Part I, Viscosity solutions to nonlinear PDEs.

This part treats modern theory of PDE, with geometric methods.

Literature: L.A. Caffarelli and X. Cabre: Fully Nonlinear Elliptic Equations. You may also find other related books/literature on the web.

Topics: Tangent paraboloids and second order differentiability, Viscosity solutions of elliptic equations; Examples, Alexandroff estimate and maximum principle, Harnack Inequality, Uniqueness of solutions, Concave equations, Dirichlet problem.

Part II, Free boundary problems and Applications

I will mostly follow my book, and notes, that I will share with you. https://bookstore.ams.org/gsm-136

Place: Lectures take place at room 3418, at Math building, Tuesdays 13:15--15:00.

Examination: HW & presentation of a topic. For your presentation, maybe it is time to decide what you want to do. I am thinking of 25 min approx (powerpoint or on the blackboard, your choice). No need to write a report, just a topic of interest to the course. You may also choose parts in Cab-Caff book or from my notes, and depend more into that. We aim for first/second week of December.

Lecture Notes:	Download HW	Upload Solutions to HW
Crash-Course in Analysis methods in PDE
Fully Nonlinear Notes 1	HW1	HW1
Fully Nonlinear Notes 2	HW2	HW2
Fully Nonlinear Notes 3	HW3	HW3
Fully Nonlinear Notes 4
Free Boundary Notes 1 Free Boundary Notes 2 Includes several weeks notes and might be updated.	HW for FB HW1 (Due Nov. 4) HW2:(Due Nov 24)	Upload for HW1 HW2
Examples of projects: around 20--25 min. In each topic you may present the problem, general idess, and heuristics with at least one proof of some part of the argument. PDE: 1) Viscosity solutions for fully nonlinear parabolic problems 2) Boundary Harnack Principle 3) Existence theory: Variational Approach 4) Existence theory Perron’s supersol method 5) C^1,a regularity of viscosity solutions 6) Proofs of Morrey Type estimates FBP: 1) ACF monotonicity formula and its applications 2) Whitney’s extension theorem and its application in FB regularity 3) Discuss the classification of global solutions for obstacle problem 4) Epiperimetric inequality (what is it and how is it used) 5) Bernoulli type FBP, You may also choose a topic closer to your taste, and we can discuss it.