Isaacs equation and Nonlocal Evans-Krylov theorem: Difference between pages

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The Isaacs equation is the equality
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.
\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
\[ \sup_\beta a_{ij}^\beta \partial_{ij} u = f \]
where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
for a family of uniformly elliptic coefficients $a_{ij}^\alpha$.


The equation appears naturally in zero sum stochastic games with [[Levy processes]].
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*}
\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 equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$.
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 any second order fully nonlinear uniformly elliptic PDE $F(D^2 u)=0$ can be written as an Isaacs equation by the following two steps:
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]].  
# $F(X)$ is Lipschitz with constant $\Lambda$, so it is the infimum of all cones $C_{X_0}(x) = F(X_0) + \Lambda|X-X_0|$.
# Each cone $C(X)$ is the supremum of all linear functions of the form $L(X) = F(X_0) + \mathrm{tr} \, A \cdot (X-X_0)$ for $||A||\leq \Lambda$.


A more general second order fully nonlinear uniformly elliptic PDE $F(D^2 u, Du, u, x)=0$ can also be written as an Isaacs equation if it is linear with respect to all parameters.
The hypothesis above are most probably not optimal. But unlike the [[differentiability estimates|C^{1,\alpha} estimates]], no extension of this result has been done.  


[[Category:Fully nonlinear equations]]
== References ==
{{reflist|refs=
<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="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>
}}

Revision as of 19:37, 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. But unlike the C^{1,\alpha} estimates, no extension of this result has been done.

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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