Complete characterization of W^{2,p} regularity on non-smooth domains for the Poisson Dirichlet problem

Determine a complete characterization of non-smooth bounded domains Ω ⊂ ℝ^N (e.g., Lipschitz or polytopic domains) and data conditions under which the weak solution u ∈ H^1_0(Ω) to the Poisson equation −Δu = f in Ω with u = 0 on ∂Ω and f ∈ L^p(Ω) for p > N/2 enjoys global W^{2,p}(Ω) regularity. The goal is to identify precise geometric and analytic criteria beyond the C^{1,1} case that ensure W^{2,p}(Ω) regularity of u.

Background

The paper’s framework requires pointwise evaluation and uniform control of couplings that are guaranteed when u ∈ W^{1,q}(Ω) with q > N. For C^{1,1} boundaries and f ∈ L^p(Ω), elliptic regularity yields u ∈ W^{2,p}(Ω), which via Sobolev embedding implies the needed W^{1,q} regularity. However, for non-smooth domains, W^{2,p} regularity may fail in general.

Although there is substantial literature on regularity in non-smooth settings (e.g., corners and edges), the authors note that a complete geometric and analytic characterization of when W^{2,p} regularity still holds is not yet known. This gap is significant for the proposed Green-representable framework because W^{2,p} regularity immediately ensures the uniform integrability conditions needed for global Green-representability.