Unified Counterexamples to Endpoint Regularity for Linear Elliptic Equations with Singular Coefficients

This paper presents unified counterexamples for which standard elliptic regularity results break down for linear elliptic equations with highly singular coefficients in dimensions d≥3. First, we establish well-posedness for the case where the drift vector field has merely L2-integrability but can be expressed as the gradient of a bounded potential function. Subsequently, we investigate the critical endpoint cases of known regularity results where coefficients or data satisfy borderline integrability conditions. By using a single, explicit function, ρ(x)=ln(2+1∥x∥), we present counterexamples to the regularity of solutions for divergence form equations and stationary Fokker–Planck equations.