My research is about the applications of variational methods to differential
equations. I am interested in the existence of homoclinic and
heteroclinic connections in time-dependent Hamiltonian systems. Recently, I've gotten interested in numerical methods for finding minimax type solutions of differential equations.
Homoclinics for a Hamiltonian with wells at different level (in Calculus of Vartiation and Differential Equations, 2007) is an improvement of this part of my thesis. In particular, the regularity assumptions are much improved over the thesis. Some of the results on semi-linear parabolic PDE that are used are proved in the notes linked below.
A Compact Embedding for Sequence Spaces This has a proof of a special case of the Rellich-Kondrachov Compactness Theorem for Sobolev spaces, in essence using Fourier series. The proof as given is accessible to an undergraduate who has had a year of real analysis. It appears in Volume 24, Issue 2 of the Missouri Journal of Mathematical Sciences.
On Mountain Pass Algorithms, Volume 20, Issue 4, in Nonlinear Differential Equations and Applications. The final publication is available at Springer Link. As the title suggests, it concerns finding minimax type critical points. In the mountain pass setting, by deforming an initial path downhill using a (negative) gradient like flow, a point on the initial path is found that flows downhill to a mountain pass point. More generally, in the setting of the saddle point theorem or a more general linking setting, an initial surface is deformed by using a negative gradient flow and a point is found on the initial path that flows down to a critical point. In addition, the paper proves that the high points along these deformed surfaces converge to a critical point of the correct type if the Palais-Smale condition is satisfied. This represents a step towards proving convergence for some mountain pass type algorithms.
Mountain Passes and Saddle Points This is an introduction to two important critical point theorems: the mountain pass theorem and the saddle point theorem, published in the June 2015 issue of SIAM Review (SIAM Rev. 57-2 (2015), pp. 275-292)
A Local Saddle Point Theorem and an Application is about a local saddle point theorem. In Rabinowitz' saddle point theorem, knowledge is assumed about an entire subspace. Here, this assumption is replaced by assumptions about the behavior of a functional close to the origin. In some sense, this saddle point theorem generalizes Rabinowitz'. I'm still working on the application, and hope to have another more standard application in the future.
Chaos in the Forced Pendulum This is a short (?) paper on how the calculus of variations can be used to show that forced pendulum is chaotic. (Submitted to Mathematics Magazine, September 2010)
Slides for PNW MAA talks:
The introduction of my thesis has some
information about what I actually did and some things I'm currently interested in.
These next two parts are about gluing certain types of solutions together. The
first part describes how certain Palais-
Smale sequences split. The second part deals
with how to find two-bump solutions that are close to concatenation of mountain
pass type critical points.
The final part deals with homoclinic solutions for a Hamiltonian with wells at different levels (as opposed to the common pendulum type Hamiltonian).
If you're a glutton for punishment, you can read all 264 pages of my thesis here.
nonlinear parabolic equations
A nonlinear Sturm comparison Theorem
The first is for solving the nonlinear parabolic equation above and seeing how it deforms curves, which is related to the 2007 paper in Calculus of Variations.
The second is a little notebook about the forced, inverted pendulum.
An algorithm for solving Fx = b when F has positive and negative eigenvalues (i.e. F is of saddle point type):