2nd Edition

# Variational-Hemivariational Inequalities with Applications

By Mircea Sofonea, Stanislaw Migorski Copyright 2025
352 Pages 2 B/W Illustrations
by Chapman & Hall

Variational-Hemivariational Inequalities with Applications, Second Edition represents the outcome of the cross-fertilization of nonlinear functional analysis and mathematical modelling, demonstrating its application to solid and contact mechanics. Based on authors’ original results, the book illustrates the use of various functional methods (including monotonicity, pseudomonotonicity, compactness, penalty and fixed-point methods) in the study of various nonlinear problems in analysis and mechanics. The classes of history-dependent operators and almost history-dependent operators are exposed in a large generality. A systematic and unified presentation contains a carefully selected collection of new results on variational-hemivariational inequalities with or without unilateral constraints. A wide spectrum of static, quasistatic, dynamic contact problems for elastic, viscoelastic and viscoplastic materials illustrates the applicability of these theoretical results.

Written for mathematicians, applied mathematicians, engineers and scientists, this book is also a valuable tool for graduate students and researchers in nonlinear analysis, mathematical modelling, mechanics of solids, and contact mechanics.

New to the second edition

• Convergence and well-posedness results for elliptic and history-dependent variational-hemivariational inequalities
• Existence results on various optimal control problems with applications in  solid and contact mechanics
• Existence, uniqueness and stability results for evolutionary and differential variational-hemivariational inequalities with unilateral constraints
• Modelling and analysis of static and quasistatic contact problems for elastic and viscoelastic materials with looking effect
• Modelling and analysis of viscoelastic and viscoplastic dynamic contact problems with unilateral constraints.

I. Variational Problems in Solid Mechanics. 1. Elliptic Variational Inequalities. 1.1. Background on functional analysis. 1.2. Existence and uniqueness results. 1.3. Convergence results. 1.4. Optimal control. 1.5. Well-posedness results. 2. History-Dependent Operators. 2.1. Spaces of continuous functions. 2.2. Definitions and basic properties. 2.3. Fixed point properties. 2.4. History-dependent equations in Hilbert spaces. 2.5. Nonlinear implicit equations in Banach spaces. 2.6. History-dependent variational inequalities. 2.7. Relevant particular cases. 3. Displacement-Traction Problems in Solid Mechanics. 3.1. Modeling of displacement-traction problems. 3.2. A displacement-traction problem with locking materials. 3.3. One-dimensional elastic examples. 3.4. Two viscoelastic problems. 3.5. One-dimensional examples. 3.6. A viscoplastic problem. II. Variational-Hemivariational Inequalities. 4. Elements of Nonsmooth Analysis. 4.1. Monotone and pseudomonotone operators. 4.2. Bochner-Lebesgue spaces. 4.3. Subgradient of convex functions. 4.4. Subgradient in the sense of Clarke. 4.5. Mixed equilibrium problem. 4.6. Miscellaneous results. 5. Elliptic Variational-Hemivariational Inequalities. 5.1. An existence and uniqueness result. 5.2. Convergence results. 5.3. Optimal control. 5.4. Penalty methods. 5.5. Well-posedness results. 5.6. Relevant particular cases. 6. History-Dependent Variational-Hemivariational Inequalities. 6.1. An existence and uniqueness result. 6.2. Convergence results. 6.3. Optimal control. 6.4. A penalty method. 6.5. A well-posedness result. 6.6. Relevant particular cases. 7. Evolutionary Variational-Hemivariational Inequalities. 7.1. A class of inclusions with history-dependent operators. 7.2. History-dependent inequalities with unilateral constraints. 7.3. Constrainted differential variational-hemivariational inequalities. 7.4. Relevant particular cases. III. Applications to Contact Mechanics. 8. Static Contact Problems. 8.1. Modeling of static contact problems. 8.2. A contact problem with normal compliance. 8.3. A contact problem with unilateral constraints. 8.4. Convergence and optimal control results. 8.5. A contact problem for locking materials. 8.6. Convergence and optimal control results. 8.7. Penalty methods. 9. Time-Dependent and Quasistatic Contact Problems. 9.1. Physical setting and mathematical models. 9.2. Two time-dependent elastic contact problems. 9.3. A quasistatic viscoplastic contact problem. 9.4. A time-dependent viscoelastic contact problem. 9.5. Convergence and optimal control results. 9.6. A frictional viscoelastic contact problem. 9.7. A quasistatic contact problem with locking materials. 10. Dynamic Contact Problems. 10.1. Mathematical models of dynamic contact. 10.2. A viscoelastic contact problem with normal damped response. 10.3. A unilateral viscoelastic frictional contact problem. 10.4. A unilateral viscoplastic frictionless contact problem.

### Biography

Mircea Sofonea earned his PhD from the University of Bucarest, Romania, and his habilitation at the Université Blaise Pascal of Clermont-Ferrand (France).He is currently a Distinguished Profesor of Applied Mathematics at the University of Perpignan Via Domitia, France and a honorary member of the Institute of Mathematics, Romanian Academy of Sciences. His areas of interest and expertise include multivalued operators, variational and hemivariational inequalities, solid mechanics, contact mechanics and numerical methods for partial differential equations. Most of his reseach is dedicated to the Mathematical Theory of Contact Mechanics, of which he is one of the main contributors. His ideas and results were published in nine books, four monographs, and more than three hundred research articles.

Stanislaw Migórski earned his PhD degree and the habilitation from the Jagiellonian University in Krakow, Poland. He is currently a Full Honorary Professor and Chair of Optimization and Control Theory at Jagiellonian University in Krakow. His areas of interest and expertise include mathematical analysis, differential equations, mathematical modelling, methods and technics of nonlinear analysis, homogenization, control theory, computational methods and pplications of partial differential equations to mechanics. His research results are internationally recognized and were published in six books, four monographs, and more than two hundred research articles.