Fine regularity of solutions of elliptic partial differential equations.

*(English)*Zbl 0882.35001
Mathematical Surveys and Monographs. 51. Providence, RI: American Mathematical Society (AMS). xiv, 291 p. (1997).

This book is an outgrowth of lectures given by the authors at the Spring School of Potential Theory and Analysis held in Paseky, Czech Republic, in June 1993. A “fine” analysis of Sobolev spaces and associate nonlinear potential theory are developed. Then it is used to prove regularity of solutions of quasilinear elliptic equations and obstacle problems. The material in the book is contained in six chapters and it may be summarized as follows.

Chapter 1 contains preliminary facts including maximal function, estimates for Riesz potential and Sobolev spaces. Chapter 2 begins with the notion of \(p\)-capacity and moves toward a comparison between the techniques due respectively to De Giorgi, Nash, Moser to prove Hölder continuity for weak solutions of a linear elliptic equation in divergence form with \(L^\infty\) coefficients. Next, a proof of the Wiener criterion for a simple equation is given. The chapter ends with a treatment of the \(p\)-fine topology as well as fine Sobolev spaces.

Chapter 3 begins the analysis of quasilinear equations with general structure with coefficients in various functional spaces. Using techniques due essentially to Moser, Serrin, and Trudinger, it is shown the connection between \(p\)-capacity and boundary regularity.

This topic is the main concern of Chapter 4. This chapter includes energy estimates and Harnack inequality for a class of supersolutions. Then, the Wiener criterion is established and it is used to analyze equations of the form \[ -\text{ div} A(x,u,\nabla u) + B(x,u,\nabla u) = \mu \] with \(\mu\) a Radon measure. Next, the weak Harnack inequality for supersolutions is used in Chapter 5 to prove continuity of the weak solutions to some double obstacle problems.

Chapter 6 begins with the analysis of existence of solutions to variational inequalities assuming a coercivity condition on the coefficients of the leading part. Existence is proved via the technique of pseudo monotone operators. Then a Dirichlet problem for equations with differentiable structure is considered. The main result is that, for continuous boundary data, there exists a solution which is locally \(C^{1,\alpha}\) assuming continuity of the boundary values at all points of the boundary where the Wiener condition is satisfied.

The book is very well written and may be read at different levels. Some parts may be used in a post graduate course in advanced PDEs but for sure it is useful for all researchers who study regularity of solutions of elliptic PDEs via real analysis techniques.

Chapter 1 contains preliminary facts including maximal function, estimates for Riesz potential and Sobolev spaces. Chapter 2 begins with the notion of \(p\)-capacity and moves toward a comparison between the techniques due respectively to De Giorgi, Nash, Moser to prove Hölder continuity for weak solutions of a linear elliptic equation in divergence form with \(L^\infty\) coefficients. Next, a proof of the Wiener criterion for a simple equation is given. The chapter ends with a treatment of the \(p\)-fine topology as well as fine Sobolev spaces.

Chapter 3 begins the analysis of quasilinear equations with general structure with coefficients in various functional spaces. Using techniques due essentially to Moser, Serrin, and Trudinger, it is shown the connection between \(p\)-capacity and boundary regularity.

This topic is the main concern of Chapter 4. This chapter includes energy estimates and Harnack inequality for a class of supersolutions. Then, the Wiener criterion is established and it is used to analyze equations of the form \[ -\text{ div} A(x,u,\nabla u) + B(x,u,\nabla u) = \mu \] with \(\mu\) a Radon measure. Next, the weak Harnack inequality for supersolutions is used in Chapter 5 to prove continuity of the weak solutions to some double obstacle problems.

Chapter 6 begins with the analysis of existence of solutions to variational inequalities assuming a coercivity condition on the coefficients of the leading part. Existence is proved via the technique of pseudo monotone operators. Then a Dirichlet problem for equations with differentiable structure is considered. The main result is that, for continuous boundary data, there exists a solution which is locally \(C^{1,\alpha}\) assuming continuity of the boundary values at all points of the boundary where the Wiener condition is satisfied.

The book is very well written and may be read at different levels. Some parts may be used in a post graduate course in advanced PDEs but for sure it is useful for all researchers who study regularity of solutions of elliptic PDEs via real analysis techniques.

Reviewer: G.Di Fazio (Catania)

##### MSC:

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

35B65 | Smoothness and regularity of solutions to PDEs |

31C15 | Potentials and capacities on other spaces |

35J25 | Boundary value problems for second-order elliptic equations |

35J70 | Degenerate elliptic equations |

35J85 | Unilateral problems; variational inequalities (elliptic type) (MSC2000) |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

35J60 | Nonlinear elliptic equations |