Classical potential theory.

*(English)*Zbl 0972.31001
Springer Monographs in Mathematics. London: Springer. xvi, 333 p. (2001).

This monograph offers many of the notions, techniques, and results which are central to classical potential theory, i.e., (analytic) potential theory related to Laplace’s equation \(\Delta h=0\). It consists of nine chapters (each about thirty pages long) on harmonic functions, harmonic polynomials, subharmonic functions, potentials, polar sets and capacity, the Dirichlet problem, the fine topology, the Martin boundary, and boundary limits (the notion of energy is not discussed). Each of the first six chapters concludes by approximately twenty exercises. A short appendix indicating references to some general mathematical tools which are employed is followed by several pages of historical notes.

The titles of the chapters suggest material familiar from other books, e.g. the monographs by L. L. Helms (1969; Zbl 0188.17203), J. L. Doob (1984; Zbl 0549.31001), reprint (2001), and S. Axler, P. Bourdon and W. Ramey (1992; Zbl 0765.31001), 2nd ed. (2001; Zbl 0959.31001). Already the presentation of the standard notions and results, however, differs in many details. Particular attention is given to the close relationship between potential theory in the plane and complex analysis. In addition, some lesser known and some fairly new results are included. Their choice is partly based on and largely motivated by the authors’ own research. The fact that the authors themselves work on problems in classical potential theory seems to have given life to the book on the whole. Moreover, the book is clearly and economically written.

On the other hand, the reader should know that this book presents classical potential theory exclusively as a theory in its own right (except for the relationship to complex analysis). Apart from half a phrase in the introduction and a few references in the historical notes there is no indication as to how classical potential theory has become a special case of more general theories during the last decades, and it seems that these developments have not influenced the presentation in the book. There is not even a hint to probabilistic aspects of classical potential theory (no word about Brownian motion).

Within these limitations the book under review will certainly be an attractive textbook for graduate students interested in the subject. In addition, it may serve researchers as a useful reference to results in classical potential theory.

The titles of the chapters suggest material familiar from other books, e.g. the monographs by L. L. Helms (1969; Zbl 0188.17203), J. L. Doob (1984; Zbl 0549.31001), reprint (2001), and S. Axler, P. Bourdon and W. Ramey (1992; Zbl 0765.31001), 2nd ed. (2001; Zbl 0959.31001). Already the presentation of the standard notions and results, however, differs in many details. Particular attention is given to the close relationship between potential theory in the plane and complex analysis. In addition, some lesser known and some fairly new results are included. Their choice is partly based on and largely motivated by the authors’ own research. The fact that the authors themselves work on problems in classical potential theory seems to have given life to the book on the whole. Moreover, the book is clearly and economically written.

On the other hand, the reader should know that this book presents classical potential theory exclusively as a theory in its own right (except for the relationship to complex analysis). Apart from half a phrase in the introduction and a few references in the historical notes there is no indication as to how classical potential theory has become a special case of more general theories during the last decades, and it seems that these developments have not influenced the presentation in the book. There is not even a hint to probabilistic aspects of classical potential theory (no word about Brownian motion).

Within these limitations the book under review will certainly be an attractive textbook for graduate students interested in the subject. In addition, it may serve researchers as a useful reference to results in classical potential theory.

Reviewer: Wolfhard Hansen (Bielefeld)

##### MSC:

31-02 | Research exposition (monographs, survey articles) pertaining to potential theory |

31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |

31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |

31B25 | Boundary behavior of harmonic functions in higher dimensions |

31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |

30C85 | Capacity and harmonic measure in the complex plane |

41A30 | Approximation by other special function classes |