The generalized Hölder and Morrey-Campanato Dirichlet problems for elliptic systems in the upper-half space

Abstract

We prove well-posedness results for the Dirichlet problem in ${\mathbb{R}}_{+}^n$ for homogeneous, second order, constant complex coefficient elliptic systems with boundary data in generalized Hölder spaces ${\mathcal{C}}^{\omega }({\mathbb{R}}^{n-1},{\mathbb{C}}^M)$ and in generalized Morrey-Campanato spaces ${\mathcal{E}}^{\omega ,p}({\mathbb{R}}^{n-1},{\mathbb{C}}^M)$ under certain assumptions on the growth function $\omega$. We also identify a class of growth functions $\omega$ for which ${\mathcal{C}}^{\omega }({\mathbb{R}}^{n-1},{\mathbb{C}}^M)={\mathcal{E}}^{\omega ,p}({\mathbb{R}}^{n-1},{\mathbb{C}}^M)$ and for which the aforementioned well-posedness results are equivalent, in the sense that they have the same unique solution, satisfying natural regularity properties and estimates.

Minor corrections