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James Bisgard's Research Page

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.

  • Papers
    • "A Local Saddle Point Theorem and an Application to a nonlocal PDE" is a generalization of Rabinowitz' 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. The following is a preliminary version, and the final publication is available at Springer Link.
    • "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)
    • "On Mountain Pass Algorithms", Volume 20, Issue 4, in Nonlinear Differential Equations and Applications. The final publication is available at Springer Link or from Springer Nature Sharing. 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.
    • "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.
    • "Homoclinics for Hamiltonians with wells at different Levels" (in Calculus of Vartiation and Differential Equations, 2007) is an improvement of a 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 appendix of my thesis linked below.
    • "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.  Sadly, I think this has gone missing during an editorial change.  Any suggestions as to where this should go would be welcome!)
  • Slides for PNW MAA Talks:
  • Slides for MathFest 2014
  • Thesis
    • 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). There is also an appendix that contains the necessary results about nonlinear parabolic equations that are used in the last section. If you're a glutton for punishment, you can read all 264 pages of my thesis here.
  • Odds and Ends

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