Variational Approach to Hyperbolic Free Boundary Problems (eBook)

This volume is devoted to the study of hyperbolic free boundary problems possessing variational structure. Such problems can be used to model, among others, oscillatory motion of a droplet on a surface or bouncing of an elastic body against a rigid obstacle. In the case of the droplet, for example, the membrane surrounding the fluid in general forms a positive contact angle with the obstacle, and therefore the second derivative is only a measure at the contact free boundary set. We will show how to derive the mathematical problem for a few physical systems starting from the action functional, discuss the mathematical theory, and introduce methods for its numerical solution. The mathematical theory and numerical methods depart from the classical approaches in that they are based on semi-discretization in time, which facilitates the application of the modern theory of calculus of variations. 


Seiro Omata was born in 1957 in Tokyo and received his Master and PhD degrees from Keio University. Since 1994 he has been working at Kanazawa University, becoming Full Professor in 2004. His research focuses on nonlinear evolutionary PDEs, including variational and free boundary problems, numerical analysis and applied analysis spanning numerous topics from fluid dynamics to mathematical finance. He devoted himself to the development of mathematics through the Mathematical Society of Japan, being invited as plenary speaker at its annual meeting in 2014 and serving as the chair of annual meeting in 2019. 

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This volume is devoted to the study of hyperbolic free boundary problems possessing variational structure. Such problems can be used to model, among others, oscillatory motion of a droplet on a surface or bouncing of an elastic body against a rigid obstacle. In the case of the droplet, for example, the membrane surrounding the fluid in general forms a positive contact angle with the obstacle, and therefore the second derivative is only a measure at the contact free boundary set. We will show how to derive the mathematical problem for a few physical systems starting from the action functional, discuss the mathematical theory, and introduce methods for its numerical solution. The mathematical theory and numerical methods depart from the classical approaches in that they are based on semi-discretization in time, which facilitates the application of the modern theory of calculus of variations.