Boundary value problems with strong nonlocalness for elliptic equations

A. B. Antonevich

Izvestiya. Mathematics , Tome 34 (1990) no. 1, pp. 1-21.

Résumé

Nonlocal boundary value problems are considered for elliptic equations of the following form. A nonperiodic mapping $g$ of the boundary to itself is given, and the boundary condition connects the values of the unknown function and its derivatives at the points $x,g(x),g(g(x)),\dots$. The author obtains necessary and sufficient conditions for the problem to be Noetherian (i.e. for the operator to be Fredholm) in terms of the invertibility of an auxiliary functional operator (the symbol of the problem), acting in a function space on the bundle of unit cotangent vectors to the boundary. Explicit necessary and sufficient conditions for the Noether property are presented for a number of examples. The main constructions and proofs are based on the theory of $C^*$-algebras generated by dynamical systems. Bibliography: 37 titles.

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