Well-Posedness of the Lord–Shulman Variational Problem of Thermopiezoelectricity
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On the basis of the initial-boundary-value Lord–Shulman problem of thermopiezoelectricity, we formulate the corresponding variational problem in terms of the vector of elastic displacements, electric potential, temperature increment, and the vector of heat fluxes. By using the energy balance equation of the variational problem, we establish sufficient conditions for the regularity of input data of the problem and prove the uniqueness of its solution. To prove the existence of the general solution to the problem, we use the procedure of Galerkin semidiscretization in spatial variables and show that the limit of the sequence of its approximations is a solution of the variational problem of Lord–Shulman thermopiezoelectricity. This fact allows us to construct a reasonable procedure for the determination of approximate solutions to this problem.
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