Conference Paper

Weighted Hardy Inequalities for Singular Elliptic Problems: Theory, Applications, and Computational Perspectives

Abstract Weighted Hardy inequalities constitute a cornerstone of modern functional analysis and the theory of partial differential equations (PDEs). By extending the classical Hardy inequality through the incorporation of weight functions, these inequalities provide a robust framework for handling singularities inherent in elliptic boundary value problems. Their applications are vast, encompassing Sobolev space theory, variational methods, mathematical physics, and numerical approximation. This paper reviews the theoretical foundations of weighted Hardy inequalities, emphasizing their critical role in establishing coercivity, stability, and a priori estimates for singular elliptic problems. Furthermore, we examine their computational implications for finite element methods (FEM) and adaptive numerical schemes. Finally, we highlight future research directions, including fractional Hardy inequalities and data-driven numerical methods. Keywords: Hardy inequality; weighted Hardy inequality; Sobolev spaces; elliptic PDEs; finite element method; numerical analysis.