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 A of the form A=A0+q(x)⋅, which is defined on a bounded domain Ω⊂ℝn, n≥1 (for n=1 we assume that Ω=(a,b) is a finite interval). Here A0=A0(x,D) is a formally self-adjoint and a uniformly elliptic differential operator of o...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2010/902638 |
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| author | Valery Serov |
| author_facet | Valery Serov |
| author_sort | Valery Serov |
| collection | DOAJ |
| description | We consider the Friedrichs self-adjoint extension for a differential
operator A of the form A=A0+q(x)⋅, which is defined on a bounded
domain Ω⊂ℝn, n≥1 (for n=1 we assume that Ω=(a,b) is a finite
interval). Here A0=A0(x,D) is a formally self-adjoint and a uniformly elliptic differential operator of order 2m with bounded smooth
coefficients and a potential q(x) is a real-valued integrable function
satisfying the generalized Kato condition. Under these assumptions
for the coefficients of A and for positive λ large enough we obtain the
existence of Green's function for the operator A+λI and its estimates
up to the boundary of Ω. These estimates allow us to prove the absolute and uniform convergence up to the boundary of Ω 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. |
| format | Article |
| id | doaj-art-657acd70ad774eb48665ac21953f608e |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-657acd70ad774eb48665ac21953f608e2025-02-03T06:12:20ZengWileyAbstract and Applied Analysis1085-33751687-04092010-01-01201010.1155/2010/902638902638Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato SpaceValery Serov0Department of Mathematical Sciences, University of Oulu, P. O. Box 3000, 90014 Oulu, FinlandWe consider the Friedrichs self-adjoint extension for a differential operator A of the form A=A0+q(x)⋅, which is defined on a bounded domain Ω⊂ℝn, n≥1 (for n=1 we assume that Ω=(a,b) is a finite interval). Here A0=A0(x,D) is a formally self-adjoint and a uniformly elliptic differential operator of order 2m with bounded smooth coefficients and a potential q(x) is a real-valued integrable function satisfying the generalized Kato condition. Under these assumptions for the coefficients of A and for positive λ large enough we obtain the existence of Green's function for the operator A+λI and its estimates up to the boundary of Ω. These estimates allow us to prove the absolute and uniform convergence up to the boundary of Ω 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.http://dx.doi.org/10.1155/2010/902638 |
| spellingShingle | Valery Serov Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space Abstract and Applied Analysis |
| title | Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space |
| title_full | Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space |
| title_fullStr | Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space |
| title_full_unstemmed | Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space |
| title_short | Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space |
| title_sort | green s function and convergence of fourier series for elliptic differential operators with potential from kato space |
| url | http://dx.doi.org/10.1155/2010/902638 |
| work_keys_str_mv | AT valeryserov greensfunctionandconvergenceoffourierseriesforellipticdifferentialoperatorswithpotentialfromkatospace |