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This procedure represents an extension of Castigliano's principle for stresses, in the formulation of Reissner [1], to thermoelasticity and heat conduction. It is also noted, that Biot's [3] variational procedure, which yields the elastic equilibrium equations and the heat conduction equation as the Euler equations, is a mixed one, in the sense that the former is one of the equations obtained in applying the extended Green procedure, while the latter belongs to the equations obtained using the extended Castigliano's procedure.6. Comparison of variational principles. Following Reissner's presentation and proof in isothermal elasticity, a comparison can be made between the values of I for functions tu , , etc. which are not solutions of SI = 0 and for functions r,-,-, etc. which are determined from 81 = 0. If both W and D are positive definite quadratic forms, the conclusion is reached that in the extended variational theorem for displacements (solid and entropy) one is concerned with a minimum problem, while in the extended variational theorem for stresses one is concerned with a maximum problem. In contrast to this, Reissner's general variational theorem, as extended to thermoelasticity and heat conduction, is only a stationary-value problem.

Existence and uniqueness of solutions in L p'q for the initial value problem for the Boussinesq equations that describe the flow of a viscous incompressible fluid subject to convective heat transfer is demonstrated via extensive use of the imbedding theorem for singular integral operators.

The identification of an unknown state-dependent source term in a reaction-diffusion equation is considered. Integral identities are derived which relate changes in the source term to corresponding changes in the measured output. The identities are used to show that the measured boundary output determines the source term uniquely in an appropriate function class and to show that a source term that minimizes an output least squares functional based on this measured output must also solve the inverse problem. The set of outputs generated by polygonal source functions is shown to be dense in the set of all admissible outputs. Results from some numerical experiments are discussed.

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