☀️ Summer School Workshop III

On By yergenaikynIn Uncategorized

 🌟Summer School Workshop III 🌟

We are excited to host a follow-up workshop to the Summer School on “Direct and Inverse Problems with Applications, and Related Topics” ( August 19-23, 2024). This workshop will feature presentations by esteemed speakers from University College London.

Speakers:

  • Ilia Kamotski
  • Valery Smyshlyaev

Joint lecture topic: Analysis of a Class of Degenerating Operators and Their Spectra: Multiscale-type Approximations

Abstract

Homogenization theory establishes that uniformly elliptic operators with rapidly oscillating periodic coefficients have operators with constant coefficients as their appropriate limits. One approach is based on applying a (scaled) Floquet-Bloch-Gelfand transform to the resolvent problem in $\mathbb{R}^n$ and then constructing uniform approximations near a singular point of the quasi-periodicity parameter. This becomes much more subtle if the coefficients are asymptotically degenerating, and the limit operator appears to be a “two-scale” one with a band-gap limit spectrum. There are diverse examples of PDE and other models displaying similar features. In this minicourse, based on [1], we review the background and introduce an abstract scheme allowing to construct such approximations for a wide class of asymptotically degenerating operators and their spectra. This is accompanied by error bounds, and illustrated this by various examples. In particular, a key example of high-contrast PDE models displays surprising links to signal processing via a novel $L^2$-isometric two-scale interpolation operator of Whittaker-Shannon type.

[1] S. Cooper, I.V. Kamotski, V.P. Smyshlyaev, Uniform asymptotics for a family of degenerating variational problems and error estimates in homogenisation theory. arXiv:2307.13151 (2023) 

📅 Dates & Times:

  • Monday, November 4, 2024: 2:30 PM – 4:30 PM
  • Tuesday, November 5, 2024: 2:30 PM – 4:30 PM

📍 Location: Lecture Room 3.2, Building S8, Campus Sterre

#SummerSchool2024 #MathematicsWorkshop #InverseProblems #PDEs #GeometricAnalysis