ON MEASURE TRANSFORMATION IN CERTAIN VARIATIONAL PROBLEMS

摘要

This paper suggests a method of "measure transformation" for justifying and studying problems of variational nature and nonstandard type. Boundary-value problems in physics are often of variational nature and inherit the kind of integration used in describing the potential energy. As a rule, in this case, one uses the traditions of the ordinary Riemann integral, or, in the worst case, the Lebesgue integral (for the historical comments, see [7]). The general theory of the integral whose ideas go back to Stieltjes is not popular in the theory of boundary-value problems, which makes problems where the physical manifold being studied can have singularities escape attention. One can remove such singularities by a successful change of variables under the sign of the differential, which is called "measure transformation" in the theory of the integral. Below we present two types of problems in which sufficiently serious structural singularities of the object being studied that do not allow one to use the standard methods of the (Schwartz-Sobolev) distribution theory turn out to be available for analysis under a successful measure transformation.