[CMC T2-2-B] 

Geometric Measure Theory and Optimal Transport

Date: August 7-10, 2017       Place: Rm. 1503, KIAS

Titles & Abstracts Home > Titles & Abstracts
Lecturer: Giovanni Alberti (University of Pisa)
Title: An introduction to Geometric Measure Theory and Finite Perimeter Sets
Abstract: In this course I will introduce the basic notions of Geometric Measure Theory (Hausdorff measures, rectifiable sets) and give an overview of the theory of Finite Perimeter Sets, including the proof of De Giorgi's rectifiability theorem. 
Then I will show how this theory can be used to prove (easy) existence result for some geometric variational problems. If time allows I will also touch some additional topics, such as the theory of integral currents, the basic regularity for minimizers, and the elliptic approximation of the area functional.

Lecturer: Kyeongsu Choi (MIT)
Pogorelov computations in Geometry and Analysis
Abstract: We will study Pogorelov computations in fully non-nlinear PDEs. And we will discuss about it applications in Geometry and Analysis such as Minkowski problems, Kahler geometry, Curvature flows, Optimal transportation, Minimal Lagrangian submanifolds, and standard Hessian equations. 

Lecturer: Aaron Palmer (UBC)
Globally Injective Equilibria in Second-Gradient Non-Linear Elasticity
Abstract: A hyper-elastic stored-energy function characterizes an elastic solid.  A stored-energy function that depends on the second-gradient of a deformation can describe materials with complex microscopic structure and provides the additional regularity that allows for a rigorous mathematical analysis of equilibria.  Local injectivity of deformations may be imposed either through assumptions on the stored-energy function or by restricting to incompressible deformations.  The additional constraint of non-interpenetration guarantees global injectivity and is both physically realistic and mathematically important.  We focus on the variational analysis of globally injective equilibria within a second-gradient model.  
The first lecture will cover some background of non-linear elasticity and present the assumptions under which we rigorously verify that the model yields physically realistic solutions. We will use the direct method of the calculus-of-variations to prove the existence of globally injective energy-minimizing deformations.  In the second lecture we will prove that the energy-minimizers satisfy a weak equilibrium equation.  This requires deriving a variational inequality and using a characterization of the admissible tangent cone to an energy-minimizing deformation to show the existence of a measure-valued surface traction.  Finally, in the third lecture we will discuss the extension of these techniques to handle domains with Lipschitz boundaries and the case of incompressible deformations if time permits.