We investigate an eigenvalue problem involving the 1-Laplacian operator on bounded domains with Neumann boundary conditions. The problem is not well posed in standard Sobolev spaces and must be studied in the space of functions of bounded variation, using tools from nonsmooth analysis. We give a geometric characterization of this eigenvalue in terms of a relative isoperimetric problem. This connection allows us to describe the shape of minimizers in domains with different geometries, and obtain results on uniqueness, multiplicity, symmetry, and symmetry breaking phenomena.

Joint work with S. McCurdy and A. Saldaña.