Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

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

“…In [28] we proved the boundary Harnack inequality and Hölder continuity for ratios of p-harmonic functions vanishing on a portion of certain Reifenberg flat and Ahlfors regular NTA-domains and we gave applications to the p-Martin boundary problem for these domains. Finally, in [29], [30] we generalized the results in [5], [6] concerning general two-phase free boundary problems for the Laplace operator to the p-Laplace operator, 1 < p < ∞. In [29] we also gave an application of our results to the free boundary-inverse type problem studied in [26].…”

“…Finally, in [29], [30] we generalized the results in [5], [6] concerning general two-phase free boundary problems for the Laplace operator to the p-Laplace operator, 1 < p < ∞. In [29] we also gave an application of our results to the free boundary-inverse type problem studied in [26]. Moreover, while the analysis in [26] is closely linked to [24], [25], and [29], this paper is more in the flavour of [28] and [30].…”

Quasi-linear PDEs and low-dimensional sets

“…Also, in [LN4] these questions were resolved for p-harmonic functions vanishing on a portion of certain Reifenberg flat and Ahlfors regular NTA-domains. The results and techniques developed in [LN], [LN1], [LN2] and [LN4] concerning p-harmonic functions have also been used and further developed in [LN5], [LN6], in the context of free boundary regularity in general two-phase free boundary problems for the p-Laplace operator, and in [LN7] in the context of regularity and free boundary regularity, below the continuous threshold, for the p-Laplace equation in Reifenberg flat and Ahlfors regular NTA-domains. In addition, in [LLuN] boundary Harnack inequalities and the Martin boundary problem was studied for more general operators of p-Laplace type with variable coefficients in Reifenberg flat domains.…”

On an inverse type problem for the heat equation in parabolic regular graph domains

“…[LN4] extends, to Reifenberg flat and Ahlfors regular domains, the results in [LN1] concerning the regularity and free boundary regularity, below the continuous threshold, for the p-Laplace equation, 1 < p < ∞, in Lipschitz and C 1 -domains. Furthermore, in [LN2,LN3] the results in [C,C1] concerning general two-phase free boundary problems for the Laplace operator are generalized to the p-Laplace operator, 1 < p < ∞, and, in particular, in [LN2] the results are also applied to the free boundaryinverse type problems below the continuous threshold in Lipschitz domains studied in [LN1]. In particular, in [LN2] the problem referred to as Problem 2 above is posed and solved in the context of the p-Laplace equation, 1 < p < ∞.…”