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A bidimensional electromagnetic problem. (English) Zbl 0709.35029
The authors study the problem \[ \Delta \phi +2i\alpha^ 2(\phi +C_ k)=0\text{ in } \Omega_ k,\quad 1\leq k\leq N, \] \[ \Delta \phi =0\text{ in } {\bar \Omega}^ c={\mathbb{R}}^ 2-\cup \Omega_ k={\mathbb{R}}^ 2- \Omega, \] where \(\Omega_ k\subset {\mathbb{R}}^ 2\) are the cross sections of cylindrical electric conductors in which a current of angular frequency \(\omega\) runs. \(\alpha\) is a real constant, i is the imaginary unit and the \(C_ k\) are given complex constants. The potential \(\phi\) is supposed to have logarithmic behavior at infinity.
They first show that the problem has unique solution \(\phi \in W^{2,p}_{loc}({\mathbb{R}}^ 2)\), for any \(\rho\in (1,\infty)\), \(\phi \in C^{1,\beta}({\mathbb{R}}^ 2)\) for any \(\beta\in (0,1)\) and \(\phi \in C^{\infty}(\Omega \cup {\bar \Omega}).\)
They also consider a variational formulation and special situations which are of practical importance.
Global asymptotic estimates and some local estimates are proved as well as a boundary layer approximation.
Reviewer: R.Sperb

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B40 Asymptotic behavior of solutions to PDEs
78A45 Diffraction, scattering
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