Topics in fully nonlinear PDE (course 858Y)

University of Maryland

Department of Mathematics

Autumn 2015






Time: Mondays, Wednesdays, Fridays at 12pm.
Room: 1313 Mathematics Building.

Teacher: Y.A. Rubinstein. Office hours: Friday 2-3pm 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 Monge-Ampere equation
- background from convex analysis
- The real Monge-Ampere operator, following Alexandrov, Rauch-Taylor.
- Solving the Dirichlet problem.
- Solving the Cauchy problem, following Rubinstein-Zelditch.

2) Dirichlet duality theory
- Subequations.
- Facts on subaffine functions.
- Maximum principle.
- Boundary defining functions, barriers.
- Solving the Dirichlet problem, following Harvey-Lawson.
- 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 Harvey-Lawson in 2) and 4), as well as on Caffarelli-Nirenberg-Spruck and Rubinstein-Solomon in 3).

Lecture notes:
Lectures 1-10 (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 second-order 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 Monge-Ampere 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:

  • Lecture 1
    Overview. The subdifferential of a convex function. The real Monge-Ampere operator - definition.

  • Lecture 2
    The real Monge-Ampere operator - construction as a Borel measure using Alexandrov's theorem. Solution of the Dirichlet problem for the homogeneous real Monge-Ampere equation using an upper envelope.

  • Lecture 3
    The Cauchy problem for the homogeneous real Monge-Ampere equation. The Legendre transform in more detail. Convex hulls and the double Legendre dual: regularity and basic properties.

  • Lecture 4
    Obstruction to the solution of the Cauchy problem for the homogeneous real Monge-Ampere equation: upper and lower bounds on the subdifferential and strict convexity of the Legendre subsolution.

  • Lecture 5
    Obstruction to the solution of the Cauchy problem for the homogeneous real Monge-Ampere equation: completion of the proof of the main theorem. Subequations: definition.

  • Lecture 6
    Subequations: basic properties. Subaffine functions. Functions of type F. Comparison with viscosity solutions.

  • Lecture 7
    Subequations: rays sets and boundary defining functions.

  • Lecture 8
    Subequations: boundary defining functions for domains with non-smooth boundary, part 1.

  • Lecture 9
    Subequations: boundary defining functions for domains with non-smooth boundary, part 2.

  • Lecture 10
    Solving the Dirichlet problem for domains with non-smooth boundary: the main theorem.

  • Lecture 11
    Solving the Dirichlet problem for domains with non-smooth boundary: boundary defining functions for boundary components.

  • Lecture 12
    The spacetime Lagrangian angle.

  • Lecture 13
    The subequation associated to the spacetime Lagrangian angle.

  • Lecture 14
    The subequation associated to the spacetime Lagrangian angle via degenerate ellipticity.

  • Lecture 15
    Constructing a solution to the Dirichlet problem associated to the spacetime Lagrangian angle.

  • Lecture 16
    Subequations on Riemannian manifolds: Overview and main results.

  • Lecture 17
    Subequations on Riemannian manifolds: F-subharmonic functions, the Riemannian Hessian.

  • Lecture 18
    Subequations on Riemannian manifolds: topological G-structures, local jet equivalence.

  • Lecture 19
    Subequations on Riemannian manifolds. Strictly F-subharmonic functions, local vs. global comparison results.

  • Lecture 20
    Subequations on Riemannian manifolds: local vs. global comparison results (continued), monotonicity subequations and strict approximation.

  • Lecture 21
    Subequations on Riemannian manifolds: affine jet equivalence, the Calabi-Yau equation.

  • Lecture 22
    Subequations on Riemannian manifolds: Strictly F-convex boundaries.

  • Lecture 23
    Subequations on Riemannian manifolds: Strictly F-convex boundaries (continued). The Dirichlet problem.

  • Lecture 24
    Subequations on Riemannian manifolds: The Dirichlet problem (continued).

  • Lecture 25
    Guest lecture by H.B. Lawson, Jr.

  • Lecture 26
    The theorem on the sums.

  • Lecture 27

  • Lecture 28

  • Lecture 29

  • Lecture 30

  • Lecture 31

  • Lecture 32

  • Lecture 33

  • Lecture 34

    Lecture 35

  • Lecture 36

    Lecture 37

  • Lecture 38

    Lecture 39

  • Lecture 40

    HW1.

    HW1 solutions (please leave comments/questions in the Forum discussion "HW1 solutions").

    HW2.

  • Lecture 5

  • Lecture 6

    HW3.

  • Lecture 7

  • Lecture 8

  • Lecture 9

    HW4.

  • Lecture 10

  • Lecture 11

  • Lecture 12

  • Lecture 13

  • Lecture 14

  • Lecture 15

    HW5.

  • Lecture 16

  • Lecture 17

  • Lecture 18

  • Lecture 19

  • Lecture 20

  • Lecture 21

  • Lecture 22

  • Lecture 23

  • Lecture 24