## 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

### MSC:

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