Regularity and free boundary regularity for the $p$-Laplace operator in Reifenberg flat and Ahlfors regular domains

Quasi-linear PDEs and low-dimensional sets

“…The purpose of this paper is to pursue the lines of thoughts outlined above in one direction by establishing certain refined boundary Harnack estimates for non-negative solutions to operators of p-Laplace type, assuming that the set K is well approximated by m-dimensional hyperplanes in the Hausdorff sense. To further put our work into perspective we recall that in [LN], [LN1], [LN2], see also [LN3], a number of results concerning the boundary behavior of positive p-harmonic functions, 1 < p < ∞, in a bounded Lipschitz domain Ω ⊂ R n were proved. In particular, the boundary Harnack inequality and Hölder continuity for ratios of positive p-harmonic functions, 1 < p < ∞, vanishing on a portion of ∂Ω were established.…”

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].…”

“…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].…”

Boundary estimates for solutions to linear degenerate parabolic equations

“…In particular, firstly in [4], [3], [47], the results are used in the study of the boundary behaviour of non-local operators exemplified by the fractional Laplacian. Secondly, in [34]- [41], a theory concerning the boundary behaviour for solutions to operators of p-Laplace type is developed. Part of the technical toolbox developed in [34]- [41], consists of techniques for establishing boundary Harnack inequalities for p-harmonic functions vanishing on a portion of the boundary of a domain which is 'flat' in the sense that its boundary is well-approximated by hyperplanes.…”

“…Secondly, in [34]- [41], a theory concerning the boundary behaviour for solutions to operators of p-Laplace type is developed. Part of the technical toolbox developed in [34]- [41], consists of techniques for establishing boundary Harnack inequalities for p-harmonic functions vanishing on a portion of the boundary of a domain which is 'flat' in the sense that its boundary is well-approximated by hyperplanes. In this case, at the final stage of the analysis, results are derived in the non-linear case by a reduction to linear degenerate elliptic equations of the form considered by Fabes et al Based on the above it is natural to attempt to develop a parabolic counterpart of the elliptic theory developed by Fabes et al, and in this case the operators of interest are second order parabolic partial differential operators of the form…”