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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.

Boundary value problems --- Differential equations, Elliptic --- Lipschitz spaces --- Smoothness of functions --- Calderâon-Zygmund operator --- Mathematics --- Civil & Environmental Engineering --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Calculus --- Operations Research --- Mathematical Theory --- Smooth functions --- Hölder spaces --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Boundary conditions (Differential equations) --- Mathematics. --- Fourier analysis. --- Integral equations. --- Partial differential equations. --- Potential theory (Mathematics). --- Potential Theory. --- Partial Differential Equations. --- Integral Equations. --- Fourier Analysis. --- Boundary value problems. --- Differential equations, Elliptic. --- Lipschitz spaces. --- Smoothness of functions. --- Calderón-Zygmund operator. --- Calderón-Zygmund singular integral operator --- Mikhlin-Calderon-Zygmund operator --- Operator, Calderón-Zygmund --- Singular integral operator, Calderón-Zygmund --- Zygmund-Calderón operator --- Linear operators --- Functions --- Function spaces --- Differential equations, Linear --- Differential equations, Partial --- Differential equations --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Differential equations, partial. --- Analysis, Fourier --- Mathematical analysis --- Equations, Integral --- Functional equations --- Functional analysis --- Partial differential equations --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mechanics

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Divergence theorem. --- Functional analysis. --- Anàlisi funcional --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Gauss-Ostrogradsky theorem --- Gauss's theorem --- Vector algebra --- Vector analysis --- Càlcul funcional --- Càlcul de variacions --- Àlgebres de Hilbert --- Àlgebres topològiques --- Anàlisi funcional no lineal --- Anàlisi microlocal --- Espais analítics --- Espais de Hardy --- Espais d'Orlicz --- Espais funcionals --- Espais vectorials normats --- Espais vectorials --- Filtres digitals (Matemàtica) --- Funcionals --- Funcions vectorials --- Multiplicadors (Anàlisi matemàtica) --- Pertorbació (Matemàtica) --- Teoria d'operadors --- Teoria de distribucions (Anàlisi funcional) --- Teoria de functors --- Teoria de l'aproximació --- Teoria del funcional de densitat --- Teoria espectral (Matemàtica) --- Equacions funcionals --- Equacions integrals

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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. Traditionally, the label “Calderón-Zygmund theory” has been applied to a distinguished body of works primarily pertaining to the mapping properties of singular integral operators on Lebesgue spaces, in various geometric settings. Volume IV amounts to a versatile Calderón-Zygmund theory for singular integral operators of layer potential type in open sets with uniformly rectifiable boundaries, considered on a diverse range of function spaces. Novel applications to complex analysis in several variables are also explored here.

Mathematical analysis. --- Integral Transforms and Operational Calculus. --- 517.1 Mathematical analysis --- Mathematical analysis --- Teoria de la mesura geomètrica --- Àlgebra vectorial --- Capa límit

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Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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. Volume III is concerned with integral representation formulas for nullsolutions of elliptic PDEs, Calderón-Zygmund theory for singular integral operators, Fatou type theorems for systems of elliptic PDEs, and applications to acoustic and electromagnetic scattering. Overall, this amounts to a powerful and nuanced theory developed on uniformly rectifiable sets, which builds on the work of many predecessors.

Algebra --- Mathematics --- algebra --- wiskunde --- Mathematical analysis. --- Integral Transforms and Operational Calculus. --- Teoria de la mesura geomètrica