MATH 579 002 ST: Part Diff Equas 3 MWF 1100-1150

Instructor: Dimiter Vassilev, Associate Professor; **
Office**:
SMLC, Office 326; ** Email**:
vassilev@unm.edu

**Description**: The course will focus on fully non-linear equations, mainly related to the Monge-Ampere equation.

ATTENDANCE: Attendance at UNM is mandatory, see policy.

**
Textbooks: **No text book required. Class notes will be your main source. Useful books: **1. Fully Nonlinear Elliptic Equations** - **Luis A. Caffarelli** and **Xavier Cabré**, *Institute for Advanced Study* - AMS, 1995, 104 pp., ** **and 2**.****The Monge—Ampère Equation**by C**ristian E. Gutiérrez**, Progress in Nonlinear Differential Equations and Their Applications Volume 44, 2001.

**Covered Topics**

- Introduction and Background.
- Bernstein & de Giorgi type problems. The Jorgen-Calabi-Pogorelov theorem
- Basic regularity theory and Calculus of Variations.
- Elliptic regularity for divergence form linear equations in Sobolev and Holder spaces.
- Non-divergence form linear elliptic equations - Cordes-Nirenberg, Schauder, Calderon-Zygmund estimates.
- The Minkowski problem.
- The Second fundamental form. The Gauss map.
- Convex hyper-surfaces - Hadamard's theorem.
- Introduction to the Minkowski problem-the results of Pogorelov & Nirenberg, Cheng-Yau and Alexandrov.
- Mixed Volumes.
- Polarization formula.
- Approximation of closed convex hypersurface by convex polyhedra.
- Surface area measure.
- Minkowski and Brunn-Minkowski inequalities.
- Minkowski problem for convex polyhedra.

- Implicit function theorem in Banach spaces.
- Some "linear algebra". Derivatives of det etc.
- Uniqueness in the Minkowski problem. Mixed discriminants - Alexandrov-Fenchel inequality.
- Existence - continuity method.
- A-priori estimates – introduction.
- Fredholm operators.
- Hyperbolic polinomials. Examples. Lorentzian geometry. Sublinearity of the largest eigenvalue. Hyperbolicity cone. Inequalities for involving hyperbolic polynomials. Constructing hyperbolic polynomials. Examples.
- A-priori estimates for the Minkowski problem.
- Estimating the extrinsic diameter.
- k+1, alpha estimate for admissible functions.
- Fully non-linear equations.
- The Evans-Krylov estimate for fully non-linear equations. Krylov-Safonov’s Harnack inequality.
- The Dirichlet problem for fully non-linear elliptic equation Recall some Interpolation inequalities.
- Maximum and comparison principles.
- Strong and weak maximum principle.
- The Alexandrov maximum principle,
*C*1 estimate for Monge-Ampere,*C*2 estimate. - Existence of solution for the Dirichlet problem for the Monge-Ampere equation. Some historical results.