Mitrea D. Geometric Harmonic Analysis V. Fredholm Theory...Finer Estimates..2023

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This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations.The ultimate goal in Volume V is to prove well-posedness and Fredholm solvability results concerning boundary value problems for elliptic second-order homogeneous constant (complex) coefficient systems, and domains of a rather general geometric nature. The formulation of the boundary value problems treated here is optimal from a multitude of points of view, having to do with geometry, functional analysis (through the consideration of a large variety of scales of function spaces), topology, and partial differential equations.
Distinguished Coefficient Tensors
Failure of Fredholm Solvability for Weakly Elliptic Systems
Quantifying Global and Infinitesimal Flatness
Norm Estimates and Invertibility Results for SIO’s on Unbounded Boundaries
Estimating Chord-Dot-Normal SIO’s on Domains with Compact Boundaries
The Radon-Carleman Problem
Fredholmness and Invertibility of Layer Potentials on Compact Boundaries
Boundary Value Problems for Elliptic Systems in Rough Domains