The generalized H\"older 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 H\"older spaces $\mathscr{C}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and in generalized Morrey-Campanato spaces $\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ under certain assumptions on the growth function $\omega$. We also identify a class of growth functions $\omega$ for which $\mathscr{C}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)=\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ 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.