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2008 HMC Mathematics Conference on Nonlinear Functional Analysis: Abstracts

Peter Bates: “Mathematical Excursions Inspired by Materials Science”

Equations for a material that can exist stably in one of two homogeneous states are derived from a microscopic or lattice viewpoint with the assumption that the evolution follows a gradient flow of the free energy with respect to some metric. Alternatively, Newtonian dynamics can be considered.The resulting lattice dynamical systems are analyzed, as are equations on the continuum where the lattice interaction energy is viewed as an approximation to a Riemann integral. Several techniques of nonlinear functional analysis are used to examine the well-posedness of the equations, while others give asymptotics or quantitative behavior of special solutions, such as traveling waves or pulses.

Robert Borrelli & Courtney Coleman (Harvey Mudd College): Forty Years of Differential Equations in the Claremont Colleges

Harvey Mudd College graduated its first class in 1961, and in the intervening 47 years the mathematics curriculum at HMC has undergone many changes. In the beginning there was no differential equations course in the mathematics core, little or no use of computers in any mathematics course and no graduate program in mathematics. HMC and the Claremont Colleges have come a long way since then. In this talk we will trace the evolution of differential equations in the HMC mathematics curriculum and cite some milestones along the way.

Mónica Clapp: “Classical and Recent Results on Elliptic Problems with Critical Nonlinerarity”

Elliptic equations with critical nonlinearity arise in fundamental questions in Differential Geometry like Yamabe's problem or the prescribed scalar curvature problem. In this talk we shall consider the model problem

\[ \begin{aligned} -\Delta u=\left\vert u\right\vert ^{2^{\ast}-2}u\text{ in }\Omega,\qquad u=0 \text{ on } \partial\Omega, \end{aligned} \]

where $ \Omega $ is a bounded smooth domain in $\mathbb{R}^{N}$ $N\geq3$, and $ 2^{\ast}:=\frac{2N}{N-2}$ is the critical Sobolev exponent.

Despite its simple form, this problem has been an amazing source of open problems and new ideas. It has a rich geometric structure: It is invariant under the group of Möbius transformations. This fact turns it into an interesting and challenging problem because, though it can be expressed as a variational problem, the usual variational methods cannot be applied to it in a straightforward manner due to the lack of compactness caused by the Möbius invariance.

We shall review some well known results about existence and nonexistence of solutions to this problem, and present some new multiplicity results for positive and sign changing solutions, recently obtained in collaboration with M. Musso, F. Pacella, A. Pistoia, and T. Weth.

Yanyan Li: A Geometric Problem and the Hopf Lemma

A classical result of A.D. Alexandrov states that a connected compact smooth $n$-dimensional manifold without boundary, embedded in $R^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1} = \mathrm{constant}$ in case $M$ satisfies: for any two points
$(X', X_{n+1}), (X', \widehat X_{n+1})$ on $M$, with $X_{n+1} > \widehat X_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, we establish it under some additional conditions. Some variations of the Hopf Lemma will also be presented. Several open problems will be described. These are joint works with Louis Nirenberg.

Ratnasingham Shivaji: Population Dynamics with Diffusion

We study the steady state distribution of reaction diffusion equations arising in population dynamics. In particular, we consider strong Allee effect type growth rates and constant yield harvesting in bounded habitats. Assuming the exterior of the habitat is completely hostile, we establish existence and multiplicity results for positive solutions. We obtain our results via the method of sub super solutions.

Zhi-Quiang Wang: A Twisting Condition, Resonances, and Periodic Solutions of Hamiltonian Systems

In this talk, we first discuss a bit of background from the Poincaré-Birkhoff theorem to its applications of finding periodic solutions of Hamiltonian systems. The Poincaré-Birkhoff theorem assures the existence of two fixed points for any area preserving homeomorphisms $F$ on a two-dimensional annulus that satisfy a boundary twist condition. This boundary twist condition states that $F$ advances points on the outer edge of the annulus positively and points on the inner edge negatively. Then we discuss analogues of the twisting condition of the Poincaré-Birkhoff theorem in the setting of higher dimensional Hamiltonian systems, which are related to resonances of the Hamiltonian functions near zero and infinity. Finally we present some recent work on the existence of periodic solutions of Hamiltonian systems under a twisting condition which resembles more in spirit of the classical one in the Poincaré-Birkhoff theorem. This talk is intended for undergraduate students.