The boundary Harnack inequality for solutions to equations of Aronsson type in the plane

Abstract

Abstract. In this paper we prove a boundary Harnack inequality for positive functions which vanish continuously on a portion of the boundary of a bounded domain Ω ⊂ R 2 and which are solutions to a general equation of p-Laplace type, 1 < p < ∞. We also establish the same type of result for solutions to the Aronsson type equationConcerning Ω we only assume that ∂Ω is a quasicircle. In particular, our results generalize the boundary Harnack inequalities in [BL] and [LN2] to operators with variable coefficients.

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

“…In the constant exponent setting, similar results were obtained by Eremenko and Lewis [26], Kilpeläinen and Zhong [43] and Bennewitz and Lewis [17]. For p = const, p-harmonic measures were employed to prove boundary Harnack inequalities, see, e.g., [17], Lewis and Nyström [46] and Lundström and Nyström [53]. The p-harmonic measure, defined as in the aforementioned papers, as well as boundary Harnack inequalities, have played a significant role when studying free boundary problems, see, e.g., Lewis and Nyström [48].…”

“…Then, we follow the steps of Lemma 2.2 in Lundström and Nyström [53] for an Aharmonic operator A(x, ∇u) := |∇u(x)| p(x)−2 ∇u(x) to obtain that u is a subsolution in B(w, r u < 1 andr < 1. Now, let η ∈ C ∞ 0 (B(w, 3r )) with 0 ≤ η ≤ 1, η ≡ 1 on B(w, 2r ) and |∇η| ≤ C r for some C > 1.…”

“…Moreover, in [52], the second author proved a boundary Harnack inequality for pharmonic functions with n < p ≤ ∞ vanishing on a m-dimensional hyperplane in R n for 0 ≤ m ≤ n − 1. We also refer to Bhattacharya [18] and Lundström and Nyström [53] for the case p = ∞, where the latter investigated A-harmonic and Aronsson-type equations in planar uniform domains. Concerning the applications of boundary Harnack inequalities, we mention free boundary problems and studies of the Martin boundary.…”

“…Indeed, by taking p = p + = p − in Theorem 6.3, we retrieve the corresponding estimates for p = const, cf. Lemma 2.7 in [53]. …”

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

“…For infinity-harmonic functions, see e.g. Bhattacharya [13], Lundström-Nyström [46] and for solutions to the variable exponent p-Laplace equation in smooth domains, see Adamowicz-Lundström [2]. Only few papers considered local estimates of positive p-harmonic functions vanishing near boundaries having dimension less than n − 1.…”