Presented by: 
Lisa Beck, Augsburg University, Germany
Date: 
Tue 19 Aug, 3:00 pm - 4:00 pm
Venue: 
Building 67, Room 442

For a smooth function u : O -> R the n-dimensional area of its graph over a domain O subset R^n is given by int(O) (1 + |Du|^2)^1/2 dx. A natural question is whether or not minimizers of this functional exist among all functions taking prescribed boundary values (one may think of minimizers as soap films realizing the least surface area among all surfaces spanned by a wire). It turns out that solutions of the least area problem exist only in a suitably generalized sense. This formulation is based on an extension of the original functional to the space of functions of bounded variation via relaxation, where attainment of the prescribed boundary values is not mandatory, but non-attainment is penalized. Consequently, such generalized minimizers do not need to be unique.
 
In my talk I will first address this non-uniqueness phenomenon for the classical least area problem and explain the crucial role of the regularity of its solutions. Then I will discuss similar convex variational problems and present some modern approaches which allow to tackle such non-uniqueness phenomena. The results presented in this talk were obtained in collaboration with Thomas Schmidt (University of Erlangen, Germany).