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

Abstract

We prove a Liouville-type theorem that is one-dimensional symmetry and classification results for non-negative L q -viscosity solutions of the equation, where M ± λ,Λ are the Pucci's operators with parameters λ, Λ ∈ R+ 0 < λ ≤ Λ and p > 1. The results are an extension of the results by Porreta and Verón in [29] for the case p ∈ (1, 2] and by o Filippucci, Pucci and Souplet in [23] for the case p > 2, both for the Laplacian case (i.e. λ = Λ = 1). As an application in the case p > 2, we prove a sharp Bernstein estimation for L q -viscosity solutions of the fully nonlinear equationx ∈ Ω, with boundary condition u = 0 on ∂Ω, where Ω ⊂ R n .

On the Liouville property for fully nonlinear equations with superlinear first-order terms

“…We emphasize that even in the quadratic case γ = 2, the result cannot be deduced through the Hopf-Cole transformation, since such change of variable leads to an inequality (as in Theorem 4.1), rather than to an equality. Still, we mention that some results in these direction recently appeared in [34].…”