Extremum Problems for Laplacian Eigenvalues

Fang-Hua Lin

Courant Institute of Mathematical Sciences

New York University

linf@cims.nyu.edu

    Eigenvalue Problems for Laplacians are among most studied ones in classical analysis, partial differential equations, calculus of variations and mathematical physics. In this lecture I shall discuss some recent progress on a couple extremum problems involving Dirichlet eigenvalues of the Laplacian. These problems have origins in shape optimization, pattern formation,..., and even the data science. We will show how they are related to harmonic maps into singular spaces and the some recent works on free boundary value problems involving vector valued functions.