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Many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator in a given domain. When the differential operator under discussion is of second order a variety of tools are available for dealing with such problems, including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. When the differential operator is of higher-order (as is the case, e.g., with anisotropic plate bending when one deals with a fourth order operator) only a few options could be successfully implemented. In the 1970s Alberto Calderón, one of the founders of the modern theory of Singular Integral Operators, advocated the use of layer potentials for the treatment of higher-order elliptic boundary value problems. The present monograph represents the first systematic treatment based on this approach.

This research monograph lays, for the first time, the mathematical foundation aimed at solving boundary value problems for higher-order elliptic operators in non-smooth domains using the layer potential method and addresses a comprehensive range of topics, dealing with elliptic boundary value problems in non-smooth domains including layer potentials, jump relations, non-tangential maximal function estimates, multi-traces and extensions, boundary value problems with data in Whitney–Lebesque spaces, Whitney–Besov spaces, Whitney–Sobolev- based Lebesgue spaces, Whitney–Triebel–Lizorkin spaces,Whitney–Sobolev-based Hardy spaces, Whitney–BMO and Whitney–VMO spaces.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
One of the main goals of the present monograph is to develop the framework of a theory for the multiple layer (or multi-layer, for short) potential operators arising in the treatment of boundary value problems associated with a higher-order, matrix-valued (complex) constant coefficient, elliptic differential operator \(\begin{array}{rcl} Lu =\sum\limits_{\vert \alpha \vert =\vert \beta \vert =m}{\partial }^{\alpha }{A}_{ \alpha \beta }\,{\partial }^{\beta }u& &\end{array}\) (where \(m \in \mathbb{N}\)) in a Lipschitz domain \(\Omega \subset {\mathbb{R}}^{n}\).
Irina Mitrea, Marius Mitrea

Chapter 2. Smoothness Scales and Calderón–Zygmund Theory in the Scalar–Valued Case

Abstract
While one of the main goals of this monograph is the systematic development of a Calderón–Zygmund theory for multi-layer type operators associated with higher-order operators with matrix-valued coefficients, the starting point is the consideration of the scalar-valued case. As such, the aim of this introductory chapter is to present an account of those aspects of the scalar theory which are most relevant for the current work.
Irina Mitrea, Marius Mitrea

Chapter 3. Function Spaces of Whitney Arrays

Abstract
Here we discuss how to adapt the traditional ways of measuring smoothness for scalar functions (defined on the boundary of a Lipschitz domain) to the case of Whitney arrays.
Irina Mitrea, Marius Mitrea

Chapter 4. The Double Multi-Layer Potential Operator

Abstract
In this chapter we take on the task of introducing and studying what we call double multi-layer potential operators, associated with arbitrary elliptic, higher-order, homogeneous, constant (complex) matrix-valued coefficients. As a preamble, we first take a look at the nature of fundamental solutions associated with such operators.
Irina Mitrea, Marius Mitrea

Chapter 5. The Single Multi-Layer Potential Operator

Abstract
The general goal in this chapter is to define and study the main properties of the single multi-layer potential operator associated with arbitrary elliptic, higher-order, homogeneous, constant (complex) matrix-valued coefficients differential operators.
Irina Mitrea, Marius Mitrea

Chapter 6. Functional Analytic Properties of Multi-Layer Potentials and Boundary Value Problems

Abstract
This chapter has a twofold goal. In a first stage, we shall study the Fredholm properties of multi-layer potentials introduced in earlier chapters, while in a second stage we shall proceed to use these results as a tool for establishing the well-posedness of boundary value problems associated with higher-order operators.
Irina Mitrea, Marius Mitrea

Backmatter

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