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 Equationby Cristian 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, C1 estimate for Monge-Ampere, C2 estimate.
• Existence of solution for the Dirichlet problem for the Monge-Ampere equation. Some historical results.