Abstract and Applied Analysis
- Abstr. Appl. Anal.
- Volume 2010 (2010), Article ID 902638, 18 pages.
Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space
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.
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 902638, 18 pages.
First available in Project Euclid: 1 November 2010
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Serov, Valery. Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space. Abstr. Appl. Anal. 2010 (2010), Article ID 902638, 18 pages. doi:10.1155/2010/902638. https://projecteuclid.org/euclid.aaa/1288620701