Foundations of Modern Potential Theory
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Foundations of Modern Potential Theory

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Naum S. Landkof
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1. Spaces of measures and signed measures. Operations on measures and signed measures (No. 1-5).-2. Space of distributions. Operations on distributions (No. 6-10)..-
3. The Fourier transform of distributions (No. 11-13).- I. Potentials and their basic properties.-
1. M. Riesz kernels (No. 1-3).-
2. Superharmonic functions (No. 4-5).-
3. Definition of potentials and their simplest properties (No. 6-9)...-
4. Energy. Potentials with finite energy (No. 10-15).-
5. Representation of superharmonic functions by potentials (No. 16-18).-
6. Superharmonic functions of fractional order (No. 19-25).- II. Capacity and equilibrium measure.-
1. Equilibrium measure and capacity of a compact set (No. 1-5).-
2. Inner and outer capacities and equilibrium measures. Capacitability (No. 6-10).-
3. Metric properties of capacity (No. 11-14).-
4. Logarithmic capacity (No. 15-18).- III. Sets of capacity zero. Sequences and bounds for potentials.-
1. Polar sets (No. 1-2).-
2. Continuity properties of potentials (No. 3-4).-
3. Sequences of potentials of measures (No. 5-8).-
4. Metric criteria for sets of capacity zero and bounds for potentials (No. 9-11).- IV. Balayage, Green functions, and the Dirichlet problem.-
1. Classical balayage out of a region (No. 1-6).-
2. Balayage for arbitrary compact sets (No. 7-11).-
3. The generalized Dirichlet problem (No. 12-14).-
4. The operator approach to the Dirichlet problem and the balayage problem (No. 15-18).-
5. Balayage for M. Riesz kernels (No. 19-23)...-
6. Balayage onto Borel sets (No. 24-25).- V. Irregular points.- 1. Irregular points of Borel sets. Criteria for irregularity (No. 1-6)...-
2. The characteristics and types of irregular points (No. 7-8).....-
3. The fine topology (No. 9-11).-
4. Properties of set of irregular points (No. 12-15).-
5. Stability of the Dirichlet problem. Approximation of continuous functions by harmonic functions (No. 16-22).- VI. Generalizations.-
1. Distributions with finite energy and their potentials (No. 1-5)...-
2. Kernels of more general type (No. 6-11).-
3. Dirichlet spaces (No. 12-15).- Comments and bibliographic references.
For a long time potential theory was necessarily viewed as only another chapter of mathematical physics. Developing in close connection with the theory of boundary-value problems for the Laplace operator, it led to the creation of the mathematical apparatus of potentials of single and double layers; this was adequate for treating problems involving smooth boundaries. A. M. Lyapunov is to be credited with the rigorous analysis of the properties of potentials and the possibilities for applying them to the 1 solution of boundary-value problems. The results he obtained at the end of the 19th century later received a more detailed and sharpened exposition in the book by N. M. Gunter, published in Paris in 1934 and 2 in New York 1967 with additions and revisions. Of fundamental significance to potential theory also was the work of H. Poincare, especially his method of sweeping out mass (balayage). At the beginning of the 20th century the work of S. Zaremba and especially of H. Lebesgue attracted the attention of mathematicians to the unsolvable cases of the classical Dirichlet problem. Through the efforts of O. Kellogg, G. Bouligand, and primarily N. Wiener, by the middle of the 20th century the problem of characterizing the so-called irregular points of the boundary of a region (i. e. the points at which the continuity of the solution of the Dirichlet problem may be violated) was completely solved and a procedure to obtain a generalized solution to the Dirichlet problem was described.

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