New Extremum Principles for Linear Problems

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Abstract
New extremum principles are derived in general terms for linear problems of physical origin. These principles are available in situations (of a heterogeneous nature, for instance) where the given linear operator can be approximated, from above or below, by one of a simpler kind. They furnish then upper and lower bounds on any linear functional of the solution, and on the physically important quadratic functional that defines the “energy”. They are compared to the familiar “classical” principles of minimum and complementary energy, and are found to offer some practical advantages.

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