Nonlinear potential theory on metric spaces.

*(English)*Zbl 1231.31001
EMS Tracts in Mathematics 17. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-099-9/hbk). xii, 403 p. (2011).

The authors give a detailed and comprehensive introduction to Newtonian (Sobolev) spaces (first six chapters) with applications to nonlinear potential theory (the remaining eight chapters). The first part presents standard material of the theory which has been scattered in several research papers so far providing the first presentation in book format. The second part of the book can be seen as an extension of [J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. Oxford: Clarendon Press (1993; Zbl 0780.31001)] to the non-Euclidean context. More precisely, the functions are defined on a metric space endowed with a doubling Borel measure. The metric measure space is also assumed to support a suitable Poincaré inequality. The doubling property reflects the finite dimensionality of the space; the Poincaré inequalities guarantee certain connectedness properties.

The \(p\)-Laplace equation is the prototype case of nonlinear partial differential equations and the corresponding variational formulation is taken as the starting point of the nonlinear potential theory on metric spaces. The arguments in the Euclidean setting do not directly generalize to the metric context since the equation is not available there, thus providing new proofs of the classical theory that depend only on the variational formulation. The emphasis is on the theory of minimizers with results on interior regularity, the Dirichlet problem, boundary regularity, removable singularities, and regular sets. In the appendices the Newtonian space is compared with classical Sobolev spaces and other Sobolev spaces defined on metric spaces. The book contains a short bibliographical note after each chapter with interesting historical details and references to the original works.

The \(p\)-Laplace equation is the prototype case of nonlinear partial differential equations and the corresponding variational formulation is taken as the starting point of the nonlinear potential theory on metric spaces. The arguments in the Euclidean setting do not directly generalize to the metric context since the equation is not available there, thus providing new proofs of the classical theory that depend only on the variational formulation. The emphasis is on the theory of minimizers with results on interior regularity, the Dirichlet problem, boundary regularity, removable singularities, and regular sets. In the appendices the Newtonian space is compared with classical Sobolev spaces and other Sobolev spaces defined on metric spaces. The book contains a short bibliographical note after each chapter with interesting historical details and references to the original works.

Reviewer: Daniel Aalto (Åbo)

##### MSC:

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

31E05 | Potential theory on fractals and metric spaces |

31C15 | Potentials and capacities on other spaces |

35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |

49J52 | Nonsmooth analysis |