Function Spaces Potential Theory (Grundlehren der mathematischen Wissenschaften)
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Von Händler/Antiquariat, J. HENRY BOOKS.
The subject of this book is the interplay between function space theory and potential theory. A crucial step in classical potential theory is the identification of the potential engergy of a charge with the square of a Hilbert space norm. This leads the Dirichlet space of locally integrable functions whose gradients are square integrable. More recently, a generalized potential theory has been developed, which has an analogous relationship to the standard Banach function spaces, Sobolev spaces, Besov spaces etc., that appear naturally in the study of partial differential equations. A surprisingly large part of classical potential theory has been extended to this nonlinear setting. The extensions are sometimes surprising, usually they are nontrivial and have required new methods. Hardcover, Label: Springer-Verlag, Springer-Verlag, Produktgruppe: Book, Publiziert: 1996-06, Studio: Springer-Verlag, Verkaufsrang: 14407740.

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"..carefully and thoughtfully written and prepared with, in my opinion, just the right amount of detail included...will certainly be a primary source that I shall turn to." Proceedings of the Edinburgh Mathematical Society2010. xi, 368 S. XI, 368 p. 235 mmVersandfertig in 3-5 Tagen, Softcover.

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Function spaces, especially those spaces that have become known as Sobolev spaces, and their natural extensions, are now a central concept in analysis. In particular, they play a decisive role in the modem theory of partial differential equations (PDE). Potential theory, which grew out of the theory of the electrostatic or gravita­ tional potential, the Laplace equation, the Dirichlet problem, etc. , had a fundamen­ tal role in the development of functional analysis and the theory of Hilbert space. Later, potential theory was strongly influenced by functional analysis. More re­ cently, ideas from potential theory have enriched the theory of those more general function spaces that appear naturally in the study of nonlinear partial differential equations. This book is motivated by the latter development. The connection between potential theory and the theory of Hilbert spaces can be traced back to C. F. Gauss [181], who proved (with modem rigor supplied almost a century later by O. Frostman [158]) the existence of equilibrium potentials by minimizing a quadratic integral, the energy. This theme is pervasive in the work of such mathematicians as D. Hilbert, Ch. -J. de La Vallee Poussin, M. Riesz, O. Frostman, A. Beurling, and the connection was made particularly clear in the work of H. Cartan [97] in the 1940'S. In the thesis of J. Deny [119], and in the subsequent work of J. Deny and J. L. Soft cover.

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