Narrow Results

Our main result concerns the behavior of bounded harmonic functions on a domain in ${ℝ}^N$ which may be represented as a strict epigraph of a Lipschitz function on ${ℝ}^{N-1}$. Generally speaking, the result says that the Hölder continuity of a harmonic function on such a domain is equivalent to the uniform Hölder continuity along the straight lines determined by the vector $e_N$, where $e_1,e_2,\dots ,e_N$ is the base of standard vectors in ${ℝ}^N$.

More precisely, let $\mathrm{\Psi }$ be a Lipschitz function on ${ℝ}^{N-1}$, and $U$ be a real-valued bounded harmonic function on $E_{\mathrm{\Psi }}=\{(x^{\prime },x_N):x^{\prime }\in {ℝ}^{N-1},x_N>\mathrm{\Psi }(x^{\prime })\}$. We show that for $\alpha \in (0,1)$ the following two conditions on $U$ are equivalent:

(a) There exists a constant $C$ such that

(b) There exists a constant $\tilde{C}$ such that

In this paper, we treat the general strongly elliptic systems with a class of singular potentials on a bounded Lipschitz domain Ω ⊂ ℝ^d, d ≥ 3. We establish the L^p resolvent estimates on Ω for the above systems with vanishing Dirichlet type or Neumann type boundary value condition, where 2d/(d + 2) - ϵ < p < 2d/(d - 2) + ϵ with some positive constant ϵ = ϵ(Ω).