Layer Potentials The Hodge Laplacian And Global Boundary Problems In Nonsmooth Riemannian Manifolds

Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

The general aim of the present monograph is to study boundary-value problems for second-order elliptic operators in Lipschitz sub domains of Riemannian manifolds. In the first part (ss1-4), we develop a theory for Cauchy type operators on Lipschitz submanifolds of co dimension one (focused on boundedness properties and jump relations) and solve the $Lp$-Dirichlet problem, with $p$ close to $2$, for general second-order strongly elliptic systems. The solution is represented in the form of layer potentials and optimal non tangential maximal function estimates are established.This analysis is carried out under smoothness assumptions (for the coefficients of the operator, metric tensor and the underlying domain) which are in the nature of best possible. In the second part of the monograph, ss5-13, we further specialize this discussion to the case of Hodge Laplacian $\Delta: =-d\delta-\delta d$. This time, the goal is to identify all (pairs of) natural boundary conditions of Neumann type. Owing to the structural richness of the higher degree case we are considering, the theory developed here encompasses in a unitary fashion many basic PDE's of mathematical physics. Its scope extends to also cover Maxwell's equations, dealt with separately in s14. The main tools are those of PDE's and harmonic analysis, occasionally supplemented with some basic facts from algebraic topology and differential geometry.

Layer Potentials, the Hodge Laplacian and Global Boundary Problems in Nonsmooth Riemannian Manifolds

The Hodge–Laplacian

The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains. The 1-st edition of the “Hodge-Laplacian”, De Gruyter Studies in Mathematics, Volume 64, 2016, is a trailblazer of its kind, having been written at a time when new results in Geometric Measure Theory have just emerged, or were still being developed. In particular, this monograph is heavily reliant on the bibliographical items. The latter was at the time an unpublished manuscript which eventually developed into the five-volume series “Geometric Harmonic Analysis” published by Springer 2022-2023. The progress registered on this occasion greatly impacts the contents of the “Hodge-Laplacian” and warrants revisiting this monograph in order to significantly sharpen and expand on previous results. This also allows us to provide specific bibliographical references to external work invoked in the new edition. Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals.