Vol. 69, No. 2, 1977

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A Sobolev space and a Darboux problem

Manda Butchi Suryanarayana

Vol. 69 (1977), No. 2, 535–550

This paper deals with a class of functions which are defined in an n-dimensional rectangle and which possess there, only the generalized partial derivatives of mixed type. It is shown that (i) this class contains as a proper subset the usual Sobolev class of order n, the dimeniion of the domain and (ii) this class can be imbedded in the space of continuous functions. In addition to the compactness of the imbedding operator, the closedness of certain nonlinear partial integro differential operators is also studied. Finally, a system of partial integro differential equations with Darboux type boundary data in a rectangle, is shown to have solutions in this class. The results of this paper are used in certain existence theorems of optimal control theory.

Mathematical Subject Classification 2000
Primary: 46E35
Secondary: 49A20
Received: 27 February 1974
Published: 1 April 1977
Manda Butchi Suryanarayana