The Application of the Essentially Infinite-Dimensional Elliptic Operator to Functions f(x,u(x)) and f(x,u_1(x),...,u_m(x))

Abstract

Background. We consider the essentially infinite-dimensional elliptic operator (of the Laplace–Lévу type) (Lu)(x)=j(u′′(x))/2, proposed by Yu.V. Bogdansky (1977), for functions on the infinite-dimensional sepa­rable real Hilbert space H. This operator doesn’t have finite-dimensional analogues. It possesses the Leibniz property and vanishes on the cylindrical functions, being the second-order differential operator. The differen­tiation rules of the composite function f(u_1(x),...,u_m(x)) for the Laplace–Lévу operator and its modificati­ons were obtained by P. Lévу, E.M. Polishchuk, G.E. Shilov, I.Ya. Dorfman and V.Ya. Sikiryavyi. The dif­ferent rule was obtained by Yu.V. Bogdansky and Ya.Yu. Sanzharevsky for the Laplacian with respect to a measure in case of Gaussian measure.

Objective. The objective is to obtain the rules of application the essentially infinite-dimensional elliptic operator to com­posite functions f(x,u(x))  and f(x,u_1(x),...,u_m(x)).

Methods. We use the semigroup theory techniques and the generalized Stone–Weierstrass theorem.

Results. We prove the rules of application the essentially infinite-dimensional elliptic operator to composite functions f(x,u(x)) and f(x,u_1(x),...,u_m(x)). We also prove the similar rule for the essentially infinite-dimensional elliptic operator in bounded L-convex domains in the space H.

Conclusions. The results obtained in the paper generalize the known results for the Laplace–Lévу operator and its modifications. They are also similar to the classical differentiation of the composite function. The results can be used in further investigations of essentially infinite-dimensional equations.