Estimates for -harmonic functions vanishing on a flat

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).…”

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.…”

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

“…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 for example Lewis-Nyström [48]. The p-harmonic measure was also used to find the optimal Hölder exponent of p-harmonic functions vanishing near the boundary, see Kilpeläinen-Zhong [42] and Lundström [52]. Moreover, a work of Peres-Sheffield [57] provides discussion of connections between p-harmonic measures, defined in a different way though, and tug-of-war games.…”

“…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].…”

“…Lewis-Nyström results have been partially generalized to operators with variable coefficients, Avelin-Lundström-Nyström [12], Avelin-Nyström [13], and to p-harmonic functions in the Heisenberg group, Nyström [55]. Moreover, in [52] the second author proved a boundary Harnack inequality for p-harmonic 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-Nyström [53] for the case p = ∞, where the latter investigated A-harmonic and Aronsson-type equations in planar uniform domains.…”