De Giorgi's regularity theory for elliptic, parabolic and kinetic equations

TL;DR

This book offers a comprehensive overview of De Giorgi's regularity theory, covering elliptic, parabolic, and kinetic equations, highlighting its historical development and recent extensions to fractional and kinetic diffusions.

Contribution

It unifies classical and modern regularity results for elliptic, parabolic, and kinetic equations, including new developments in fractional and kinetic diffusions.

Findings

Extension of De Giorgi's techniques to fractional Laplacian equations

Development of regularity theory for kinetic equations with local and integral diffusions

Historical synthesis of regularity theory from 1956 to recent advances

Abstract

This book presents a comprehensive regularity theory for solutions of elliptic, parabolic, and kinetic equations. The foundation of this theory was laid by E. De Giorgi's groundbreaking resolution of Hilbert's nineteenth problem in 1956. The innovative tools he developed to tackle this problem proved to be remarkably versatile. In 1957, just one year later, J. Nash independently developed analogous techniques for parabolic equations, concurrently with De Giorgi's research. By the year 2000, these techniques had been extended to address elliptic and parabolic equations featuring integral diffusion, such as the fractional Laplacian. More recently, the theory has evolved to encompass kinetic equations, accommodating both local and integral diffusions. This book aims to present these results in a unified and coherent manner, beginning with the classical elliptic framework and progressing through to the most recent advancements in kinetic equations.