Glasgow Mathematical Journal

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Glasgow Mathematical Journal (2010), 52:619-633 Cambridge University Press
Copyright © Glasgow Mathematical Journal Trust 2010
doi:10.1017/S0017089510000479

Research Article

THE BOUNDARY CONDITIONS DESCRIPTION OF TYPE I DOMAINS


MOHAMED EL-GEBEILYa1 and DONAL O'REGANa2

a1 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia e-mail: mgebeily@kfupm.edu.sa
a2 Department of Mathematics, National University of Ireland, University Road, Galway, Ireland
Article author query
el-gebeily m [Google Scholar]
o'regan d [Google Scholar]

Abstract

Type I domains are the domains of the self-adjoint operators determined by the weak formulation of formally self-adjoint differential expressions ℓ. This class of operators is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts agrees with the inner product 〈ℓu, v〉. A complete characterisation of the boundary conditions assumed by functions in these domains for second-order differential expressions is given in this paper. In the singular case, the boundary conditions are stated in terms of certain ‘boundary condition’ functions and in the regular case they are given in terms of classical function values.

(Received July 24 2009)

(Accepted June 16 2010)

2010 Mathematics Subject Classification34B15; 34A34