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.

State of the art, original, expansive account of elliptic boundary value problems Elaborate treatment of Fredholm theory for boundary layer potential operators The main results, in print here for the first time, bridge between geometry and partial differential equations

Autorentext

Inhalt
Introduction and Statement of Main Results Concerning the Divergence Theorem.- Examples, Counterexamples, and Additional Perspectives.- Tools from Geometric Measure Theory, Harmonic Analysis, and functional Analysis.- Open Sets with Locally Finite Surface Measures and Boundary Behavior.- Proofs of the Main Results Pertaining to the Divergence Theorem.- Applications to Singular Integrals, Function Spaces, Boundary Problems, and Further Results.