##
**Nonsmooth variational problems and their inequalities. Comparison principles and applications.**
*(English)*
Zbl 1109.35004

Springer Monographs in Mathematics. New York, NY: Springer (ISBN 978-0-387-30653-7/hbk; 978-0-387-46252-3/ebook). xi, 395 p. (2007).

This monograph contains seven chapters, the bibliography of 223 entries, and an index.

Chapter 1, Introduction, pp. 1–10, explains the contents of the book.

Chapter 2, Mathematical preliminaries, pp. 11–80, contains a brief summary, mostly without proofs, of a large body of results from functional analysis.

Chapter 3, Variational equations, pp. 81–142, deals with semilinear and quasilinear elliptic and parabolic boundary value problems. Sign-changing solutions are studied via Fu\(\check{c}\)ik spectrum.

Chapters 4–7 contain many results due to the authors.

Chapter 4, Multivalued variational equations, pp. 143–210, deals with differential inclusions and nonlinear boundary value problems with discontinuous nonlinearities. The notion of sub- and supersolutions is defined and used, Clarke’s gradient is defined and plays an important role in the theory, and a comparison principle for parabolic inclusions with local growth is given.

Chapter 5, Variational inequalities, pp. 211–278, deals with the method of sub- and supersolutions. Solvability of noncoercive inequalities and the existence of extremal solutions for these inequalities are established under suitable assumptions. Evolutionary variational inequalities are treated, monotone penalty approximations are considered, and systems of variational inequalities are studied.

Chapter 6, Hemivariational inequalities, pp. 279–318, deals with some generalization of variational inequalities involving Clarke’s generalized gradient. The notion of sub- and supersolutions is defined, a comparison principle is obtained, extremal solutions are studied, evolutionary hemivariational inequalities are treated.

Chapter 7, Variational- hemivariational inequalities, pp. 319-378, deals with the inequalities of the form: \[ (Au-f, v-u)+\psi(v)-\psi(u)+\int_{\Omega}j^o(u;v-u)\,dx\geq 0, \quad \forall v\in V_0, \] where \(\Omega\in \mathbb R^N\) is a bounded domain with Lipschitz boundary, \(V_0\) is a Sobolev space \(W^{1,p}_0(\Omega)\), \(j^o\) is a generalized directional derivative of a locally Lipschitz function \(j: \mathbb R\to \mathbb R\), and \(A\) is an operator of Leray-Lions type. For these inequalities the sub- and supersolutions are defined, a comparison principle is derived, the existence of extremal solutions is proved, evolutionary variational-hemivariational inequalities are studied, nonsmooth critical points theory is discussed, a constraint hemivariational inequality is considered, and an eigenvalue problem for a variational-hemivariational inequality is discussed.

This well-written book contains a large number of material. It can be useful for graduate students and researchers interested in variational methods. In the beginning of each chapter the authors give some motivation for the material of the chapter.

In the formulation of Theorem 2.9 (the principle of uniform boundedness) the assumption that \(F\) are linear operators is missing.

Chapter 1, Introduction, pp. 1–10, explains the contents of the book.

Chapter 2, Mathematical preliminaries, pp. 11–80, contains a brief summary, mostly without proofs, of a large body of results from functional analysis.

Chapter 3, Variational equations, pp. 81–142, deals with semilinear and quasilinear elliptic and parabolic boundary value problems. Sign-changing solutions are studied via Fu\(\check{c}\)ik spectrum.

Chapters 4–7 contain many results due to the authors.

Chapter 4, Multivalued variational equations, pp. 143–210, deals with differential inclusions and nonlinear boundary value problems with discontinuous nonlinearities. The notion of sub- and supersolutions is defined and used, Clarke’s gradient is defined and plays an important role in the theory, and a comparison principle for parabolic inclusions with local growth is given.

Chapter 5, Variational inequalities, pp. 211–278, deals with the method of sub- and supersolutions. Solvability of noncoercive inequalities and the existence of extremal solutions for these inequalities are established under suitable assumptions. Evolutionary variational inequalities are treated, monotone penalty approximations are considered, and systems of variational inequalities are studied.

Chapter 6, Hemivariational inequalities, pp. 279–318, deals with some generalization of variational inequalities involving Clarke’s generalized gradient. The notion of sub- and supersolutions is defined, a comparison principle is obtained, extremal solutions are studied, evolutionary hemivariational inequalities are treated.

Chapter 7, Variational- hemivariational inequalities, pp. 319-378, deals with the inequalities of the form: \[ (Au-f, v-u)+\psi(v)-\psi(u)+\int_{\Omega}j^o(u;v-u)\,dx\geq 0, \quad \forall v\in V_0, \] where \(\Omega\in \mathbb R^N\) is a bounded domain with Lipschitz boundary, \(V_0\) is a Sobolev space \(W^{1,p}_0(\Omega)\), \(j^o\) is a generalized directional derivative of a locally Lipschitz function \(j: \mathbb R\to \mathbb R\), and \(A\) is an operator of Leray-Lions type. For these inequalities the sub- and supersolutions are defined, a comparison principle is derived, the existence of extremal solutions is proved, evolutionary variational-hemivariational inequalities are studied, nonsmooth critical points theory is discussed, a constraint hemivariational inequality is considered, and an eigenvalue problem for a variational-hemivariational inequality is discussed.

This well-written book contains a large number of material. It can be useful for graduate students and researchers interested in variational methods. In the beginning of each chapter the authors give some motivation for the material of the chapter.

In the formulation of Theorem 2.9 (the principle of uniform boundedness) the assumption that \(F\) are linear operators is missing.

Reviewer: Alexander G. Ramm (Manhattan)

### MSC:

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

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

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

35K85 | Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |