Title: Well-posedness concepts and convergence results

Time: 9:00 － 10:00 on June 19, 2024

Meeting Room: Math Building 2-1 Room

Speaker: Mircea Sofonea，University of Perpignan Via Domitia

Abstract: We start by recalling the concepts of Tykhonov and Levitin-Polyak well-posedness for minimization and variational inequality problems. These concepts are based on two main ingredients: the existence of a unique solution to the considered problem and the conver- gence to it of a special class of sequences, the so-called approximating sequences. Inspired by these properties, we define a new mathematical object, the so-called Tykhonov triple, denoted by T . Then, we introduce a new concept of well-posedness for abstract problems in metric spaces, the T -well-posedness concept, which extends both the Tykhonov and Levitin- Polyak well-posedness concepts, among others. The theory of T -well-posedness problems we construct gives necessary and sufficient conditions which guarantee the convergence to the solution of a nonlinear problem, unifies different convergence results and provides a frame- work in which the link between different problems can been established. It can be used in the study of a large class of nonlinear problems like fixed point problems, minimization problems, inequality problems and various inclusions, for instance. Details can be found in [1]. We illustrate the theory in the study of elliptic variational inequalities and history-dependent inclusions in Hilbert spaces. For these problems we state and prove T -well-posedness results and convergence criteria, as well. Finally, we apply our abstract results in the study of two mathematical models which describe the equilibrium of an elastic body with an obstacle. We also present numerical simulations which validate the corresponding convergence results. In this way we illustrate the cross fertilization between the abstract mathematical concepts, on one hand, and their applications in Contact Mechanics, on the other hand.

Introduction: Mircea Sofonea is a Distinguished Professor of Applied Mathematics at the University of Perpignan Via Domitia (UPVD) in France. He is also an Honorary Member of the Institute of Mathematics of the Romanian Academy of Sciences and previously served as the Chair of the Laboratory of Mathematics and Physics at UPVD from 2010 to 2020. Professor Sofonea is widely recognized as an expert in nonlinear analysis, partial differential equation theory, and its applications. He has been a member of the editorial board for more than 10 journals. In addition to his role at UPVD, Professor Sofonea has been a visiting professor at various universities and institutes in Europe, China, and the U.S.A. He has supervised twenty-eight Ph.D. students and has organized multiple international congresses and workshops. As a principal investigator, he has been involved in various research projects. Professor Sofonea has authored eight books, four monographs, and more than 300 research papers. Notably, he published a recent paper titled “Numerical analysis of hemivariational inequalities in contact mechanics” in Acta Numerica 2019. For more information, please visit his web page at http://perso.univ-perp.fr/sofonea/.

References

[1] Sofonea M., Well-posed Nonlinear Problems. A Study of Mathematical Models of Contact. Birkha ̈user, Cham (2023).

Inviter：Professor Fei Wang