Time: Mondays, Wednesdays, Fridays at 12pm.
Room: 1313 Mathematics Building.
Teacher: Y.A.
Rubinstein. Office hours: Friday 23pm or by appointment.
Course plan:
This will be a course in fully nonlinear elliptic PDE with a slant towards
equations that arise naturlly in geometry. The course should be useful to
*BEGINNING* graduate students even with little or no background in PDE. No
background in geometry is needed.
Prerequisites: Permission of the instructor.
1) The real MongeAmpere equation
 background from convex analysis
 The real MongeAmpere operator, following Alexandrov, RauchTaylor.
 Solving the Dirichlet problem.
 Solving the Cauchy problem, following RubinsteinZelditch.
2) Dirichlet duality theory
 Subequations.
 Facts on subaffine functions.
 Maximum principle.
 Boundary defining functions, barriers.
 Solving the Dirichlet problem, following HarveyLawson.
 Generalization to Riemannian manifolds.
3) Subequations arising in Lagrangian geometry
 The special Lagrangian equation.
 The degenerate special Lagrangian equation.
4) Further topics (time permitting)
 Potential theory for general subequations.
 Removable singularities for subequations.
 Restriction theorems for subequations.
 Geometric plurisubharmonicity.
The references for this course will be mainly based on the works of
HarveyLawson in 2) and 4), as well as on CaffarelliNirenbergSpruck
and RubinsteinSolomon in 3).
Lecture notes:
Lectures 110
(by X. Na)
References:
F.R. Harvey, H.B. Lawson, Jr.,
Dirichlet duality and the nonlinear Dirichlet problem.
F.R. Harvey, H.B. Lawson, Jr.,
Dirichlet duality and the nonlinear Dirichlet problem
on Riemanninan Manifolds.
Z. Slodkowski,
Pseudoconvex classes of functions. I. Pseudoconcave and pseudoconvex
sets.
Z. Slodkowski,
Pseudoconvex classes of functions. II. Affine pseudoconvex classes on
R^N.
Z. Slodkowski,
Pseudoconvex classes of functions. III. Characterization of dual
pseudoconvex classes on complex homogeneous spaces.
L. Caffarelli, L. Nirenberg, J. Spruck,
The Dirichlet problem for nonlinear secondorder elliptic equations.
III. Functions of the eigenvalues of the Hessian.
Y.A. Rubinstein, J.P. Solomon
The degenerate special Lagrangian equation.
Y.A. Rubinstein, S. Zelditch
The Cauchy problem
for the homogeneous MongeAmpere equation, II. Legendre transform.
F.R. Harvey, H.B. Lawson, Jr.,
Removable singularities for nonlinear subequations.
F.R. Harvey, H.B. Lawson, Jr.,
The restriction theorem for fully nonlinear subequations.
F.R. Harvey, H.B. Lawson, Jr.,
Geometric plurisubharmonicity and convexity: an introduction.
Requirements:
Occasional homework will be assigned in class. It will be beneficial for you to try to do all the
homework on your own or with fellow students but you are not required to submit it, with one
exception: if you are taking this course for credit you will be expected to type up solutions for
one of the homeworks (the instructor will assign each homework to a different student). These
solutions will then be posted for the benefit of the other students.
Schedule:
