Original Title: “The BMO-Dirichlet problem for elliptic systems in the upper half space and quantitative characterizations of VMO”
Authors: José María Martell (ICMATCSIC), Dorina Mitrea (Universidad de Missouri), I. Mitrea (Universidad de Temple) y Marius Mitrea (Universidad de Missouri)
Source: Analysis & PDE 12 (2019), no. 3, 605720
Date of online publication: 2019
E.B. Fabes, R.L. Johnson y U. Neri established in 1976 the well posedness –that is, there is a unique solution– of the socalled Dirichlet problem for the Laplacian in the upper halfspace n+, con n ≥ 2, and with boundary data in the space of functions with bounded mean oscillation (BMO), when the first order derivatives of the solutions satisfy a Carleson measure estimate. Furthermore, they also obtained a Fatoutype result on which BMO can be described as the collection of non-tangential pointwise traces of harmonic functions in the upper half space whose derivatives satisfy the aforementioned Carleson measure condition. In this recent paper, J.M. Martell, D. Mitrea, I. Mitrea, and M. Mitrea extend these results in several ways. First, they consider elliptic systems with constant complex coefficients, including for instance complex versions of the Lamé system of elasticity. Second, besides working with boundary data in BMO, they treat work with boundary data belonging to its subspace of vanishing mean oscillation functions VMO –where the mean oscillation converges to 0 uniformly, when the cubes have arbitrarily small size. In such a case, the natural class of solutions is that where the first order derivatives of the solutions additionally satisfy a vanishing Carleson measure condition (that is, the Carleson measure constant is uniformly small when considered in scales that converge to 0).
Furthermore, Fatou type results are obtained for both BMO and VMO. Namely, BMO and VMO can be identified with the pointwise non-tangential traces of the null solutions whose first order derivatives satisfy a Carleson measure condition, which is additionally “vanishing” in the VMO case. A remarkable consequence is that smooth functions in BMO are shown to be dense in VMO, fact that improves on Sarason’s classical result describing VMO as the closure in BMO of the space of uniformly continuous functions with bounded mean oscillations. Even though this density result in VMO is of a purely real variable nature, Martell, Mitrea, Mitrea, and Mitrea’s argument makes essential use of the PDE-rooted results obtained in the paper. Indeed, any given VMO function is approximated in BMO, by precisely using the solution to the associated Dirichlet problem.