Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem

A note on one-dimensional symmetry for Hamilton-Jacobi equations with extremal Pucci operators and application to Bernstein type estimate

“…As an application to their Liouville result, Filipucci, Pucci and Souplet proved an improvement of the Bernstein estimation in [28], now with the boundary value, for the inhomogeneous Dirichlet problem (specifically Theorem 1.6 in [23]). The result establishes a more precise constant in the Bernstein estimate, which is also valid for equation of the form (7) −M ± λ,Λ (D 2 u) = |Du| p + f (x), in Ω, u = 0, on ∂Ω, this by using our Theorem 1.2 and similar ideas as in [23].…”

“…From these theorems, classification results can be established by solving the ODE, see the beginning of section 3. Notice that the only work we know of one-dimensional symmetry or rigidity results in the half space for fully non-linear operator is in [7], for explosive boundary condition. Other symmetry type results in bounded domain can be found in [20].…”