Geodesic
On the Hersch--Payne--Schiffer inequalities for Steklov eigenvalues
Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 2, pp. 33-47

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We prove that the Hersch–Payne–Schiffer isoperimetric inequality for the $n$th nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all $n\ge 1$. The equality is attained in the limit by a sequence of simply connected domains degenerating into a disjoint union of $n$ identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch–Payne–Schiffer inequality for $n=2$ and show that it is strict in this case.
Keywords: Steklov eigenvalue problem, eigenvalue, isoperimetric inequality. 
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     author = {A. Girouard and I. V. Polterovich},
     title = {On the {Hersch--Payne--Schiffer} inequalities for {Steklov} eigenvalues},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {33--47},
     publisher = {mathdoc},
     volume = {44},
     number = {2},
     year = {2010},
     language = {ru},
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A. Girouard; I. V. Polterovich. On the Hersch--Payne--Schiffer inequalities for Steklov eigenvalues. Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 2, pp. 33-47. http://geodesic.mathdoc.fr/item/FAA_2010_44_2_a3/