T17:39:50Z

The classical Alexsandroff-Bakelman-Pucci Theorem

← Older revision		Revision as of 12:39, 6 June 2011
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	Moreover, the function $f$ is assumed to be continuous. Then, the ABP Theorem says that		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 "~~concave~~ envelope" of $u$: it is the ~~smallest~~ non-~~nonnegative concave~~ 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 ~~concave~~ envelope is an important feature of the estimate and it is not to be overlooked <ref name="CC"/>.		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"/>.

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

imported>Nestor at 14:56, 6 June 2011

2011-06-06T14:56:31Z

← Older revision		Revision as of 09:56, 6 June 2011
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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 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 celebrated "Alexandroff-Bakelman-Pucci Maximum Principle" (often abbreviated 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 ==

imported>Nestor at 19:17, 4 June 2011

2011-06-04T19:17:44Z

← Older revision		Revision as of 14:17, 4 June 2011
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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		The coefficients $a_{ij}(x)$ are only assumed to be measurable functions such that for positive constants $\lambda<\Lambda$ we have

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

	Moreover, the function $f$ is assumed to be continuous. Then, the ABP Theorem says that		Moreover, the function $f$ is assumed to be continuous. Then, the ABP Theorem says that
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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		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

	\[ (\lambda \leq a(x,y) \leq \Lambda\]		\[ \lambda \leq a(x,y) \leq \Lambda \;\;\forall\;x,y\in \R^n\]

	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		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