Quasi-linear PDEs and low-dimensional sets

Abstract

In this paper we establish new results concerning boundary Harnack inequalities and the Martin boundary problem, for non-negative solutions to equations of p -Laplace type with variable coefficients. The key novelty is that we consider solutions which vanish only on a low-dimensional set \Sigma in \mathbb R^n and this is different compared to the more traditional setting of boundary value problems set in the geometrical situation of a bounded domain in \mathbb R^n having a boundary with (Hausdorff) dimension in the range [n-1,n) . We establish our quantitative and scale-invariant estimates in the context of low-dimensional Reifenberg flat sets.

Note on an eigenvalue problem for an ODE originating from a homogeneous $p$-harmonic function

“…Proof. Existence and uniqueness of u i , i = 1, 2, in K(α), 0 < α ≤ π, follows from boundary Harnack inequalities proved in [15] for Reifenberg flat domains and arguments similar to those in section 4 of [19]. The proof that λ 1 (π) = 1 − (n − 1)/p is essentially the same as the proof we outlined in the p harmonic setting for n−1 < p < n. Indeed Lemmas 2.2, 2.4, 2.6, 2.7 are proved in Proposition 9.7, Lemma 10.9, Lemma 13.7, and display (13.86), respectively, of [1] in the A− harmonic setting when A ∈ M p (δ), 1 < p < n, satisfies (3.18).…”

“…Uniqueness of v with the above properties, can be shown using boundary Harnack inequalities proved by Lewis and Nyström in [16], [19]. Indeed in [16], Theorem 2, the authors proved a boundary Harnack theorem for domains with a Lipschitz boundary which tailored to K(α), 0 < α < π, is stated as follows:…”

“…To prove uniqueness of v in K(π), we use arguments from [19], section 4, to prove (2.7) when p > n − 1 and v 1 , v 2 are positive p− harmonic functions in K(π) with continuous boundary value 0 on ∂K(π). Uniqueness of v implies, as in the argument following (2.8), that v has the form (1.3).…”

“…for some c * = c * (p, n) which gives the upper estimate for λ 1 (α) in Theorem 1.1 when n − 1 < p < n. . The proof of the lower estimate for λ 1 (α) as α→π when n − 1 < p < n is similar,just using Lemmas 2.6, 2.7, and Theorem 1.9 in [19], but somewhat more tedious to prove. This completes our outline of the proof of Theorem 1.1 when n − 1 < p < n. 2…”

“…To outline our efforts in proving Theorem 1, we began by trying to use the DE in (1.4) to solve for λ 1 (π), ψ. From a boundary Harnack inequality in [19] (see Theorem 1.9 and Lemma 5.3), we knew that…”