Differentiability estimates and Nonlocal Evans-Krylov theorem: Difference between pages

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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 classical [[Evans-Krylov theorem]] <ref name="E"/> <ref name="K"/> says that convex or concave fully nonlinear elliptic equations have $C^{2,\alpha}$ (therefore classical) solutions. This type of equations can be written as a Hamilton-Jacobi-Bellman equation.
\[ \sup_\beta a_{ij}^\beta \partial_{ij} u = f \]
for a family of uniformly elliptic coefficients $a_{ij}^\alpha$.


'''Theorem'''. Let $u \in L^\infty(R^n) \cap C(\overline B_1)$ solve the equation \[Iu = 0 \ \ \text{in } B_1.\]
A purely integro-differential version of this theorem<ref name="CS3"/> says that solutions of an integro-differential [[Bellman equation]] of the form
Then $u \in C^{1,\alpha}(B_{1/2})$ and the following estimate holds
\[ \sup_\beta \int_{\R^n} (u(x+y) - u(x)) K_\beta (y) \mathrm d y = 0 \qquad \text{in } B_1\]
\[ ||u||_{C^{1,\alpha}(B_{1/2})} \leq C ||u||_{L^\infty}. \]
are $C^{s+\alpha}(B_{1/2})$ (which implies that they are classical) if the kernels satisfy the following assumptions
 
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]].
 
== 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 <ref name="kriventsov2013c" /> and <ref name="serra2014regularity" />.
 
==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 <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" />.
 
=== Isaacs equation with variable coefficients but close to constant ===
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 $x$.
\[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)(a_0(y) + a_{\alpha \beta}(x,y))}{|y|^{n+s}} \mathrm d y =0\]
such that we have for $\eta$ small enough and any $\alpha$, $\beta$,
\begin{align*}
\begin{align*}
|a_{\alpha \beta}(x,y)| &< \eta \qquad \text{ for every } \alpha, \beta \\
\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{uniform ellipticity of order $s$} \\
\lambda &\leq a_0(y) \leq \Lambda \\
D^2 K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s+2}} && \text{Decay of the tails of $K$ in $C^2$} \\
|\nabla a_0(y)| &\leq C |y|^{-1}
K(y) &= K(-y) && \text{symmetry}
\end{align*}
\end{align*}
(note that this $C^{1,\alpha}$ estimate is nontrivial in the linear case as well)


=== Isaacs equation with continuous coefficients ===
The $C^{s+\alpha}$ estimate '''does not blow up as $s \to 2$'''. Thus, the result is a true generalization of Evans-Krylov theorem.
If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates <ref name="CS2"/>.
\[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)a_{\alpha \beta}(x,y)}{|y|^{n+s}} \mathrm d y = 0\]
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*}


Note that the result is relevant only if $s>1$, otherwise it is a weaker result compared to the [[differentiability estimates|$C^{1,\alpha}$ estimates]].


The hypothesis above are most probably not optimal. Most likely a similar estimate would hold for kernels with $C^\alpha$ dependence respect $x$. Unlike the [[differentiability estimates|$C^{1,\alpha}$ estimates]], no variation of this result is known.


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
<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="CS3">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=The Evans-Krylov theorem for non local fully non linear equations | year=to appear | journal=[[Annals of Mathematics]] | issn=0003-486X}}</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="E">{{Citation | last1=Evans | first1=Lawrence C. | title=Classical solutions of fully nonlinear, convex, second-order elliptic equations | url=http://dx.doi.org/10.1002/cpa.3160350303 | doi=10.1002/cpa.3160350303 | year=1982 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=35 | issue=3 | pages=333–363}}</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="K">{{Citation | last1=Krylov | first1=N. V. | title=Boundedly inhomogeneous elliptic and parabolic equations | year=1982 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=46 | issue=3 | pages=487–523}}</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 19:40, 29 May 2011

The classical Evans-Krylov theorem [1] [2] says that convex or concave fully nonlinear elliptic equations have $C^{2,\alpha}$ (therefore classical) solutions. This type of equations can be written as a Hamilton-Jacobi-Bellman equation. \[ \sup_\beta a_{ij}^\beta \partial_{ij} u = f \] for a family of uniformly elliptic coefficients $a_{ij}^\alpha$.

A purely integro-differential version of this theorem[3] says that solutions of an integro-differential Bellman equation of the form \[ \sup_\beta \int_{\R^n} (u(x+y) - u(x)) K_\beta (y) \mathrm d y = 0 \qquad \text{in } B_1\] are $C^{s+\alpha}(B_{1/2})$ (which implies that they are classical) if the kernels satisfy the following assumptions \begin{align*} \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{uniform ellipticity of order $s$} \\ D^2 K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s+2}} && \text{Decay of the tails of $K$ in $C^2$} \\ K(y) &= K(-y) && \text{symmetry} \end{align*}

The $C^{s+\alpha}$ estimate does not blow up as $s \to 2$. Thus, the result is a true generalization of Evans-Krylov theorem.

Note that the result is relevant only if $s>1$, otherwise it is a weaker result compared to the $C^{1,\alpha}$ estimates.

The hypothesis above are most probably not optimal. Most likely a similar estimate would hold for kernels with $C^\alpha$ dependence respect $x$. Unlike the $C^{1,\alpha}$ estimates, no variation of this result is known.

References

  1. ↑ Evans, Lawrence C. (1982), "Classical solutions of fully nonlinear, convex, second-order elliptic equations", Communications on Pure and Applied Mathematics 35 (3): 333–363, doi:10.1002/cpa.3160350303, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.3160350303 
  2. ↑ Krylov, N. V. (1982), "Boundedly inhomogeneous elliptic and parabolic equations", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 46 (3): 487–523, ISSN 0373-2436 
  3. ↑ Caffarelli, Luis; Silvestre, Luis (to appear), "The Evans-Krylov theorem for non local fully non linear equations", Annals of Mathematics, ISSN 0003-486X 
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