## On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues.(English. Russian original)Zbl 1217.35125

Funct. Anal. Appl. 44, No. 2, 106-117 (2010); translation from Funkts. Anal. Prilozh. 44, No. 2, 33-47 (2010).
The Hersch-Payne-Schiffer inequalities for the Steklov eigenvalues are studied in this paper. If $$\Omega$$ is a simply connected bounded planar domain with Lipschitz boundary $$\partial\Omega$$ and $$p\geq 0$$ with $$p\in L^\infty(\partial\Omega)$$, we consider the problem $$\Delta u = 0$$ in $$\Omega$$, $$\partial u/\partial\nu= \sigma p u$$ on $$\partial\Omega$$ ($$\nu$$ unit outward normal vector) with $$M=\int_{\partial\Omega} p$$.
The main result ( Theorem 1.3.1) states that if $$\sigma(\Omega)$$ is the $$n$$th eigenvalue, there exists a family $$\Omega_\varepsilon$$ of such domains verifying $$\lim\sigma_n(\Omega_\varepsilon)M(\Omega_\varepsilon)= 2\pi n$$, $$n= 2,3,\dots$$ and $$\lim\sigma_n(\Omega_\varepsilon) \sigma_{n+ 1}(\Omega_\varepsilon) M(\Omega_\varepsilon)^2= 4\pi^2 n^2$$. These domains degenerate into a union of $$n$$ identical disks when $$\varepsilon$$ goes to $$0$$. This implies that the inequalities are sharp for all $$n> 1$$. It is also proved that $$\sigma_2(\Omega)M(\Omega)< 4\pi$$ (Theorem 1.3.6) by using 2 the Riemann mapping theorem.

### MSC:

 35P05 General topics in linear spectral theory for PDEs 49R05 Variational methods for eigenvalues of operators 35J25 Boundary value problems for second-order elliptic equations 47A75 Eigenvalue problems for linear operators
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### References:

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