Variational Inequalities: Theory and Applications
Description
This book provides a comprehensive and unified treatment of variational inequalities, quasi-variational inequalities, and related impulse control problems, combining rigorous mathematical analysis with effective numerical methods and practical applications. Such models arise naturally in constrained partial differential equations, free boundary problems, optimal control, reliability engineering, and mathematical finance.
The main objective of the book is to develop a coherent framework that connects functional analytic foundations, qualitative properties, numerical approximation, and long-time behavior. Particular attention is given to the analysis of existence, uniqueness, and regularity of solutions for elliptic and parabolic variational inequalities, as well as to quasi-variational problems involving solution-dependent constraints.
The book also presents robust and convergent numerical schemes, with a focus on finite element methods, penalty techniques, and approximation strategies that preserve the qualitative features of the continuous models. Special emphasis is placed on applications to free boundary problems, impulse control, and non-smooth phenomena, which play a central role in reliability modeling and American option pricing.
This volume is intended for graduate students, researchers, and practitioners in applied mathematics, numerical analysis, and related fields who are interested in both the theoretical and computational aspects of variational inequalities.
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Persons
Professor Salah Boulaaras is a Professor of Mathematics at the Department of Mathematics, College of Science, Qassim University, Saudi Arabia. His research interests include numerical analysis, partial differential equations, variational and quasi-variational inequalities, and mathematical modeling. He obtained his Ph.D. in numerical and functional analysis with distinction and was awarded the HDR (Habilitation à Diriger des Recherches) in numerical analysis from Annaba University, Algeria. He has more than 15 years of experience in teaching and research and has held academic positions at several universities in Algeria and Saudi Arabia. Professor Boulaaras is ranked among the top 2% of scientists worldwide over the past five years, according to internationally recognized bibliometric rankings. He has authored and coauthored more than 597 scientific publications, including journal articles and books, with many of his works published in internationally indexed journals. His research is characterized by a strong combination of theoretical analysis and numerical methods, with applications to nonlinear partial differential equations, variational inequalities, and complex physical models.
Professor Mohamed Haiour is a Professor of Mathematics at the Department of Mathematics, Badji Mokhtar University, Annaba, Algeria. He holds a Ph.D. in applied mathematics and his research focuses on variational inequalities, free boundary problems, impulse control, and numerical methods for constrained partial differential equations. His work covers both theoretical and computational aspects of elliptic and parabolic variational and quasi-variational inequalities, including regularity analysis, subsolution and supersolution techniques, finite element methods, and asymptotic behavior. He has published extensively in these areas and has supervised several graduate students. Professor Haiour has contributed to the development of reliable numerical approaches for applications in control theory, reliability engineering, and mathematical finance, particularly in the study of American options and impulse control models.
Content
General Introduction.- Foundations of Variational Inequalities.- Parabolic Variational Inequalities and Time Discretization.- Finite Element Approximation of Variational Inequalities.- Asymptotic Behavior and Long-Time Analysis.