  In this paper, we consider the motion of incompressible magnetohydrodynamics
(MHD) with resistivity in a domain bounded by a free surface. The free boundary
problem for MHD is an important problem not only for mathematical fluid
dynamics but also some application to the field of engineering In fact, when a
thermonuclear reaction is caused artificially, a high-temperature plasma is
sometimes subjected to a magnetic field and held in the air, and the boundary
of the fluid at this time is a free one. In this paper, an electromagnetic
field generated by some currents in an external domain keeps an MHD flow in a
bounded domain. On the free surface, free boundary conditions for MHD flow and
transmission conditions for electromagnetic fields are imposed. We proved the
local well-posedness in the general setting of domains from a mathematical
point of view. The solutions are obtained in the maximal regularity class, and
in particular, the regularity class of velocity fields is one more higher than
the regularity class of magnetic fields. To prove our main result, we use Lp-Lq
maximal regularity theorem for the Stokes equations with free boundary
conditions and for the magnetic field equations with transmission conditions.
