Higher order elliptic equations on nonsmooth domains

In 1995, D. Jerison and C. Kenig in [10] considered the inhomogeneous Dirichlet problem Δ ⁢ 𝑢 = 𝑓 in Ω, 𝑢 = 0 on ∂Ω in Lipschitz domains. One of their main results shows that the 𝑊 1 , 𝑝 estimate holds for the sharp range 3 2 − 𝜀 < 𝑝 < 3 + 𝜀 if 𝑑 ≥ 3 and 4 3 − 𝜀 < 𝑝 < 4 + 𝜀 if 𝑑 = 2 . Although the methods employed in [10] yield optimal results, they rely in an essential fashion on the maximum principle and, as such, do not readily adapt to higher order case. By using a new method, the aim of this paper is to establish an extension of their theorem for higher order inhomogeneous elliptic equations on bounded Lipschitz domains.