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| 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.
| | The Evans-Krylov theorem says that if $u$ is a solution to a uniformly elliptic, fully nonlinear, convex or concave, equation |
| | \[ F(D^2 u) = 0 \text{ in } B_1,\] |
| | then $u \in C^{2,\alpha}(B_{1/2})$ and there is an estimate |
| | \[ \|u\|_{C^{2,\alpha}(B_{1/2})} \leq C \|u\|_{L^\infty(B_1)}, \] |
| | where $C$ and $\alpha>0$ depend only on the ellipticity constants and dimension. |
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| '''Theorem'''. Let $u \in L^\infty(R^n) \cap C(\overline B_1)$ solve the equation \[Iu = 0 \ \ \text{in } B_1.\]
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| Then $u \in C^{1,\alpha}(B_{1/2})$ and the following estimate holds
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| \[ ||u||_{C^{1,\alpha}(B_{1/2})} \leq C ||u||_{L^\infty}. \]
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| A theorem as above is known to hold under some assumptions on the [[nonlocal operator]] $I$. A list of valid assumptions is provided below.
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| 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]].
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| == Idea of the proof ==
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| 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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| \[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \leq 0 \]
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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 $u \in C^{1,\alpha}$.
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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 $u_e \in L^\infty(\R^n)$. 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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| ==Examples for which the estimate holds ==
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| === Translation invariant, uniformly elliptic 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.
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| \[
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| \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) \leq \frac{(2-s)\Lambda}{|y|^{n+s}} \ \ \text{(standard unif. ellipticity of order $s$)}
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| \]
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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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| === Isaacs equation with variable coefficients but close to constant ===
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| If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates <ref name="CS2"/>. 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 $x$.
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| \[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)(a_0(y) + a_{\alpha \beta}(x,y))}{|y|^{n+s}} \mathrm d y =0\]
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| such that we have for $\eta$ small enough and any $\alpha$, $\beta$,
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| \begin{align*}
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| |a_{\alpha \beta}(x,y)| &< \eta \qquad \text{ for every } \alpha, \beta \\
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| \lambda &\leq a_0(y) \leq \Lambda \\
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| |\nabla a_0(y)| &\leq C |y|^{-1}
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| \end{align*}
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| (note that this $C^{1,\alpha}$ estimate is nontrivial in the linear case as well)
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| === Isaacs equation with continuous coefficients ===
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| If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates <ref name="CS2"/>.
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| \[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)a_{\alpha \beta}(x,y)}{|y|^{n+s}} \mathrm d y = 0\]
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| such that for every $\alpha$, $\beta$ we have
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| \begin{align*}
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| \lambda \leq a_{\alpha \beta}(x,y) &\leq \Lambda \\
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| \nabla_y a_{\alpha \beta}(x,y) &\leq C_1/((2-s)|y|)\\
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| |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 $c$}.
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| \end{align*}
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| == References ==
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| {{reflist|refs=
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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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| <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>
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| <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>
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| <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>
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| <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>
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