A problem related to the Hall effect in a semiconductor with an electrode of an arbitrary shape.

*(English)*Zbl 0929.35154Summary: A problem concerning the electric current in a semiconductor film from an electrode of an arbitrary shape is studied in the presence of a magnetic field. This situation describes the Hall effect, which indicates the deflection of electric current from electric field in a semiconductor. From a mathematical standpoint we consider the skew derivative problem for harmonic functions in the exterior of an open arc in a plane. By means of potential theory the problem is reduced to the Cauchy singular integral equation and next to the Fredholm equation of the second kind which is uniquely solvable. The solution of the integral equation can be computed by standard codes by discretization and inversion of the matrix. The uniqueness and existence theorems are formulated.

##### MSC:

35Q60 | PDEs in connection with optics and electromagnetic theory |

81V70 | Many-body theory; quantum Hall effect |

78A35 | Motion of charged particles |

31A10 | Integral representations, integral operators, integral equations methods in two dimensions |

82D37 | Statistical mechanical studies of semiconductors |