Dr Mat Langford, The University of Newcastle
The course will provide a firm foundation for further study in the theory and application of nonlinear elliptic and parabolic partial differential equations.
Our primary goal will be a complete proof of the existence (and uniqueness) of solutions to a large class of quasilinear and fully nonlinear elliptic equations, motivated by a number of natural problems arising in engineering, pure and applied mathematics, and the sciences.
It should be emphasized however that, while the existence of solutions is certainly of fundamental importance, the analytical and geometrical methods which will be developed along the way have a far broader impact than the literal statement of our primary existence theorems.
Topics covered in this course include:
- PDE everywhere: motivating problems from biology, chemistry, computer science, geology, geometry, economics, engineering, materials science, physics (theoretical and applied), sociology, topology, and more!
- (Re)introduction to PDE: review of potential theory (the Laplace and Poisson equations).
- Introduction to nonlinear PDE. Or: How I learned to stop worrying and love a priori estimates (Hölder spaces, the Arzelà–Ascoli theorem and the method of continuity.)
- Weak and strong maximum principles; first a priori estimates
- Schauder’s estimate — Hölder continuity of the second derivatives of solutions to linear elliptic equations with Hölder continuous coefficients.
- Solubility in Hölder spaces of the Dirichlet problem for linear elliptic equations.
- Introduction to quasilinear equations and their applications.
- The Harnack inequality of de Giorgi, Nash and Moser — Hölder continuity of solutions to linear elliptic equations of divergence form with bounded measurable coefficients. The resolution of Hilbert’s nineteenth problem.
- Application to equations of mean curvature type.
- Introduction to fully nonlinear equations and their applications.
- Alexandrov’s maximum principle.
- Unlocking the door to fully nonlinear equations: the Harnack inequality of Krylov and Safanov — Hölder continuity of solutions to linear elliptic equations of nondivergence form with bounded measurable coefficients.
- Hölder continuity of the second derivatives of solutions to concave Hessian elliptic equations.
- Application to equations of Monge–Ampère type.
- The (Riemannian) Penrose inequality in general relativity — a foray into degenerate nonlinear PDE.
Students will certainly benefit from having taken an introductory course on ordinary and partial differential equations (at the undergraduate level). Absent such an introduction, they will be expected to have a very strong background in multivariable calculus. Some familiarity with metric spaces and Banach spaces will be expected. Some exposure to point-set topology, measure theory and the differential geometry of curves and surfaces may be helpful (but will not be necessary).
- 3 assignments 16.6% each (50% total)
- Final take home exam – 50%
- For those completing the subject for their own knowledge/interest, all students must complete all three assignments with a 30% pass grade as an attendance requirement.
- Gilbarg, David and Trudinger, Neil S.
Elliptic partial differential equations of second order.
Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp. ISBN: 3-540-41160-7
- Han, Qing
Nonlinear elliptic equations of the second order.
Graduate Studies in Mathematics, 171. American Mathematical Society, Providence, RI, 2016. viii+368 pp. ISBN: 978-1-4704-2607-1
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