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.


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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