Variational Principles for Problems with Linear Constraints: Prescribed Jumps and Continuation Type Restrictions

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Abstract
An abstract theory of problems subjected to linear constraints is developed. It supplies a general framework for boundary methods that are the subject of extensive research at present, as a tool to study numerically partial differential equations associated with many problems of science and engineering. Two kinds of general problem are considered, one for which the solutions are required to satisfy prescribed jumps and the other for which solutions can be continued smoothly into neighbouring regions as functions that satisfy given equations. The general theory is developed systematically, but only applications to variational principles are reported here. In previous papers the possibility of using this theory to discuss more general questions has been suggested; such applications will be discussed thoroughly in a further paper that is being prepared.

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