On the Phragmén–Lindelöf principle for second-order elliptic equations

Abstract

This paper is concerned with the maximum principle for subsolutions of second-order elliptic equations in non-divergence form in unbounded domains. Eventually the zero-order term can change sign and the involved functions can be unbounded at infinity with an admissible growth depending on the geometric properties of the domain. Following Gilbarg and Hopf, we also show a Phragmén-Lindelöf principle in angular sectors and give an example of an interesting field of application to nonlinear equations, deriving comparison principles for quasi-linear operators.

A qualitative Phragmèn–Lindelöf theorem for fully nonlinear elliptic equations

“…Analogously, for η > 0 the above extends the "variant" of ABP estimate of [18] for cones and for the much more general class of G * domains (see again Example 2).…”

“…The techniques employed to establish the result, which rely in an essential way on the Caffarelli-Cabré [6] boundary weak Harnack inequality and the local maximum principle for viscosity solutions, are partially mutuated from the papers [8,18]. In both papers the focus is on general unbounded domains, in [18] the target being the Phragmèn-Lindelöf principle for linear elliptic equations while [8] deals with the ABP Maximum Principle for fully nonlinear equations satisfying conditions (1.2) and (1.3). The general geometric condition G * is precisely defined in Section 3, with examples and counterexamples, and generalized by an iteration process, starting from the globalization of a local geometric condition.…”

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

“…Horgan [27], Quintanilla [52], has been frequently studied during the last century. To mention few papers, Ahlfors [4] extended results from [51] to the upper half space of R n , Gilbarg [21] and Serrin [53] considered more general elliptic equations of second order and Vitolo [54] considered the problem in angular sectors. Kurta [38] and Jin-Lancaster [29,30,31] considered quasilinear elliptic equations and non-hyperbolic equations while Capuzzo-Vitolo [18] and Armstrong-Sirakov-Smart [6] considered fully nonlinear equations.…”

Directional ellipticity on special domains: Weak Maximum and Phragmén–Lindelöf principles

“…, k, provided that sup Ω c(x) is small enough. This fact is well-known for the case of uniformly elliptic linear operators of the form Tr(A(x)D 2 u) + c(x)u if c + is small enough with respect to the ellipticity constant of the matrix A; see for instance [6], [23].…”