Volume 1, Issue 2

In this paper we prove 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. Applications are given to the p -Martin boundary problem for these domains.

We prove a lower semicontinuity theorem for a polyconvex functional of integral form, related to maps u : Ω ⊂ ℝ n → ℝ m in W 1, n (Ω;ℝ m ) with n ≥ m ≥ 2, with respect to the weak W 1, p -convergence for p > m – 1, without assuming any coercivity condition.

Suppose that a domain Ω ⊂ ℝ n admits the ( p, β 0 )-Hardy inequality, i.e. that holds for all . Here . We show that then there exists ε > 0 such that Ω admits ( q, β )-Hardy inequalities for all p – ε < q < p + ε and all β 0 – ε < β < β 0 + ε .

We prove that all bounded weak solutions of the fourth order system where the nonlinearity grows critically with the gradient of a solution, i.e., , are regular once an appropriate smallness condition (expressed in terms of Morrey norms and guaranteeing that u is small in BMO) is satisfied. The result holds in every dimension.