Boundary Harnack Principle for p-harmonic Functions in Smooth Euclidean Domains

Abstract

We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions in Euclidean C 1,1 -domains and obtain estimates for the decay rates of positive p-harmonic functions vanishing on a segment of the boundary in terms of the distance to the boundary. We use these estimates to study the behavior of conformal Martin kernel functions and positive p-superharmonic functions near the boundary of the domain.

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

“…parts (i) and (ii) of Lemma 5.1. As a corollary, we also obtain a decay estimate for supersolutions (a counterpart of Proposition 6.1 in Aikawa et al [7]). For w ∈ ∂ , we denote by A r (w) a point satisfying d(A r (w), ∂ ) = r and |A r (w)−w| = r .…”

“…4, the comparison principle and Harnack's inequality. Our approach extends arguments from Aikawa et al [7] to the case of variable exponents. We point out that the constants in (1.2), and thus also in the boundary Harnack inequality, depend on u and v. Such a dependence is expected for variable exponent PDEs and difficult to avoid, as, e.g., parameters in the Harnack inequality Lemma 3.1 and the barrier functions depend on solutions as well.…”

“…See Aikawa et al [7,Lemma 2.2] for a proof. We also note that if ⊂ R n satisfies the ball condition then is a NTA domain and hence also a uniform domain.…”

“…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].…”

“…In the analysis of PDEs, barrier functions appear, for example, in comparison arguments and in establishing growth conditions for functions, see, e.g., Aikawa et al [7], Lundström [52], Lundström and Vasilis [54] for the setting of p-harmonic functions. Furthermore, barriers can be applied in the solvability of the Dirichlet problem, especially in studies of regular points, see, e.g., Chapter 6 in Heinonen et al [38] and Chapter 11 in Björn and Björn [19].…”