Potential Theory on Harmonic Spaces

ISBN-13: 9783642654343 / Angielski / Miękka / 2012 / 360 str.

Corneliu Constantinescu;Aurel Cornea

Opis

There has been a considerable revival of interest in potential theory during the last 20 years. This is made evident by the appearance of new mathematical disciplines in that period which now-a-days are considered as parts of potential theory. Examples of such disciplines are: the theory of Choquet capacities, of Dirichlet spaces, of martingales and Markov processes, of integral representation in convex compact sets as well as the theory of harmonic spaces. All these theories have roots in classical potential theory. The theory of harmonic spaces, sometimes also called axiomatic theory of harmonic functions, plays a particular role among the above mentioned theories. On the one hand, this theory has particularly close connections with classical potential theory. Its main notion is that of a harmonic function and its main aim is the generalization and unification of classical results and methods for application to an extended class of elliptic and parabolic second order partial differential equations. On the other hand, the theory of harmonic spaces is closely related to the theory of Markov processes. In fact, all important notions and results of the theory have a probabilistic interpretation.

Spis treści

Terminology and Notation.- One.- 1. Harmonic Sheaves and Hyperharmonic Sheaves.- § 1.1. Convergence Properties.- §1.2. Resolutive Sets.- § 1.3. Minimum Principle.- 2. Harmonic Spaces.- §2.1. Definition of Harmonic Spaces.- § 2.2. Superharmonic Functions and Potentials.- § 2.3. G-Harmonic Spaces and B-Harmonic Spaces.- § 2.4. Resolutive Sets on Harmonic Spaces.- 3. Bauer Spaces and Brelot Spaces.- §3.1. Definitions and Fundamental Results.- § 3.2. The Laplace Equation.- § 3.3. The Heat Equation.- Two.- 4. Convex Cones of Continuous Functions on Baire Topological Spaces.- §4.1. Natural Order and Specific Order.- §4.2. Balayage.- 5. The Convex Cone of Hyperharmonic Functions.- §5.1. The Fine Topology.- §5.2. Capacity.- § 5.3. Supplementary Results on the Balayage of Positive Super-harmonic Functions.- 6. Absorbent Sets, Polar Sets, Semi-Polar Sets.- §6.1. Absorbent Sets.- § 6.2. Polar Sets.- § 6.3. Thinness and Semi-Polar Sets.- 7. Balayage of Measures.- §7.1. General Properties of the Balayage of Measures.- § 7.2. Fine Properties of the Balayage of Measures.- 8. Positive Superharmonic Functions. Specific Order.- §8.1. Abstract Carriers.- § 8.2. Sets of Nonharmonicity.- § 8.3. The Band M.- §8.4. Quasi-Continuity.- Three.- 9. Axiom of Polarity and Axiom of Domination.- §9.1. Axiom of Polarity.- § 9.2. Axiom of Domination.- 10. Markov Processes on Harmonic Spaces.- § 10.1. Sub-Markov Semi-Groups.- § 10.2. Sub-Markov Semi-Groups on Harmonic Spaces.- 11. Integral Representation of Positive Superharmonic Functions.- § 11.1. Locally Convex Vector Spaces of Harmonic Functions.- § 11.2. Locally Convex Topologies on the Convex Cone of Positive Superharmonic Functions.- §11.3. Abstract Integral Representation.- §11.4. Riesz-Martin Kernels.- § 11.5. Integral Representation of Positive Superharmonic Functions.- References.- Notation.