Differentiability estimates - Revision history

imported>Luis at 18:09, 4 March 2014

2014-03-04T18:09:14Z

← Older revision		Revision as of 13:09, 4 March 2014
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	where $M^\pm_{\mathcal L}$ are the [[extremal operators]] with respect to the corresponding class of operators $\mathcal L$. If the [[Holder estimates]] apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.		where $M^\pm_{\mathcal L}$ are the [[extremal operators]] with respect to the corresponding class of operators $\mathcal L$. If the [[Holder estimates]] apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.

	There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.		There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$. Because of this difficulty, the first versions of the proof assume an extra regularity condition on the family of kernels. This regularity condition can be removed following the methods in <ref name="kriventsov2013c" /> and <ref name="serra2014regularity" />.

	==Examples for which the estimate holds ==		==Examples for which the estimate holds ==

	=== Translation invariant, uniformly elliptic of order $s$~~, and some smoothness in the tails of the kernels~~ ===		=== Translation invariant, uniformly elliptic of order $s$ ===

	The first situation in which the interior $C^{1,\alpha}$ estimate was proved for a nonlocal equation was if $I$ is translation invariant and [[uniformly elliptic]] with respect to the class of kernels satisfying the following hypothesis ~~for some $\rho_0$ small enough<ref name="CS"/>~~.		The first situation in which the interior $C^{1,\alpha}$ estimate was proved for a nonlocal equation was if $I$ is translation invariant and [[uniformly elliptic]] with respect to the class of kernels satisfying the following hypothesis.
	\~~begin{align*}~~		\[
	\frac{(2-s)\lambda}{\|y\|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{\|y\|^{n+s}} && \text{(standard unif. ellipticity of order $s~~$)}\\~~		\frac{(2-s)\lambda}{\|y\|^{n+s}} \leq K(y) \leq \frac{(2-s)\Lambda}{\|y\|^{n+s}} \ \ \text{(standard unif. ellipticity of order $s$)}
	~~\int_{\R^n \setminus B_{\rho_0}} \frac{\|K(y)-K(y-h)\|}{\|h\|} \mathrm d y &\leq C \qquad \text{every time $\|h\|<\frac {\rho_0} 2$} && \text{(kernel tails in $W^{1,1}~~$)}		\]
	\~~end{align*}~~

	=~~== Variant if the kernel tails are $C^1$ ===~~		The result was first proved in <ref name="CS"/> assuming an extra regularity condition in the family of kernels. This condition was later removed in <ref name="kriventsov2013c" />. For the parabolic version of the problem, it was first done in <ref name="changdavila" /> with the extra smoothness assumption on the kernel, which was later removed in <ref name="serra2014regularity" />.

	~~A small variation of the previous result is to assume~~ the ~~class~~ of kernels ~~satisfying the slightly stronger assumptions~~. ~~A scale invariant class for which interior $C^{1,\alpha}$ regularity holds is~~ <ref name="~~CS2~~"/>
	~~\begin{align*}~~
	~~\frac{(2-s)\lambda}{\|y\|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{\|y\|^{n+s}} && \text{(standard unif~~. ~~ellipticity of order $s$)}\\~~
	~~\nabla K(y) &\leq \frac{\Lambda}{\|y\|^{n+s+1}} && \text{appropriate decay~~ of the ~~kernel in $C^1$.}~~
	~~\end{align*}~~

	~~Then~~, ~~any solution of $Iu=0$~~ in ~~$B_r$ satisfies the estimate~~
	~~\[ [u]_{C^{1,\alpha}(B_{r/2})} \leq C \left(\frac 1 {r^{1+\alpha}} \|\|u\|\|_{L^\infty(B_r)} + \frac 1 {r^{1+\alpha-s}} \int_{\R^n \setminus B_r} \frac{\|u(y)\|}{\|y\|^{n+s}} \mathrm d y \right). \]~~
	~~Other $C^{1,\alpha}$ estimates are obtained from this one using [[perturbation methods]]~~ <ref name="~~CS2~~"/>.

	~~=== A class of non-differentiable kernels ===~~
	~~A scale invariant class for which interior $C^{1,\alpha}$ regularity holds and~~ the ~~kernels can be very irregular is given by~~ the ~~following hypothesis~~<ref name="~~CS2~~"/>
	~~\begin{align*}~~
	~~K(y) &= (2-s) \frac{a_1(y) + a_2(y)}{\|y\|^{n+s}} \\~~
	~~\lambda &\leq a_1(y) \leq \Lambda \\~~
	~~\|a_2\| &\leq \eta \\~~
	~~\|\nabla a_1(y)\| &\leq \frac{C_1}{\|y\|} \qquad \text{in } \R^n \setminus \{0\}~~
	~~\end{align*}~~
	~~for $s>1$ and $\eta$ small enough (depending on $\lambda$, $\Lambda$, $C_1$ and dimension)~~

	=== Isaacs equation with variable coefficients but close to constant ===		=== Isaacs equation with variable coefficients but close to constant ===
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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>		<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="CS2">{{Citation \| last1=Caffarelli \| first1=Luis \| last2=Silvestre \| first2=Luis \| title=Regularity results for nonlocal equations by approximation \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| year=2009 \| journal=Archive for Rational Mechanics and Analysis \| issn=0003-9527 \| pages=1–30}}</ref>		<ref name="CS2">{{Citation \| last1=Caffarelli \| first1=Luis \| last2=Silvestre \| first2=Luis \| title=Regularity results for nonlocal equations by approximation \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| year=2009 \| journal=Archive for Rational Mechanics and Analysis \| issn=0003-9527 \| pages=1–30}}</ref>
			<ref name="kriventsov2013c">{{Citation \| last1=Kriventsov \| first1= Dennis \| title=C 1, $\alpha$ Interior Regularity for Nonlinear Nonlocal Elliptic Equations with Rough Kernels \| journal=Communications in Partial Differential Equations \| year=2013 \| volume=38 \| pages=2081--2106}}</ref>
			<ref name="serra2014regularity">{{Citation \| last1=Serra \| first1= Joaquim \| title=Regularity for fully nonlinear nonlocal parabolic equations with rough kernels \| journal=arXiv preprint arXiv:1401.4521}}</ref>
			<ref name="changdavila">{{Citation \| last1=Lara \| first1= HéctorChang \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of non local parabolic equations \| url=http://dx.doi.org/10.1007/s00526-012-0576-2 \| journal=Calculus of Variations and Partial Differential Equations \| publisher=Springer Berlin Heidelberg \| issn=0944-2669 \| volume=49 \| pages=139-172 \| doi=10.1007/s00526-012-0576-2}}</ref>
	}}		}}

imported>Luis: /* Idea of the proof */

2014-03-04T17:53:54Z

Idea of the proof

← Older revision		Revision as of 12:53, 4 March 2014
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	There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.		There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.

	The first solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$.

	==Examples for which the estimate holds ==		==Examples for which the estimate holds ==

imported>Luis: /* Idea of the proof */

2013-02-18T22:49:16Z

Idea of the proof

← Older revision		Revision as of 17:49, 18 February 2013
Line 16:		Line 16:
	There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.		There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.

	The first solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$~~. A solution to this difficulty which removed the smoothness assumption on the kernels is given in <ref name="DK">~~.		The first solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$.

	==Examples for which the estimate holds ==		==Examples for which the estimate holds ==

imported>Luis: /* Idea of the proof */

2013-02-18T22:47:28Z

Idea of the proof

← Older revision		Revision as of 17:47, 18 February 2013
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	There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.		There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.

	The ~~only known~~ solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$. It is ~~an interesting [[open problems\|open problem]] whether a better solution exist~~.		The first solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$. A solution to this difficulty which removed the smoothness assumption on the kernels is given in <ref name="DK">.

	==Examples for which the estimate holds ==		==Examples for which the estimate holds ==

imported>Luis: /* Idea of the proof */

2012-02-07T17:03:22Z

Idea of the proof

← Older revision		Revision as of 12:03, 7 February 2012
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	There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.		There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.

	The only known solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$. It is an interesting [[open problem]] whether a better solution exist.		The only known solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$. It is an interesting [[open problems\|open problem]] whether a better solution exist.

	==Examples for which the estimate holds ==		==Examples for which the estimate holds ==

imported>Luis at 18:43, 29 May 2011

2011-05-29T18:43:54Z

← Older revision		Revision as of 13:43, 29 May 2011
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	== Idea of the proof ==		== Idea of the proof ==
	The idea to prove a $C^{1,\alpha}$ estimate is to apply [[Holder estimates]] to the derivatives of the solutions $u$. The directional derivatives $u_e$ satisfy the two inequalities		The idea to prove a $C^{1,\alpha}$ estimate is to apply [[Holder estimates]] to the derivatives of the solutions $u$. The directional derivatives $u_e$ satisfy the two inequalities
	\[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \~~geq~~ 0 \]		\[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \leq 0 \]
	where $M^\pm_{\mathcal L}$ are the [[extremal operators]] with respect to the corresponding class of operators $\mathcal L$. If the [[Holder estimates]] apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.		where $M^\pm_{\mathcal L}$ are the [[extremal operators]] with respect to the corresponding class of operators $\mathcal L$. If the [[Holder estimates]] apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.

69.217.124.45 at 17:49, 29 May 2011

2011-05-29T17:49:45Z

← Older revision		Revision as of 12:49, 29 May 2011
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	=== Isaacs equation with continuous coefficients ===		=== Isaacs equation with continuous coefficients ===
	If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates <ref name="CS2"/>.		If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates <ref name="CS2"/>.
	\[ \inf_\alpha \sup_\beta \int_{\R^n} ~~\delta~~ u(x,y) \frac{(2-s)a_{\alpha \beta}(x,y)}{\|y\|^{n+s}} \mathrm d y = 0\]		\[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)a_{\alpha \beta}(x,y)}{\|y\|^{n+s}} \mathrm d y = 0\]
	such that for every $\alpha$, $\beta$ we have		such that for every $\alpha$, $\beta$ we have
	\begin{align*}		\begin{align*}

69.217.124.45 at 17:48, 29 May 2011

2011-05-29T17:48:58Z

← Older revision		Revision as of 12:48, 29 May 2011
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	The only known solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$. It is an interesting [[open problem]] whether a better solution exist.		The only known solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$. It is an interesting [[open problem]] whether a better solution exist.

	==~~Classes of kernels~~ for which the estimate holds ==		==Examples for which the estimate holds ==

	=== Translation invariant, uniformly elliptic of order $s$, and some smoothness in the tails of the kernels ===		=== Translation invariant, uniformly elliptic of order $s$, and some smoothness in the tails of the kernels ===

69.217.124.45 at 17:48, 29 May 2011

2011-05-29T17:48:08Z

← Older revision		Revision as of 12:48, 29 May 2011
Line 12:		Line 12:
	The idea to prove a $C^{1,\alpha}$ estimate is to apply [[Holder estimates]] to the derivatives of the solutions $u$. The directional derivatives $u_e$ satisfy the two inequalities		The idea to prove a $C^{1,\alpha}$ estimate is to apply [[Holder estimates]] to the derivatives of the solutions $u$. The directional derivatives $u_e$ satisfy the two inequalities
	\[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \geq 0 \]		\[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \geq 0 \]
	where $M^\pm_{\mathcal L}$ are the [[extremal ~~operatos~~]] with respect to the corresponding class of operators $\mathcal L$. If the [[Holder estimates]] apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.		where $M^\pm_{\mathcal L}$ are the [[extremal operators]] with respect to the corresponding class of operators $\mathcal L$. If the [[Holder estimates]] apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.

	There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.		There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.

69.217.124.45 at 17:45, 29 May 2011

2011-05-29T17:45:45Z

@@ Line 1: / Line 1: @@
-Given a [[fully nonlinear integro-differential equation]] $Iu=0$, [[uniformly elliptic]] with respect to certain [[class of operators]], sometimes an interior $C^{1,\alpha}$ estimate holds. Assume $I0=0$. The $C^{1,\alpha}$ estimate is a result like the following.
+Given a [[fully nonlinear integro-differential equation]] $Iu=0$, [[uniformly elliptic]] with respect to certain [[class of operators]], sometimes an interior $C^{1,\alpha}$ estimate holds for some $\alpha>0$ (typically very small). Assume $I0=0$. The $C^{1,\alpha}$ estimate is a result like the following.
 '''Theorem'''. Let $u \in L^\infty(R^n) \cap C(\overline B_1)$ solve the equation \[Iu = 0 \ \ \text{in } B_1.\]
@@ Line 39: / Line 39: @@
 \[ [u]_{C^{1,\alpha}(B_{r/2})} \leq C \left(\frac 1 {r^{1+\alpha}} ||u||_{L^\infty(B_r)} + \frac 1 {r^{1+\alpha-s}} \int_{\R^n \setminus B_r} \frac{|u(y)|}{|y|^{n+s}} \mathrm d y \right). \]
 Other $C^{1,\alpha}$ estimates are obtained from this one using [[perturbation methods]] <ref name="CS2"/>.
 == References ==