Consider a curve on a closed surface endowed with a Riemannian metric. The Steklov transmission problem is to find continuous functions which are harmonic away from the curve, and such that the jump of the normal derivative across the curve is proportional to the value of the function. Such functions are called Steklov transmission eigenfunctions, and the corresponding proportionality coefficients are called Steklov transmission eigenvalues. We will discuss isoperimetric inequalities for these eigenvalues and highlight some similarities and differences compared to the usual Steklov case, in two main cases : the uncontrained setting, and the case where the metric are invariant under the action of some group. The talk is based on a joint work with Mikhail Karpukhin (UCL).
