Quadratic growth of solutions of fully nonlinear second order equations in \({\mathbb{R}{}}^ n\) (Q1174372)

The authors consider problems of the form \[ u_ t+F(D^ 2u)=0,\quad u(0,x)=\psi(x), \leqno(1) \] and the stationary case (2) \(u+F(D^ 2u)=f(x)\). Here \(F\) is a nonincreasing function of an \(N\times N\) matrix (with the usual ordering in \(\mathbb{R}\)) and \(x\in\mathbb{R}^ N\). It is further assumed that an ellipticity condition of the form \(F(B)\leq F(A)\) if \(A\leq B\) holds for \(A\), \(B\) real symmetric \(N\times N\) matrices. They prove existence and uniqueness of viscosity solutions in (1) and (2). In problem (1) they also study an associated inverse problem by considering an appropriate semigroup.