A decomposition for Borel measures $μ\le \mathcal{H}^{s}$

TL;DR

This work extends Delaware's notion of $s$-straight sets to finite Borel measures by showing that any μ with μ ≤ $\\mathcal{H}^s$ can be decomposed into countably many disjoint pieces μ|_{E_k} that satisfy μ|_{E_k} ≤ $\\mathcal{H}_\\infty^s$. It proves that every nonzero μ ≤ $\\mathcal{H}^s$ contains a positive-measure $s$-straight part, enabling a measure-wise decomposition and, in turn, applications to nonlinear PDEs. Using the decomposition, the authors provide a streamlined existence proof for distributional solutions of the Dirichlet problem $-\\Delta u + (e^{u}-1)=ν$ under ν ≤ $\\mathcal{H}^s$ bounds, including a refined BLOP-type result for the exponential nonlinearity. The approach leverages density estimates, an atomless intermediate-value lemma for measures, and Zorn's lemma for non-σ-finite cases, offering a versatile framework with potential impact on measure theory and nonlinear elliptic PDEs.

Abstract

We prove that every finite Borel measure $μ$ in $\mathbb{R}^N$ that is bounded from above by the Hausdorff measure $\mathcal{H}^s$ can be split in countable many parts $μ\lfloor_{E_k}$ that are bounded from above by the Hausdorff content $\mathcal{H}_\infty^s$. Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to show the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.

Theorems & Definitions (27)

• Definition 1.1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Example 2.1
• Proposition 2.1
• proof
• Example 2.2
• Proposition 2.2