Geometric harmonic analysis V. Fredholm theory and finer estimates for integral operators, with applications to boundary problems (Q6162160)

The present book is the last installment in a series of five volumes, at the confluence of harmonic analysis, geometric measure theory, function space theory, and partial differential equations. The series is generically branded as `Geometric harmonic analysis' with the individual volumes carrying the following subtitles: Volume~I: A sharp divergence theorem with nontangential pointwise traces [Zbl 1517.42001]; Volume~II: Function spaces measuring size and smoothness on rough sets [Zbl 1521.42003]; Volume~III: Integral representations, Calderón-Zygmund theory, Fatou theorems, and applications to scattering [1523.35001]; Volume~IV: Boundary layer potentials in uniformly rectifiable domains, and applications to complex analysis [Zbl 1526.42001]; Volume~V: Fredholm theory and finer estimates for integral operators, with applications to boundary problems. The main objective of the series is to produce tools having the ability to treat efficiently boundary value problems for elliptic systems in inclusive geometric settings, beyond the category of Lipschitz domains. The present volume focuses on proving well-posedness and Fredholm solvability results regarding boundary value problems for elliptic second-order homogeneous constant complex coefficient systems, and domains of a rather general geometric character. The formulation of the boundary value problems treated here is optimal from a multitude of aspects, pertaining to Functional Analysis, Geometry, Partial Differential Equations, and Topology. The work in the present volume aligns with the program stemming from A.P. Calderón's 1978 ICM address which advocates the use of layer potentials for much more general elliptic systems than the Laplacian. In Chapter~1, the authors introduce the notion of distinguished coefficient tensor, which is central to all subsequent developments in this volume. Relevant examples of weakly elliptic homogeneous constant (complex) coefficient second-order systems possessing distinguished coefficient tensors are given, including certain Lamé-like systems, and the entire class of scalar weakly elliptic homogeneous constant (complex) coefficient second-order operators in dimensions \(\geq 3\). Also, the authors study whether the quality of being a distinguished coefficient tensor is stable under transposition, and discuss the issue of the existence and uniqueness of a distinguished coefficient tensor. Chapter~2 focuses on providing examples of weakly elliptic homogeneous constant coefficient second-order systems with the property that their associated \(L^p\) Dirichlet Problems in the upper half-space fail to be Fredholm solvable. The manner in which this connects with earlier work in Chapter~1 is that one looks for such pathological weakly elliptic systems in the class of those which lack a distinguished coefficient tensor. Chapter~3 is devoted to the task of quantifying global and infinitesimal flatness in classes of Euclidean sets of locally finite perimeter which may otherwise lack structural qualities which have traditionally been used to describe regularity. Chapter~4 deals with the class of singular integral operators (SIO's) of chord-dot-normal type (as defined in Volume~IV, \S 5.2) associated with Ahlfors regular domains \(\Omega\subset\mathbb{R}^n\) with unbounded boundaries. These operators are sensitive to flatness, in the sense that they become identically zero when the underlying domain is a half-space in \(\mathbb{R}^n\). In Chapter~5, the authors clarify how the flatness of a surface is related to the functional analytic properties of singular integral operators defined on it by setting up a two-way street between geometry and analysis. Working now in domains with compact boundaries, the authors establish estimates for the essential norm of SIO's of chord-dot-normal type in terms of the proximity of the unit normal to the surface in question to Sarason's space VMO. Chapter~6 deals with the Radon-Carleman Problem, which the authors define as the task of computing and/or estimating the essential norm and/or Fredholm radius of singular integral operators of double layer type, associated with elliptic PDE, on function spaces naturally intervening in the formulation of boundary problems for said PDE. In Chapter~7, the authors prove Fredholm and invertibility properties of boundary layer potentials on compact surfaces, which are subsequently used to treat boundary value problems in domains with compact boundaries. This body of work points to the fact that the category of Ahlfors regular domains with a compact and sufficiently flat boundary at infinitesimal level is a most natural geometric environment where Fredholm theory becomes applicable to boundary layer potential operators. In Chapter~8, as a culmination of the work in this volume, the authors formulate and solve boundary value problems for elliptic second-order systems, involving a large variety of boundary conditions and function spaces, in a geometric setting displaying new levels of generality and inclusiveness. Also, they take the first steps in the direction of combining geometric measure theory with scattering theory, by solving the basic boundary value problems in acoustic theory in novel geometric settings. Significantly, the authors successfully implement the method of boundary layer potentials for the Dirichlet and Neumann Problems for elliptic systems with data in Muckenhoupt weighted Lebesgue spaces, as well as Hardy, Sobolev, BMO, VMO, Hölder, Morrey, Besov, Triebel-Lizorkin spaces, and Generalized Banach Function Spaces.