2010-2011 Analysis Applied Math Seminar

This is joint work with M. Peletier (TU Eindhoven). If time permits, I will also discuss work in progress with N. Le (Columbia) and Peletier which exploits the Gamma-limit structure of the energy to prove convergence of the associated gradient flows.

condensate, in the framework of a mean-field description (Gross-Pitaevskii), using a highly efficient and fast converging numerical scheme. For repulsive interactions, we find that the average spatial extension of the stationary density profile decreases with an increasing disorder strength. The numerical long-time behavior of the condensate will be addressed.

oceans, and large amplitude, long wavelength nonlinear waves can be produced in the interface and propagate over large distances. In some instances, the visible signature of internal waves on the surface of the ocean is a band of roughness which propagates at the same velocity as the internal wave. Several of the earliest observations are the most striking, consisting of long brightly shining strips of many kilometers in extent and photographed from the Space Shuttle.

I will discuss the asymptotic analysis of the coupling between the interface and the free surface of a two layers fluid in a scaling regime chosen to capture these observations, in which the internal mode is treated as a long wavelength nonlinear internal wave, while the surface mode is smaller and taken in a modulational regime.

This talk is based on joint work with Walter Craig and Philippe Guyenne.

space enjoys certain rigidity properties if $1\le k < n$. For example, its image cannot be contained in a ball of radius much less than 1. This is easily seen to be false if $k\ge n$, and there are dramatic counterexamples, due to Nash and Kuiper, showing that it is also false if one considers isometric embeddings that are merely $C1$. We consider isometric embedding of the Euclidean $n$-ball into $\mathbb{R}^{n+k}$ that belong to the Sobolev space $W^{2,p}$ for certain choices of $p\le n$. Maps with this regularity may fail to be $C1$. Nonetheless, we show that if $p \ge k+1$ then an isometric embedding is $C^{1,\alpha}$ for some positive $\alpha$, and enjoys rigidity properties similar to those of $C2$ isometric embeddings. These results are believed, but not known, to be optimal. This is joint work with Reza Pakzad.

The full and detailed analysis: Chapter 16 of My Future Monster Book

biharmonic equation $\Delta^2 u =\frac{\lambda}{(1-u)^2}$ on a ball $B \subset R^N$, under Navier boundary conditions $u=\Delta u=0$ on $\partial B$, where $\lambda >0$ is a parameter. It is known that there exists a $\lambda^{*}$ such that for $\lambda>\lambda^{*}$ there is no solution while for $\lambda<\lambda^{*}$ there is a branch of minimal solutions. Our main result asserts that the extremal solution $u^{*}$ is regular ($\sup_{B}u^{*}<1$) for $N\leq 8$ and it is singular ($\sup_{B}u^{*}=1$) for $N\geq 9$. Our proof for the singularity of extremal solutions in dimensions $N\geq 9$ is based on certain improved Hardy-Rellich inequalities.