## An isoperimetric inequality for eigenvalues of the Stekloff problem.(English)Zbl 0971.35055

Summary: Let $$\Omega$$ be a bounded smooth domain in $$\mathbb{R}^n$$ and let $$0= \lambda_1\leq \lambda_2\leq\cdots$$ denote the eigenvalues of the Stekloff problem: $$\Delta u=0$$ in $$\Omega$$ and $$(\partial u)/(\partial v)=\lambda u$$ on $$\partial\Omega$$. We show that $$\sum^{n+1}_{i=2} \lambda^{-1}_i\geq n/\lambda^*_2$$, where $$\lambda^*_2$$ denotes the second eigenvalue of the Stekloff problem in a ball having the same measure as $$\Omega$$. The proof is based on a weighted isoperimetric inequality.

### MSC:

 35P15 Estimates of eigenvalues in context of PDEs 26D15 Inequalities for sums, series and integrals 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

### References:

 [1] : Isoperimetric inequalities and applications. Pitman, Boston 1980. · Zbl 0436.35063 [2] ; ; ; : A weighted isoperimetric inequality and applications to symmetrization. J. Inequ. Appl. (to appear), 20 pp. · Zbl 1029.26018 [3] Hersch, Z. Angew. Math. Phys. 19 pp 82– (1968) · Zbl 0165.12603 [4] Stekloff, Ann. Sci. Ecole Norm. Sup. 19 pp 455– (1902) [5] Szegö, J. Rat. Mech. Anal. 3 pp 343– (1954) [6] Weinberger, J. Rat. Mech. Anal. 5 pp 633– (1956) [7] Weinstock, J. Rat. Mech. Anal. 3 pp 745– (1954)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.