Given two planar, conformal, smooth open sets Ω \Omega and ω \omega , we prove the existence of a sequence of smooth sets ( Ω ϵ ) (\Omega _\epsilon ) which geometrically converges to Ω \Omega and such that the (perimeter normalized) Steklov eigenvalues of ( Ω ϵ ) (\Omega _\epsilon ) converge to the ones of ω \omega . As a consequence, we answer a question raised by Girouard and Polterovich on the stability of the Weinstock inequality and prove that the inequality is genuinely unstable. However, under some a priori knowledge of the geometry related to the oscillations of the boundaries, stability may occur.

We study a shape optimization problem involving a solid $$K\subset {\mathbb {R}}^n$$ K ⊂ R n that is maintained at constant temperature and is enveloped by a layer of insulating material $$\Omega $$ Ω which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all $$(K,\Omega )$$ ( K , Ω ) with prescribed measure for K and $$\Omega $$ Ω , and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on $$\partial \Omega $$ ∂ Ω ) we obtain a full description of minimizers, while for general heat transfer conditions, we prove the existence and regularity of solutions and give a partial description of minimizers.

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We prove stability estimates for the isoperimetric inequalities for the first and the second nonzero Laplace eigenvalues on surfaces, both globally and in a fixed conformal class. We employ the notion of eigenvalues of measures and show that if a normalized eigenvalue is close to its maximal value, the corresponding measure must be close in the Sobolev space W −1,2 to the set of maximizing measures. In particular, this implies a qualitative stability result: metrics almost maximizing the normalized eigenvalue must be W −1,2 -close to a maximal metric. Following this approach, we prove sharp quantitative stability of the celebrated Hersch's inequality for the first eigenvalue on the sphere, as well as of its counterpart for the second eigenvalue. Similar results are also obtained for the precise isoperimetric eigenvalue inequalities on the projective plane, torus, and Klein bottle. The square of the W −1,2 distance to a maximizing measure in these stability estimates is controlled by the difference between the normalized eigenvalue and its maximal value, indicating that the maxima are in a sense nondegenerate. We construct examples showing that the power of the distance can not be improved, and that the choice of the Sobolev space W −1,2 is optimal.

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