论文标题
Borel测量$μ\ le \ Mathcal {h}^{s} $的分解
A decomposition for Borel measures $μ\le \mathcal{H}^{s}$
论文作者
论文摘要
我们证明，每个有限的鲍勒级尺寸$μ$ in $ \ mathbb {r}^n $由hausdorff mouse $ \ mathcal {h}^s $从上方限制，可以在可数的许多零件中划分$μ\ lfloor_ $ \ mathcal {h} _ \ infty^s $。由于特拉华州的R. Delaware，这一结果概括了一个定理，该定理说，任何具有有限豪斯多夫措施的鲍雷尔套件都可以分解为可计数的直接组合。我们将这种分解应用以显示涉及指数非线性的Dirichlet问题的解决方案的存在。
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.
