(Difference between pages)
imported>Luis |
imported>Nestor |
| Line 1: |
Line 1: |
| The Isaacs equation is the equality
| | Broadly speaking, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that minimize (within a class of given admissible configurations) the following energy functional: |
| \[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \] | |
| where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
| |
|
| |
|
| The equation appears naturally in zero sum stochastic games with [[Levy processes]].
| |
|
| |
|
| The equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$.
| | \[ J_K(E)= \int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy, s\ in (0,1) \] |
| | |
| 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:
| |
| # $F(X)$ 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.
| |
Revision as of 11:55, 31 May 2011
Broadly speaking, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that minimize (within a class of given admissible configurations) the following energy functional:
\[ J_K(E)= \int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy, s\ in (0,1) \]