Analysis & Applied Math

Event Information Estimates of first and second order shape derivatives in non smooth multi-dimensional domains. Applications to shape optimization problems with convexity constraint
13:10 on Friday March 27, 2015
14:00 on Friday March 27, 2015
BA6183, Bahen Center, 40 St. George St.
Arian Novruzi
http://mysite.science.uottawa.ca/novruzi/
University of Ottawa

Since Newton's study of the shape of the body of least resistance (1685), which he thought “... will be not without application in the building of ships”, it was believed that the optimal shape was smooth. Contrary to the general belief, recently it was discovered that this problem, which involves a convexity constraint for the domain, has a non smooth optimal shape solution.

I will present a general shape optimization problem with a convexity constraint for the domain variable, depending on a general second order linear elliptic PDE. I will show that, under appropriate assumptions, the optimal shape is a polygon (in dimension two) or a domain whose boundary has zero Gauss curvature wherever it is smooth (dimension greater than two).

These results are based on some sharp estimates for first and second order shape derivatives of shape functionals depending on second order linear elliptic PDEs around non smooth domains, essentially either Lipschitz or convex domains, or satisfying a uniform exterior ball condition, in any dimension greater or equal than two.