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A limit model for thermoelectric equations. (English) Zbl 1255.35207

Summary: 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.

MSC:

35Q79 PDEs in connection with classical thermodynamics and heat transfer
35Q60 PDEs in connection with optics and electromagnetic theory
35D30 Weak solutions to PDEs
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