Explicit subsolutions and a Liouville theorem for fully nonlinear uniformly elliptic inequalities in halfspaces

2011/11/04 by Fabiana Leoni, Leoni, Fabiana
Computer Science · Mathematics · #35B53 #35J60 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.AP #msc:35B53 #msc:35J60

paper · pdf · doi:10.48550/arxiv.1111.1083

18 pages

openalex publication_date 2011/11/04 · arxiv created 2011/12/06 · arxiv updated 2011/12/07 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We prove a Liouville type theorem for arbitrarily growing positive viscosity supersolutions of fully nonlinear uniformly elliptic equations in halfspaces. Precisely, let M^-_{λ, Λ} be the Pucci's inf- operator, defined as the infimum of all linear uniformly elliptic operators with ellipticity constants Λ≥ λ>0. Then, we prove that the inequality M^-_{λ, Λ}(D²u) +u^p ≤ 0 does not have any positive viscosity solution in a halfspace provided that -1≤ p ≤ (Λ/λn+1)/(Λ/λn-1), whereas positive solutions do exist if either p < -1 or p > (Λ/λ(n-1)+2)/(Λ/λ(n-1)). This will be accomplished by constructing explicit subsolutions of the homogeneous equation M^-_{λ, Λ}(D²u)=0 and by proving a nonlinear version in a halfspace of the classical Hadamard three-circles Theorem for entire superharmonic functions.

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