Boundary behavior and the Martin boundary problem forpharmonic functions in Lipschitz domains

Abstract

In a previous article, we proved a boundary Harnack inequality for the ratio of two positive p harmonic functions, vanishing on a portion of the boundary of a Lipschitz domain. In the current paper we continue our study by showing that this ratio is Hölder continuous up to the boundary. We also consider the Martin boundary of certain domains and the corresponding question of when a minimal positive p harmonic function (with respect to a given boundary point w) is unique up to constant multiples. In particular we show that the Martin boundary can be identified with the topological boundary in domains that are convex or C 1 . Minimal positive p harmonic functions relative to a boundary point w in a Lipschitz domain are shown to be unique, up to constant multiples, provided the boundary is sufficiently flat at w.

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

“…The extension of these results to the more general setting of p-harmonic operators turned out to be difficult, largely due to the nonlinearity of p-harmonic functions for p = 2. However, recently, there has been a substantial progress in studies of boundary Harnack inequalities for nonlinear Laplacians: Aikawa et al [7] studied the case of p-harmonic functions in C 1,1 -domains, while in the same time, Lewis and Nyström [45,47,48] began to develop a theory applicable in more general geometries such as Lipschitz and Reifenberg-flat domains. Lewis-Nyström results have been partially generalized to operators with variable coefficients, Avelin et al [12], Avelin and Nyström [13], and to p-harmonic functions in the Heisenberg group, Nyström [55].…”

Boundary estimates for solutions to linear degenerate parabolic equations

“…In a subsequent paper we intend to consider the case of time-dependent weights as part of an ambition to understand the boundary behaviour of non-negative solutions to non-linear parabolic equations of p-parabolic type somehow along the lines of the elliptic theory developed in [34], [35], [38], [36]. However, already the case of time-independent weights λ(x, t) = λ(x) ∈ A 1+2/n (R n ) forces us to revisit essentially all the relevant arguments used in the corresponding context of uniformly parabolic equations.…”

Optimal doubling, Reifenberg flatness and operators of -Laplace type

“…In section 2 we state a number of estimates for nonnegative A-harmonic functions in a bounded domain Ω ⊂ R n , assuming that Ω is a NTA-domain and, at instances, also assuming that Ω is (δ, r 0 )-Reifenberg flat. All of these results can be found in [LLuN] where the second and third author, together with John Lewis, began the generalization of the results in [LN,LN1,LN2] to a class of p-Laplace type operators allowing, in particular, for operators with variable coefficients. In particular, in [LLuN] new results concerning boundary Harnack inequalities and the Martin boundary problem, in Reifenberg flat domains, for operators of p-Laplace type of the form ∇ · A(x, ∇u) = 0 were established.…”