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

TL;DR

This paper demonstrates a method to decompose finite Borel measures in Euclidean space into parts controlled by Hausdorff content, and applies this to establish solutions for a nonlinear Dirichlet problem.

Contribution

It generalizes Delaware's theorem by decomposing measures bounded by Hausdorff measure into parts bounded by Hausdorff content, and applies this to nonlinear PDEs.

Findings

Decomposition of measures into parts bounded by Hausdorff content.

Existence of solutions for a nonlinear Dirichlet problem.

Generalization of measure decomposition theorems.

Abstract

We prove that every finite Borel measure $\mu$ in $\mathbb{R}^N$ that is bounded from above by the Hausdorff measure $\mathcal{H}^s$ can be split in countable many parts $\mu\lfloor_{E_k}$ 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.