Colloquium: Mark Allen (Purdue University)

 

Title:  Free Boundary on a Cone

 

Abstract:   The first problems a student encounters in a partial differential equations class involve looking for an unknown function which is a solution to an equation and has prescribed boundary data over the boundary of a fixed domain.However, in many applications in the physical sciences, biology, and even geometry, it becomes necessary to study a PDE with both an a priori unknown function and unknown domain. These types of problems have come to be called "free boundary problems". In this talk I will give some examples of free boundary problems that are notable both for their applications and for their importance in the historical development of this area of mathematics. I will discuss the importance of studying free boundary problems on manifolds that are not necessarily "smooth". I will also discuss recent work on studying the Alt-Caffarelli free boundary problem on a two dimensional cone generated by a smooth curve γ on the sphere. We show that when length(γ) < 2π the free boundary avoids the vertex of the cone. When length(γ) ≥ 2π we provide examples of minimizers such that the vertex belongs to the free boundary. This result is analogous to a well known result for distance minimizing geodesics on a cone. Indeed, the Alt-Caffarelli free boundary is known to behave similarly to minimal surfaces, and our result establishes another similarity between this free boundary problem and minimal surfaces. This is joint work with H. Chang Lara.

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