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

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 $d\mu (x^{\prime },t):=|\nabla u(x){|}^{2}td{x}^{\prime }dt$ 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.

References in corpus (4)

• The Dirichlet problem for elliptic systems with data in Köthe function spaces
• The BMO-Dirichlet problem for elliptic systems in the upper-half space and quantitative characterizations of VMO
• The generalized Hölder and Morrey-Campanato Dirichlet problems for elliptic systems in the upper-half space
• Fatou-Type Theorems and Boundary Value Problems for Elliptic Systems in the Upper Half-Space