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Asymptotic average solutions for second order hypoelliptic PDEs

Alessia E. Kogoj

TL;DR

The paper develops a framework for asymptotic average solutions of second-order hypoelliptic operators with nonnegative characteristic form by introducing ${\mathcal{L}}$-balls via a fundamental solution $\Gamma$ and a Poisson–Jensen representation. It proves that the Newtonian-type potentials $u_f(x)=\int \Gamma(x,y) f(y)\,dy$ solve the Poisson problem in the asymptotic sense, via a Pizzetti-type theorem that links local averages to the forcing term $f$; moreover, for compactly supported data, these asymptotic-average solutions align with weak (distributional) solutions, underscoring a robust bridge between mean-value formulas and distribution theory in a broad hypoelliptic setting. The work also establishes structural properties of ${\mathcal{L}}$-balls and provides a representation framework that unifies classical, weak, and asymptotic notions of solvability. By extending Pizzetti-type characterizations to operators including those on Carnot groups and Kolmogorov-type ultraparabolic operators, the results offer a solid solvability paradigm for Poisson problems in a wide range of geometric and analytical contexts.

Abstract

Following an analogous procedure with that used in \cite{kogoj_lanconelli_pizzetti}, in turn inspired by a 1909 paper by Pizzetti \cite{pizzetti}, we introduce the notion of {\it asymptotic average solutions} for hypoelliptic linear partial differential operators with non-negative characteristic form. This notion makes every Poisson equation $\elle u(x)=-f(x)$ with continuous data $f$ pointwise solvable.

Asymptotic average solutions for second order hypoelliptic PDEs

TL;DR

The paper develops a framework for asymptotic average solutions of second-order hypoelliptic operators with nonnegative characteristic form by introducing -balls via a fundamental solution and a Poisson–Jensen representation. It proves that the Newtonian-type potentials solve the Poisson problem in the asymptotic sense, via a Pizzetti-type theorem that links local averages to the forcing term ; moreover, for compactly supported data, these asymptotic-average solutions align with weak (distributional) solutions, underscoring a robust bridge between mean-value formulas and distribution theory in a broad hypoelliptic setting. The work also establishes structural properties of -balls and provides a representation framework that unifies classical, weak, and asymptotic notions of solvability. By extending Pizzetti-type characterizations to operators including those on Carnot groups and Kolmogorov-type ultraparabolic operators, the results offer a solid solvability paradigm for Poisson problems in a wide range of geometric and analytical contexts.

Abstract

Following an analogous procedure with that used in \cite{kogoj_lanconelli_pizzetti}, in turn inspired by a 1909 paper by Pizzetti \cite{pizzetti}, we introduce the notion of {\it asymptotic average solutions} for hypoelliptic linear partial differential operators with non-negative characteristic form. This notion makes every Poisson equation with continuous data pointwise solvable.
Paper Structure (5 sections, 5 theorems, 61 equations)

This paper contains 5 sections, 5 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. The ${\mathcal{L}}$-balls: some properties
  3. A representation formula
  4. A Pizzetti-type theorem
  5. Classical, weak and asymptotic average solutions

Key Result

Theorem 1.2

For every $f:{{\mathbb {R}}^{ {n} }}\rightarrow{\mathbb {R}}$ compactly supported continuous function, the function is an asymptotic average solution to

Theorems & Definitions (8)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 4.1
  • proof
  • Theorem 5.1
  • Remark 5.2