A bidimensional electromagnetic problem. (English) Zbl 0709.35029

Crouzeix, Michel; Descloux, Jean

SIAM J. Math. Anal. 21, No. 3, 577-592 (1990).

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

Keywords:

cylindrical electric conductors; unique solution; Global asymptotic estimates; boundary layer