Liu Bie Ju
Centre for Mathematical Sciences
City University of Hong Kong
Mathematical Analysis and its Applications
Colloquium
Organized by Prof. Philippe G. Ciarlet and Prof. Roderick Wong
Minimal Surfaces and Optimal Control
by
Professor Claude Vallee
University of Poitiers, France
Date: April 20, 2005 (Wednesday)
Time: 4:30 pm to 5:30 pm
Venue: Room B6605 (Faculty Conference Room)
Blue Zone, Level 6
Academic Building
City University of Hong Kong
Abstract: We revisit the Plateau problem: Given a closed curve, find the surface with this curve as its boundary and whose area is minimal. The question we want to discuss is how to actually construct such a surface.
According to Pierre Ossian Bonnet's theorem (1867), a surface is known up to a rigid body displacement from the data of its first and second fundamental forms. Instead of defining a minimal surface by finding directly its parametric equations, we prefer to first identify its first and second fundamental forms, and then to construct the surface.
In this lecture, we will develop the following variational plan: "Minimize the area regarded as a functional of the two fundamental forms, taking into account the Gauss-Coddazzi-Mainardi conditions by means of Lagrange multipliers". We will show that these multipliers satisfy by duality a conjugate partial differential equation, as in optimal control when one applies the Pontryagin principle.
The well-known pioneering discovery of Jean-Baptiste Meusnier de La Place (1785) asserting that "the mean curvature (as defined later by Sophie Germain) has to be naught" will be identified as a compatibility condition of this conjugate equation.
(Tea, coffee and cookies will be provided at the Faculty Common Room in B6501 before the colloquium from 4:00 to 4:30 pm. Please come and join us!)
** All interested are welcome **
For enquiry: 2788-9816
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