Zeros of Steklov eigenfunctions

A Steklov eigenfunction in a bounded domain is a harmonic function whose normal derivative at the boundary is proportional to the function itself. I will tell you most of what I know about the zero sets of such functions. A nice fact is that there are many zeros near the boundary: I will give a gentle proof of this in the first part of the talk. In the second, perhaps a little less gentle, part I will discuss some lower and upper bounds for the Hausdorff measure of the zero set; several questions remain unanswered. Comparisons with the (slightly) better understood case of eigenfunctions of the Laplace-Beltrami operator will also be provided.