The generalized H\"older and Morrey-Campanato Dirichlet problems for elliptic systems in the upper-half space

TL;DR

This paper establishes well-posedness for elliptic systems in the upper-half space with boundary data in generalized H"older and Morrey-Campanato spaces, identifying conditions under which these spaces coincide and solutions are unique.

Contribution

It extends well-posedness results to generalized function spaces for elliptic systems and characterizes when these spaces are equivalent, ensuring unique solutions.

Findings

Well-posedness results for elliptic systems with generalized boundary data.

Identification of conditions for equivalence of generalized H"older and Morrey-Campanato spaces.

Characterization of growth functions ensuring unique solutions in both spaces.

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.