Fourier decomposition of properly elliptic boundary value problems in a half-plane

For a general class of second-order elliptic boundary value problems in the lower half-plane, we show that the existence and uniqueness of solutions in Lp Sobolev spaces is reduced to the invertibility of the ordinary differential operators obtained by Fourier decomposition. This terminology refers to the partial Fourier series expansion in the case of horizontally periodic solutions and to the partial Fourier transform otherwise. The problem is straightforward when p = 2 and, in the periodic case, the same question on a strip with finite width can also be quickly settled by indirect arguments irrespective of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p \in (1, \infty )}$$\end{document}. However, in the half-plane, the infinite depth raises serious difficulties when p ≠ 2. These difficulties are overcome by writing the problem as a first-order system and using existing abstract results about operator valued Fourier multipliers. In that approach, the randomized boundedness of the resolvent becomes the central issue.