Department of Mathematical Sciences, University of Oulu, P. O. Box 3000, 90014 Oulu, Finland
Academic Editor: Martin D. Schechter
Copyright © 2010 Valery Serov. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the Friedrichs self-adjoint extension for a differential
operator of the form , which is defined on a bounded
domain , (for we assume that is a finite
interval). Here is a formally self-adjoint and a uniformly elliptic differential operator of order 2m with bounded smooth
coefficients and a potential is a real-valued integrable function
satisfying the generalized Kato condition. Under these assumptions
for the coefficients of and for positive large enough we obtain the
existence of Green's function for the operator and its estimates
up to the boundary of . These estimates allow us to prove the absolute and uniform convergence up to the boundary of of Fourier
series in eigenfunctions of this operator. In particular, these results
can be applied for the basis of the Fourier method which is usually
used in practice for solving some equations of mathematical physics.