Variational principles in potential theory

Summary Although the theoretical and practical importance of variational techniques in potential theory has long been secure, fresh instances of their utility are not without interest. Two cases in point are detailed, one relating to the capacitance of a condenser formed by a centrally placed strip within a circular shell and the other relating to the torsional rigidity of a circular cylindrical bar with a radial slit. The high degree of geometrical symmetry reflected in these configurations affords a corresponding diversity in approaches to the concomitant boundary value problems; and it is noteworthy that the exercise of different procedural options furnishes variational characterizations with a complementary nature. Thus, in the foregoing examples, a pair of newly devised variational principles are shown to provide opposite bounds for the capacitance and torsional rigidity, respectively, from those associated with alternative (though more readily established) principles. It appears, furthermore, that the new variational formulations are especially suited to the circumstances wherein the larger values of the ratio of the strip, or slit, width to the circular diameter obtain, whereas the others enjoy a corresponding fitness if the ratio is small compared with unity.This work was supported in part by the Office of Naval Research Contract Nonr-225(74). Reproduction in whole or in part is permitted for any purpose of the United States Government.