Title: On a class of linear parabolic equations with degenerate coefficients
Abstract: We investigate a class of linear parabolic equations in divergence form with degenerate diffusion coefficients. Our study is motivated on the questions about the finer regularity of solutions to a class of degenerate viscous Hamilton-Jacobi equations. Suitable weighted spaces are found in which the existence and regularity estimates of solutions are proved under some partially VMO condition on the leading coefficients. The talk is based on the joint work with Hongjie Dong (Brown University) and Hung Tran (University of Wisconsin – Madison).

Title: Solitons and collapses in the models based on focusing nonlinear Schrodinger Equation.
Abstract: Nonlinear Schrodinger Equations with focusing nonlinearity plays an important role in nonlinear optics. It describes the self-focusing of light due to the Kerr effect. In this talk, I will discuss dynamics and stability properties of soliton solutions of models based on coupled NLS equations. These solutions were obtained numerically for various frequency ratios in both resonant and non-resonant regimes. These findings could lead to a better understanding of critical power for self-similar multi-color collapse and could lead to novel types of laser filamentation.

Title: Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary
Abstract: We consider the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution. To prove the result, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn’s lemma and a version of Kato’s inequality.

Title: A Constructive Definition of the Fourier Transform over a Separable Banach Space
Abstract: Gill and Myers proved that every separable Banach space, denoted $\mathcal{B}$, has an isomorphic, isometric embedding in $\mathbb{R}^{\infty}=\mathbb{R}\times\mathbb{R}\times\cdots$\ . They used this result and a method due to Yamasaki to construct a sigma-finite Lebesgue measure $\lambda_{\mathcal{B}}$\ for $\mathcal{B}$\ and defined the associated integral $\int_{\mathcal{B}}\cdot\ d\lambda_{\mathcal{B}}$\ in a way that equals a limit of finite-dimensional Lebesgue integrals. \\ \hspace{2in}\\ The objective of this talk is to apply this theory to developing a constructive definition of the Fourier transform on $L^1[\mathcal{B}]$. Our approach is constructive in the sense that this Fourier transform is defined as an integral on $\mathcal{B}$, which, by the aforementioned definition, equals a limit of Lebesgue integrals on Euclidean space as the dimension $n\to\infty$. Thus with this theory we may evaluate infinite-dimensional quantities, such as the Fourier transform on $\mathcal{B}$, by means of finite-dimensional approximation. As an application, we will apply the familiar properties of the transform to solving the heat equation on $\mathcal{B}$.