Brown Analysis Seminar
Abstract: We discuss a multiple layer potential approach for the treatment of boundary value problems associated with higher-order, constant coefficient, elliptic differential operators on smooth and Lipschitz domains. This study falls within the scope of the program outlined by A.P. Calderon in his 1978 ICM plenary address in which he advocates the use of layer potentials for much more general elliptic systems [than the Laplacian].
Scientific Computing Seminar
Abstract: A class of model prototype hybrid systems comprised of a microscopic Arrhenius surface process describing adsorption/desorption and/or surface diffusion of particles coupled to an ordinary or partial differential equation displaying bifurcations triggered by the microscopic process is presented and analyzed. These models arise from diverse applications ranging from surface processes and catalysis to atmospheric and oceanic sciences.
We derive and study closures for this hybrid system by employing different closure methods. We analyze the impact of microscopic fluctuations and interactions on the overall system transient and long time dynamics.
The effects are further demonstrated by Monte Carlo simulations which are employed to this purpose.
PDE Seminar
Abstract: Inspired by a result of Ambrosio, Lecumberry and Maniglia, in a joint work with Gianluca Crippa we show a simple derivation of integral logarithmic bounds for solutions of ordinary differential equations
$$ \left{ \frac {d \Phi(t,x)} {dt} } = b(t, \Phi (t,x))
(1)
\Phi (0, x) = x. $$
These bounds depend only on the ${\bf L}^\infty$ and $W ^{1,p} (p > 1) $ norm of $b $, and on the compressibility constant of $\Phi$ (which in turn can be bound by ||div^ {b}||\infty). These a-priori estimates allow to recover in a simple way many old and new results about existence, uniqueness, stability, and differentiability properties of solutions of ODE's with Sobolev coefficients. As new corollaries, we conclude that the Cauchy problem for transport equations with Sobolev coefficients preserve a mild regularity property of the initial data and we give an affirmative answer to the ${\bf L}^p$ version of a Conjecture of Bressan.
<--- 2006 Index