Differentiability estimates and Nonlocal Evans-Krylov theorem: Difference between pages
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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. | |||
\[ \sup_\beta a_{ij}^\beta \partial_{ij} u = f \] | |||
for a family of uniformly elliptic coefficients $a_{ij}^\alpha$. | |||
A purely integro-differential version of this theorem<ref name="CS3"/> 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*} | \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*} | \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 [[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=" | <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=" | <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="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=" | |||
}} | }} |
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
- ↑ 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
- ↑ Krylov, N. V. (1982), "Boundedly inhomogeneous elliptic and parabolic equations", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 46 (3): 487–523, ISSN 0373-2436
- ↑ Caffarelli, Luis; Silvestre, Luis (to appear), "The Evans-Krylov theorem for non local fully non linear equations", Annals of Mathematics, ISSN 0003-486X