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In the present paper we introduce the perturbed two-dimensional Robinbi-Laplacian in the exterior of a bounded simply-connected $C^2$-smooth openset. The considered perturbation is of lower order and corresponds to tension.We prove that the essential spectrum of this operator coincides with thepositive semi-axis and that the negative discrete spectrum is non-empty if, andonly if, the boundary parameter is negative. As the main result, we obtain anisoperimetric inequality for the lowest eigenvalue of such a perturbed Robinbi-Laplacian with a negative boundary parameter in the exterior of a boundedconvex planar set under the constraint on the maximum of the curvature of theboundary with the maximizer being the exterior of the disk. The isoperimetricinequality is proved under the additional assumption that to the lowesteigenvalue for the exterior of the disk corresponds a radial eigenfunction. Weprovide a sufficient condition in terms of the tension parameter and the radiusof the disk for this property to hold.

An isoperimetric inequality for the perturbed Robin bi-Laplacian in a planar exterior domain