The $\mathrm{CMO}$-Dirichlet problem for elliptic systems in the upper half-space

TL;DR

This paper establishes well-posedness of the Dirichlet problem for elliptic systems in the upper half-space with boundary data in CMO, using Carleson measure conditions and Fatou-type theorems.

Contribution

It introduces a new approach linking Carleson measure conditions with boundary trace characterizations for elliptic systems in the upper half-space.

Findings

Proves well-posedness of Dirichlet problem with CMO boundary data.

Provides a Poisson integral representation for solutions.

Characterizes CMO and XMO spaces via traces of elliptic system solutions.

Abstract

We prove that for any second-order, homogeneous, $N \times N$ elliptic system $L$ with constant complex coefficients in $\mathbb{R}^n$, the Dirichlet problem in $\mathbb{R}^n_+$ with boundary data in $\mathrm{CMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ is well-posed under the assumption that $d\mu(x', t) := |\nabla u(x)|^2\, t \, dx' dt$ is a strong vanishing Carleson measure in $\mathbb{R}^n_+$ in some sense. This solves an open question posed by Martell et al. The proof relies on a quantitative Fatou-type theorem, which not only guarantees the existence of the pointwise nontangential boundary trace for smooth null-solutions satisfying a strong vanishing Carleson measure condition, but also includes a Poisson integral representation formula of solutions along with a characterization of $\mathrm{CMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ in terms of the traces of solutions of elliptic systems. Moreover, we are able to establish the well-posedness of the Dirichlet problem in $\mathbb{R}^n_+$ for a system $L$ as above in the case when the boundary data belongs to $\mathrm{XMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$, which lines in between $\mathrm{CMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ and $\mathrm{VMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$. Analogously, we formulate a new brand of strong Carleson measure conditions and a characterization of $\mathrm{XMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ in terms of the traces of solutions of elliptic systems.