Work

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

Year: 2026

Type: article

Abstract: 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.

Source: Mathematics

DOI: 10.3390/math14071130

Author: Haesung Lee

Institution: Kumoh National Institute of Technology

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: Ministry of Education, India