Differentiability estimates

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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
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\[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \geq 0 \]  
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\[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \leq 0 \]  
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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 $u_e \in L^\infty(\R^n)$. 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" />.
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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.
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==Examples for which the estimate holds ==
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==Classes of kernels for which the estimate holds ==
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=== Translation invariant, uniformly elliptic of order $s$ ===
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=== Translation invariant, uniformly elliptic of order $s$, and some smoothness in the tails of the kernels ===
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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.
 +
\[
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\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) \leq \frac{(2-s)\Lambda}{|y|^{n+s}} \ \ \text{(standard unif. ellipticity of order $s$)}
 +
\]
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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"/>.
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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" />.
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\begin{align*}
+
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\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\
+
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\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 $W^{1,1}$)}
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\end{align*}
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=== Variant if the kernel tails are $C^1$ ===
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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"/>
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\begin{align*}
+
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\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\
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\nabla K(y) &\leq \frac{\Lambda}{|y|^{n+s+1}} && \text{appropriate decay of the kernel in $C^1$.}
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\end{align*}
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+
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Then, any solution of $Iu=0$ in $B_r$ satisfies the estimate
+
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\[ [u]_{C^{1,\alpha}(B_{r/2})} \leq C \left(\frac 1 {r^{1+\alpha}} ||u||_{L^\infty(B_r)} + \frac 1 {r^{1+\alpha-s}} \int_{\R^n \setminus B_r} \frac{|u(y)|}{|y|^{n+s}} \mathrm d y \right). \]
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Other $C^{1,\alpha}$ estimates are obtained from this one using [[perturbation methods]] <ref name="CS2"/>.
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=== A class of non-differentiable kernels ===
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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"/>
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\begin{align*}
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K(y) &= (2-s) \frac{a_1(y) + a_2(y)}{|y|^{n+s}} \\
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\lambda &\leq a_1(y) \leq \Lambda \\
+
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|a_2| &\leq \eta \\
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|\nabla a_1(y)| &\leq \frac{C_1}{|y|} \qquad \text{in } \R^n \setminus \{0\}
+
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\end{align*}
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for $s>1$ and $\eta$ small enough (depending on $\lambda$, $\Lambda$, $C_1$ and dimension)
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=== Isaacs equation with variable coefficients but close to constant ===
=== Isaacs equation with variable coefficients but close to constant ===
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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"/>.
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\[ \inf_\alpha \sup_\beta \int_{\R^n} \delta u(x,y) \frac{(2-s)a_{\alpha \beta}(x,y)}{|y|^{n+s}} \mathrm d y = 0\]
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\[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)a_{\alpha \beta}(x,y)}{|y|^{n+s}} \mathrm d y = 0\]
such that for every $\alpha$, $\beta$ we have
such that for every $\alpha$, $\beta$ we have
\begin{align*}
\begin{align*}
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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>
}}
}}

Latest revision as of 18:09, 4 March 2014

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 \[Iu = 0 \ \ \text{in } B_1.\] Then $u \in C^{1,\alpha}(B_{1/2})$ and the following estimate holds \[ ||u||_{C^{1,\alpha}(B_{1/2})} \leq C ||u||_{L^\infty}. \]

A theorem as above is known to hold under some assumptions on the nonlocal operator $I$. A list of valid assumptions is provided below.

Note that the result is stated for general fully nonlinear integro-differential equations, but the most important cases to apply it are the Isaacs equation and Bellman equation.

Contents

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 \[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \leq 0 \] 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 $u \in C^{1,\alpha}$.

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 [1] and [2].

Examples for which the estimate holds

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. \[ \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) \leq \frac{(2-s)\Lambda}{|y|^{n+s}} \ \ \text{(standard unif. ellipticity of order $s$)} \]

The result was first proved in [3] assuming an extra regularity condition in the family of kernels. This condition was later removed in [1]. For the parabolic version of the problem, it was first done in [4] with the extra smoothness assumption on the kernel, which was later removed in [2].

Isaacs equation with variable coefficients but close to constant

If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates [5]. The family of integro-differential operators has kernels which are the sum of a fixed term $a_0$ (the same for all kernels in the class) and a small term which can depend on $x$. \[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)(a_0(y) + a_{\alpha \beta}(x,y))}{|y|^{n+s}} \mathrm d y =0\] such that we have for $\eta$ small enough and any $\alpha$, $\beta$, \begin{align*} |a_{\alpha \beta}(x,y)| &< \eta \qquad \text{ for every } \alpha, \beta \\ \lambda &\leq a_0(y) \leq \Lambda \\ |\nabla a_0(y)| &\leq C |y|^{-1} \end{align*} (note that this $C^{1,\alpha}$ estimate is nontrivial in the linear case as well)

Isaacs equation with continuous coefficients

If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates [5]. \[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)a_{\alpha \beta}(x,y)}{|y|^{n+s}} \mathrm d y = 0\] such that for every $\alpha$, $\beta$ we have \begin{align*} \lambda \leq a_{\alpha \beta}(x,y) &\leq \Lambda \\ \nabla_y a_{\alpha \beta}(x,y) &\leq C_1/((2-s)|y|)\\ |a_{\alpha \beta}(x_1,y) - a_{\alpha \beta}(x_2,y)| &\leq c(|x_1-x_2|) && \text{for some uniform modulus of continuity $c$}. \end{align*}


References

  1. 1.0 1.1 Kriventsov, Dennis (2013), "C 1, $\alpha$ Interior Regularity for Nonlinear Nonlocal Elliptic Equations with Rough Kernels", Communications in Partial Differential Equations 38: 2081--2106 
  2. 2.0 2.1 Serra, Joaquim, "Regularity for fully nonlinear nonlocal parabolic equations with rough kernels", arXiv preprint arXiv:1401.4521 
  3. 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 
  4. Lara, HéctorChang; Dávila, Gonzalo, "Regularity for solutions of non local parabolic equations", Calculus of Variations and Partial Differential Equations (Springer Berlin Heidelberg) 49: 139-172, doi:10.1007/s00526-012-0576-2, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-012-0576-2 
  5. 5.0 5.1 Caffarelli, Luis; Silvestre, Luis (2009), "Regularity results for nonlocal equations by approximation", Archive for Rational Mechanics and Analysis (Berlin, New York: Springer-Verlag): 1–30, ISSN 0003-9527 
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