MATH 5010: Topics in Analysis I: Geometric Measure Theory (Badger)
Description: This course will give an introduction to topics from geometric measure theory with a view towards applications in geometric and harmonic analysis. Course material will be from Evans and Gariepy's monograph, "Measure Theory and Fine Property of Functions", and additional papers from the literature. Main topics will include: (1) Covering theorems, Hausdorff measures, elementary structure theory (rectifiable versus purely unrectifiable sets) (2) Rademacher's theorem, functions of bounded variation, sets of finite perimeter (3) Reifenberg's algorithm -- i.e. parameterization tool used to prove regularity in the Plateau problem (4) Jones' / Okikiolu's traveling salesman theorem. The workload: attend class, no written work required
Prerequisites: Measure theory at the level of Math 5111
MATH 5010 - Section 1: Geometric Inequalities and Applications to Partial Differential Equations
Description: The first half of this course will focus on proofs of geometric inequalities including Sobolev, Poincare and Trudinger-Moser inequalities, etc. We will present some basic tools such as the Marcinkiewicz interpolation theorem, weak and strong type estimates for fractional integrals and use them to prove Poincare and Sobolev inequalities and Trudinger-Moser inequalities. If time allows, we will also prove the best constants for Sobolev and Trudinger-Moser inequalities. The prerequisite for this half is Lebesgue integration theory.
The second half will cover the critical point theory and the mountain pass theorem and use them to study nonlinear partial differential equations with nonlinearity of polynomial growth by Sobolev inequalities and of exponential growth by Trudinger-Moser inequalities.
Instructor: Guozhen Lu
Sections: Spring 2017 on Storrs Campus
|19437||5010||001||Lecture||TuTh 11:00:00 AM-12:15:00 PM||MONT 245||Lu, Guozhen|