Optimisation of the lowest Robin eigenvalue in the exterior of a compact   set
  We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact planar set, subject to attractive Robin boundary conditions. Under either a constraint of fixed perimeter or area, we show that the maximiser within the class of exteriors of convex sets is always the exterior of a disk. We also argue why the results fail without the convexity constraint and in higher dimensions. 
David Krejcirik and Vladimir Lotoreichik
math.SP math-ph math.AP math.MP math.OC