Geometric regularity estimates for elliptic equations

TL;DR

This paper introduces geometric tangential analysis, a new systematic approach for deriving regularity estimates in elliptic and parabolic partial differential equations, enhancing understanding across various mathematical and applied fields.

Contribution

It presents a novel geometric tangential analysis method for regularity estimates, providing a unified framework for elliptic and parabolic PDEs.

Findings

Development of geometric tangential analysis framework

Improved regularity estimates for elliptic equations

Potential applications to broader PDE classes

Abstract

Regularity theory for diffusive operators is among the finest treasures of the modern mathematical sciences. It appears in several different fields, such as, differential geometry, topology, numerical analysis, dynamical systems, mathematical physics, economics, etc. Within the general field of Partial Differential Equations, regularity theory places itself in the very core, by bridging the notion of weak solutions (often found by energy methods or probabilistic interpretations) to the classical concept of solutions. In this article we discuss about a new systematic approach, termed geometric tangential analysis, for addressing regularity estimates for elliptic and parabolic problems.