##
**Introduction to heat potential theory.**
*(English)*
Zbl 1251.31001

Mathematical Surveys and Monographs 182. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4998-9/hbk). xiii, 266 p. (2012).

Classical potential theory is in fact the theory of the Laplace operator. It can be considered as a study of harmonic and superharmonic functions, potentials, balayage, the Dirichlet problem and other topics. Gradually it became clear that important properties related to Laplace’s equation are also shared by extended classes of second order partial differential equations of elliptic type and also parabolic equations, like the heat equation. (Among these properties we can list, for instance, the maximum principle, the Poisson integral, showing that there is a base of open sets for which the Dirichlet problem has a solution for every continuous boundary function, and Harnack’s convergence theorem.) From the 1950s onwards, this led M. Brelot, J. L. Doob, H. Bauer and others to the creation of abstract potential theory, the theory of harmonic spaces. An excellent survey article is [H. Bauer, Conf. Semin. Mat. Univ. Bari 197, 34 p. (1984; Zbl 0572.31005)]. The first systematic exposition of abstract potential theory including heat potential theory was presented in [H. Bauer, Harmonische Räume und ihre Potentialtheorie. Berlin etc.: Springer (1966; Zbl 0142.38402)]. For parabolic equations, the proof that it is possible to solve the Dirichlet problem for the sufficiently large class of open sets is difficult. Alternative approaches to circumventing this difficulty have been proposed in [C. Constantinescu and A. Cornea, Potential theory on harmonic spaces. Berlin etc.: Springer (1972; Zbl 0248.31011)] and [J. Bliedtner and W. Hansen, Potential theory. An analytic and probabilistic approach to balayage. Berlin etc.: Springer (1986; Zbl 0706.31001)]. So, for several decades, a multitude of results for the heat equation has been hidden under the umbrella of harmonic or balayage spaces.

Since the 1970s, quite independently of abstract potential theory, Neil Watson has published a long series of papers on potential theory for the heat equation. The book under review is a systematic exposition of heat potential theory based largely on author’s own research contributions. The book is essentially a self-contained, well organized and carefully written monograph in which a large part of caloric (= heat) potential theory stands as a subject in its own right. The presentation does not depend on relevant results from harmonic space theory obtained previously and does not offer any excursion into probabilistic interpretation. For the latter see [J. L. Doob, Classical potential theory and its probabilistic counterpart. Berlin: Springer (2001; Zbl 0990.31001)].

Chapter 1 of the book deals with basic properties of temperatures and smooth subtemperatures (these correspond to harmonic and subharmonic functions in classical potential theory) and to mean values over heat spheres and heat balls, which are related to level sets of the fundamental solution of the heat equation. The kernel for the heat ball is unbounded near the centre of the ball, which is inconvenient for certain purposes. Therefore, modified heat balls are introduced and the corresponding mean value theorems are used to prove Harnack’s theorem. As a corollary, the Harnack inequality for temperatures, as well as compactness properties, are established. In caloric potential theory, there is no easy way to obtain an analogue of the Poisson integral for classical harmonic functions. In the book, the usual representation of solutions of the heat equation on rectangular domains is omitted. Instead, in Chapter 2, caloric measure is investigated on a circular cylinder. Double layer heat potentials and the contraction mapping principle are tools here. In Chapter 3, a caloric form of the maximum principle for (upper semicontinuous) subtemperatures is proved, but its relationship to maximum principles known in general potential theory is not discussed. Then the Perron-Wiener-Brelot solution for convex domains of revolution is investigated in detail. This yields that the cone is a regular set for the heat Dirichlet problem (p. 70). The theory developed makes it possible to provide further characterizations of subtemperatures. This shows that the notion of a subtemperature introduced by Watson coincides with subharmonic functions in abstract potential theory, when specialized to the heat equation. Temperatures and various classes of subtemperatures on an infinite strip are studied in Chapters 4 and 5. Green functions and heat potentials are dealt with in Chapter 6. The distributional approach is used for establishing the Riesz decomposition theorem. In Chapter 7, general heat potential theory is developed (polar sets, a refinement of the maximum principle, the natural order decomposition, reduction and balayage of supertemperatures, thermal capacity). The capacitability theorem is proved. Chapter 8 is devoted to the generalized Dirichlet problem for arbitrary open sets, in which the prescribed function is defined only on the essential boundary, and only one-sided regularity is sought at points of the semi-singular boundary. It would be interesting to compare the solution obtained with that usually adopted in previous theories. Since the Keldych theorem (uniqueness of an operator of the generalized Dirichlet problem) no longer holds for the heat equation, the coincidence with the standard PWB-solution is not guaranteed; cf. [J. Lukeš et al., Integral representation theory. Applications to convexity, Banach spaces and potential theory. Berlin: Walter de Gruyter (2010; Zbl 1216.46003)]. The approach adopted acknowledges a specific feature of the heat equation concerning the fact that the temporal variable behaves differently from the spatial variables. This is also reflected by a finer classification of various types of boundary points. This leads to a new version of the boundary maximum principle. No comparison with similar investigations in the context of harmonic spaces is mentioned, cf., for instance, [J. Bliedtner and W. Hansen, Math. Z. 151, 71–87 (1976; Zbl 0319.31011)]. The PWB-method follows the usual pattern and leads to harmonic measures, the resolutivity theorem and a study of the boundary behavior of solutions. It would be useful to compare the results obtained with those in [J. Lukeš and J. Malý, Math. Ann. 257, 355–366 (1981; Zbl 0461.31003)]. The proof of the Wiener criterion of regularity [L. C. Evans and R. F. Gariepy, Arch. Ration. Mech. Anal. 78, 293–314 (1982; Zbl 0508.35038)] is not included, but its statement is mentioned at the end of Chapter 8. A geometric test (tusk condition) for regularity is proved. Finally, in Chapter 9, the thermal fine topology is discussed and the fundamental convergence theorem is proved. Each chapter concludes with bibliographical notes and comments.

A positive feature of the book is that several open problems are proposed. The title of the book indicates that it was not the intention of the author to include all that is known on the heat equation. However, I feel that slightly more could have been done. Namely, a brief survey of closely related results of more recent origin with good references would have been helpful and desirable, and these could have been put in larger numbers into the Notes and Comments. For instance, a discussion of absorbing sets, of the Choquet theory aspects of the heat equation, the weak Dirichlet problem, simpliciality, the essential base, more on semipolar sets, balayage of measures, etc. could have been at least briefly mentioned. In general, a discussion of the interplay between concrete heat potential theory and heat potential theory from the harmonic spaces perspective could have been discussed. This objection is, of course, not overly serious. In all, Watson’s book is the first monograph devoted entirely to potential theory of the heat equation. It is mathematically precise, well written and represents a valuable account of time-dependent potential theory.

Since the 1970s, quite independently of abstract potential theory, Neil Watson has published a long series of papers on potential theory for the heat equation. The book under review is a systematic exposition of heat potential theory based largely on author’s own research contributions. The book is essentially a self-contained, well organized and carefully written monograph in which a large part of caloric (= heat) potential theory stands as a subject in its own right. The presentation does not depend on relevant results from harmonic space theory obtained previously and does not offer any excursion into probabilistic interpretation. For the latter see [J. L. Doob, Classical potential theory and its probabilistic counterpart. Berlin: Springer (2001; Zbl 0990.31001)].

Chapter 1 of the book deals with basic properties of temperatures and smooth subtemperatures (these correspond to harmonic and subharmonic functions in classical potential theory) and to mean values over heat spheres and heat balls, which are related to level sets of the fundamental solution of the heat equation. The kernel for the heat ball is unbounded near the centre of the ball, which is inconvenient for certain purposes. Therefore, modified heat balls are introduced and the corresponding mean value theorems are used to prove Harnack’s theorem. As a corollary, the Harnack inequality for temperatures, as well as compactness properties, are established. In caloric potential theory, there is no easy way to obtain an analogue of the Poisson integral for classical harmonic functions. In the book, the usual representation of solutions of the heat equation on rectangular domains is omitted. Instead, in Chapter 2, caloric measure is investigated on a circular cylinder. Double layer heat potentials and the contraction mapping principle are tools here. In Chapter 3, a caloric form of the maximum principle for (upper semicontinuous) subtemperatures is proved, but its relationship to maximum principles known in general potential theory is not discussed. Then the Perron-Wiener-Brelot solution for convex domains of revolution is investigated in detail. This yields that the cone is a regular set for the heat Dirichlet problem (p. 70). The theory developed makes it possible to provide further characterizations of subtemperatures. This shows that the notion of a subtemperature introduced by Watson coincides with subharmonic functions in abstract potential theory, when specialized to the heat equation. Temperatures and various classes of subtemperatures on an infinite strip are studied in Chapters 4 and 5. Green functions and heat potentials are dealt with in Chapter 6. The distributional approach is used for establishing the Riesz decomposition theorem. In Chapter 7, general heat potential theory is developed (polar sets, a refinement of the maximum principle, the natural order decomposition, reduction and balayage of supertemperatures, thermal capacity). The capacitability theorem is proved. Chapter 8 is devoted to the generalized Dirichlet problem for arbitrary open sets, in which the prescribed function is defined only on the essential boundary, and only one-sided regularity is sought at points of the semi-singular boundary. It would be interesting to compare the solution obtained with that usually adopted in previous theories. Since the Keldych theorem (uniqueness of an operator of the generalized Dirichlet problem) no longer holds for the heat equation, the coincidence with the standard PWB-solution is not guaranteed; cf. [J. Lukeš et al., Integral representation theory. Applications to convexity, Banach spaces and potential theory. Berlin: Walter de Gruyter (2010; Zbl 1216.46003)]. The approach adopted acknowledges a specific feature of the heat equation concerning the fact that the temporal variable behaves differently from the spatial variables. This is also reflected by a finer classification of various types of boundary points. This leads to a new version of the boundary maximum principle. No comparison with similar investigations in the context of harmonic spaces is mentioned, cf., for instance, [J. Bliedtner and W. Hansen, Math. Z. 151, 71–87 (1976; Zbl 0319.31011)]. The PWB-method follows the usual pattern and leads to harmonic measures, the resolutivity theorem and a study of the boundary behavior of solutions. It would be useful to compare the results obtained with those in [J. Lukeš and J. Malý, Math. Ann. 257, 355–366 (1981; Zbl 0461.31003)]. The proof of the Wiener criterion of regularity [L. C. Evans and R. F. Gariepy, Arch. Ration. Mech. Anal. 78, 293–314 (1982; Zbl 0508.35038)] is not included, but its statement is mentioned at the end of Chapter 8. A geometric test (tusk condition) for regularity is proved. Finally, in Chapter 9, the thermal fine topology is discussed and the fundamental convergence theorem is proved. Each chapter concludes with bibliographical notes and comments.

A positive feature of the book is that several open problems are proposed. The title of the book indicates that it was not the intention of the author to include all that is known on the heat equation. However, I feel that slightly more could have been done. Namely, a brief survey of closely related results of more recent origin with good references would have been helpful and desirable, and these could have been put in larger numbers into the Notes and Comments. For instance, a discussion of absorbing sets, of the Choquet theory aspects of the heat equation, the weak Dirichlet problem, simpliciality, the essential base, more on semipolar sets, balayage of measures, etc. could have been at least briefly mentioned. In general, a discussion of the interplay between concrete heat potential theory and heat potential theory from the harmonic spaces perspective could have been discussed. This objection is, of course, not overly serious. In all, Watson’s book is the first monograph devoted entirely to potential theory of the heat equation. It is mathematically precise, well written and represents a valuable account of time-dependent potential theory.

Reviewer: Ivan Netuka

### MathOverflow Questions:

\(\mathcal{C}^1(\overline{\Omega})\) gradient bounds for the Dirichlet problem of the heat equation on general domains### MSC:

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

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

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

31B20 | Boundary value and inverse problems for harmonic functions in higher dimensions |

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

31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |

31C15 | Potentials and capacities on other spaces |

31D05 | Axiomatic potential theory |

35K05 | Heat equation |

35K08 | Heat kernel |