The UCF Mathematics Colloquium, held every Monday from 3:30pm to 4:30pm in MSB 318, offers a diverse platform for research scholars, faculty, students, and industry experts to share and exchange ideas, fostering discussion and networking across various areas of mathematics.
Dr. Stanley Snelson, Associate Professor from Florida Institute of Technology, will speak at this week's colloquium on shapes that optimize the eigenvalues of a differential operator.
Abstract: Geometric optimization problems for eigenvalues of the Laplacian have a long history, going back at least to the work of Rayleigh in the 19th century. The most quintessential result is the Faber-Krahn Theorem, which says that a ball minimizes the first Dirichlet eigenvalue of the Laplacian over all open subsets of R^n of a given volume. One can try to generalize this result in many different directions: higher eigenvalues, different boundary conditions, different constraints, or replacing the Laplacian with a more general operator. Apart from their theoretical interest, many of these questions have implications in applied fields such as structural design. In this talk, we will give a broad overview of this large and active body of research, and discuss some easy-to-state questions that surprisingly remain open. Finally, we will discuss some recent results, obtained in collaboration with E. Teixeira, on the existence of an optimal shape for the first Dirichlet eigenvalue of an elliptic operator with irregular coefficients.
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