  We analyze the asymptotic behavior corresponding to the arbitrary high
conductivity of the heat in the thermoelectric devices. This work deals with a
steady-state multidimensional thermistor problem, considering the Joule effect
and both spatial and temperature dependent transport coefficients under some
real boundary conditions in accordance with the Seebeck-Peltier-Thomson
cross-effects. Our first purpose is that the existence of a weak solution holds
true under minimal assumptions on the data, as in particular nonsmooth domains.
Two existence results are studied under different assumptions on the electrical
conductivity. Their proofs are based on a fixed point argument, compactness
methods, and existence and regularity theory for elliptic scalar equations. The
second purpose is to show the existence of a limit model illustrating the
asymptotic situation.
