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

Abstract

We prove estimates of a p-harmonic measure, p ∈ (n − m, ∞], for sets in R n which are close to an m-dimensional hyperplane Λ ⊂ R n , m ∈ [0, n − 1]. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains Ω ⊂ R n \ Λ for p-subharmonic functions. Moreover, we give local and global growth estimates for p-harmonic functions, vanishing on sets in R n , which are close to an m-dimensional hyperplane.

Estimates of $p$-harmonic functions in planar sectors

“…Indeed, we conclude that solutions must vanish at the same rate as |x| k(v,p) as x approaches the apex (Corollary 5.1). Similar growth estimates where proved in the setting of C 1,1 -domains in [8] and for wider classes of equations and other geometric settings in [37,38,39,45,46,5,48]. An immediate consequence of Corollary 5.1 is the boundary Harnack inequality for p-harmonic functions in planar sectors, see (5.5).…”

“…Estimates for the p-harmonic measure of a small spherical cap and of small axially symmetric sets are proved in [20,21], and in [22] estimates for the p-harmonic measure is given of the part of the boundary of an infinite slab outside a cylinder. In [46] estimates of p-harmonic measure, n − m < p ≤ ∞, for sets in R n which are close to an mdimensional hyperplane, 0 ≤ m ≤ n − 1 are proved, and in [42] it is proved that the p-harmonic measure in R n + of a ball of radius 0 < δ ≤ 1 in R n−1 is bounded above and below by a constant times δ α , and explicit estimates for α are given. For more on possible applications of p-harmonic measure see e.g.…”

“…In [40], Phragmén-Lindelöf's theorem for n-subharmonic functions, when the boundary is an m-dimensional hyperplane in R n , 0 ≤ m ≤ n − 1 is proved. This was extended to p-subharmonic functions, n − m < p ≤ ∞, in [46]. In [16] it is showed that solutions of a generalized p-Laplace equation in the upper halfplane, vanishing on {x n = 0}, equals u(x) = x n (modulo normalization) and the growth of solutions of the minimal surface equation over domains containing a halfplane was considered in [43].…”

Strong maximum principle and boundary estimates for nonhomogeneous elliptic equations

“…To mention previous results related to our boundary estimates for nonlinear PDEs, Aikawa-Kilpeläinen-Shanmugalingam-Zhong [2] proved similar results in the setting of positive pharmonic functions while in the same year Lewis-Nyström [15,16,17] started to develop a theory for proving boundary estimates such as the boundary Harnack inequality for p-Laplace type equations in more general geometries such as Lipschitz and Reifenberg-flat domains, and Lundström [18,19] estimated the growth of positive p-harmonic functions, n−m < p ≤ ∞, vanishing on an m-dimensional hyperplane in R n , 0 ≤ m ≤ n − 1. For nonhomogeneous equations Adamowicz-Lundström [1] proved similar boundary estimates as we do here but for positive p(x)-harmonic functions, and Avelin-Julin [3] proved a sharp boundary Harnack inequality for operators similar to those considered in this paper but without dependence on u.…”

Growth of subsolutions to fully nonlinear equations in halfspaces

“…Lindqvist [28] established Phragmén-Lindelöf's theorem for n-subharmonic functions when the boundary is an m-dimensional hyperplane in R n , 0 ≤ m ≤ n − 1. This was extended to p-subharmonic functions, n − m < p ≤ ∞, in Lundström [31]. We also mention that recently, Braga-Moreira showed that nonnegative solutions to a generalized p-Laplace equation in the upper halfplane, vanishing on {x n = 0}, equals u(x) = x n (modulo normalization) and Lundberg-Weitsman [29] studied the growth of solutions to the minimal surface equation over domains containing a halfplane.…”