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

TL;DR

De Giorgi's regularity theory provides a unified, robust framework for proving Hölder continuity and related regularity properties for elliptic, parabolic, and kinetic equations with rough coefficients. The approach hinges on local energy estimates, De Giorgi's classes, the expansion of positivity, and the intermediate value principle, extended from elliptic to parabolic and kinetic settings, including equations with local and integral diffusions. The work connects Hilbert's 19th problem, Nash's parabolic theory, and modern kinetic models (Kolmogorov, Landau, Boltzmann), delivering universal, scale-invariant results such as Hölder regularity and Harnack-type inequalities across disparate models. By transferring regularity from velocity to spatial variables and accommodating nonlocal diffusions, the framework broadens applicability to kinetic Fokker–Planck and Boltzmann-type equations, offering versatile tools for regularity analysis in high-dimensional and nonlocal contexts.

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.

Figures (13)

• Figure 1: Hölder regularity at each point of $\Omega$.
• Figure 2: Zooming in: Two consecutive shrinking cylinders are scaled to $\mathcal{N}_{\frac{1}{2}}$ and $\mathcal{N}_1$.
• Figure 3: Neighborhoods: balls (elliptic equations), straight cylinders (parabolic equations), slanted cylinders (kinetic equations)
• Figure 4: Expansion of positivity. A lower bound on the super-level set on $\mathcal{N}_{\mathrm{pos}}$ implies a point-wise lower bound in $\mathcal{N}_1$. On the left, illustration of the expansion of positivity for elliptic equations. On the right, the parabolic and kinetic cases. For time-dependent equations, positivity is expanded as time increases.
• Figure 5: Geometric setting of the intermediate value principle.

Theorems & Definitions (258)

• Theorem 2.1.1: E. De Giorgi -- DeG56
• Definition 1: Oscillation
• Proposition 4.1.1: Characterization of Hölder continuity
• proof
• Definition 2: The space $H^1 (\Omega)$
• Definition 3: The space $H^1_0(\Omega)$
• Proposition 4.2.1: Density of smooth functions in $H^1(\Omega)$
• Lemma 4.2.2
• proof
• Definition 4: Weak solutions