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| 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. | | The Evans-Krylov theorem says that if $u$ is a solution to a uniformly elliptic, fully nonlinear, convex or concave, equation |
| \[ \sup_\beta a_{ij}^\beta \partial_{ij} u = f \] | | \[ F(D^2 u) = 0 \text{ in } B_1,\] |
| for a family of uniformly elliptic coefficients $a_{ij}^\alpha$.
| | 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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| A purely integro-differential version of this theorem<ref name="CS3"/> says that solutions of an integro-differential [[Bellman equation]] of the form
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| \[ \sup_\beta \int_{\R^n} (u(x+y) - u(x)) K_\beta (y) \mathrm d y = 0 \qquad \text{in } B_1\]
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| are $C^{s+\alpha}(B_{1/2})$ (which implies that they are classical) if the kernels satisfy the following assumptions
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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{uniform ellipticity of order $s$} \\
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| D^2 K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s+2}} && \text{Decay of the tails of $K$ in $C^2$} \\
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| K(y) &= K(-y) && \text{symmetry}
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| \end{align*}
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| The $C^{s+\alpha}$ estimate '''does not blow up as $s \to 2$'''. Thus, the result is a true generalization of Evans-Krylov theorem.
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| 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]].
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| 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.
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| == References ==
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| {{reflist|refs=
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| <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>
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| <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>
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| <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>
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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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