The maximum principle for viscosity solutions of elliptic differential functional equations (Q2842006)

Let \(\Omega\) be an open subset of \(\mathbb{R}^n\). Define \(I_\Omega: C_0(\Omega)\rightarrow C(\mathbb{R}^n)\), \(R:C(\mathbb{R}^n)\rightarrow C(\mathbb{R}^n)\), \(P_\Omega: C(\mathbb{R}^n)\rightarrow C(\Omega)\) and \(R_\Omega: C_0(\Omega)\rightarrow C(\Omega)\) by NEWLINE\[NEWLINE (I_\Omega u)(x)=\begin{cases} u(x) &\text{for } x\in\Omega, \\ 0 &\text{for } x\not\in \Omega, \end{cases}NEWLINE\]NEWLINE NEWLINE\[NEWLINERu(x)=u(\alpha(x)),\qquad P_\Omega u=u|_\Omega,\qquad R_\Omega=P_\Omega RI_\Omega. NEWLINE\]NEWLINE The author discuss the Maximum Principle for viscosity solutions of the following functional differential elliptic problem: NEWLINE\[NEWLINE \begin{cases} F(x,u(x),R_\Omega u(x),Du(x),D^2u(x))=0, \qquad &\text{in}\,\,\Omega, \\ u=0,\qquad &\text{ on}\,\, \mathbb{R}^n\backslash\Omega, \end{cases} NEWLINE\]NEWLINE where \(F:\Omega\times \mathbb{R}\times C(\Omega)\times \mathbb{R}^n\times S(n)\rightarrow \mathbb{R}\) is a given function. Here \(S(n)\) is the set of symmetric \(n\times n\) matrices.