Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem
The authors prove gradient boundary blow up rates for ergodic functions in bounded domains related to fully nonlinear degenerate/singular elliptic operators. Consider the following second order, fully nonlinear infinite boundary value problem \[ \left \{\begin{array}{ll} -F(D^2u)+|\nabla u|^\beta=0\ &\mbox{in}\ \mathbb{R}^N_+;\\ u(x',0)=+\infty \quad & \mbox{on} \ \partial \mathbb{R}^N_+, \end{array} \right.\tag{1} \] where \(\beta>1\), \(F:\mathcal{S}_N\to\mathbb{R}\) is a positively homogeneous uniformly elliptic operator, \(D^2u\) is the hessian matrix of the unknown function \(u\), and \(\nabla u\) is the gradient of \(u\). The first result is a Liouville-type classication theorem. Then the uniqueness, up to constants, of the ergodic functions, is obtained.
scientific article; zbMATH DE number 7356227
scientific article