Differential Equations with Operator Coefficients

with Applications to Boundary Value Problems for Partial Differential Equations

Vladimir Kozlov, Vladimir Maz'ya (Autoren)

Buch | Softcover

XX, 444 Seiten

2010
Springer Berlin (Verlag)
978-3-642-08453-9 (ISBN)

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lt;p>The first systematic, self-contained presentation of a theory of arbitrary order ODEs with unbounded operator coefficients in a Hilbert or Banach space. Developed over the last 10 years by the authors, it deals with conditions of solvability, classes of uniqueness, estimates for solutions and asymptotic representations of solutions at infinity.

I. Differential Equations with Constant Operator Coefficients.- 1. Power-Exponential Zeros.- 2. Differential Operator Equations in Weighted Sobolev Spaces.- 3. Solutions in a Local Sobolev Space.- 4. Two-Weight L2-Estimates.- II. Differential Equations with Variable Operator Coefficients.- 5. Existence, Uniqueness and "Pointwise" Estimates.- 6. Corollaries of Previous Results Under Special Assumptions on L(t, Dt).- 7. Two-Weight L2-Estimates for Equations with Variable Coefficients.- 8. Connection of Solutions Corresponding to Different Strips.- 9. Applications to the Case of Perturbations Vanishing at Infinity.- 10. Variants and Extensions of the Previous Theory.- III. Asymptotic Theory of Operator Differential Equations.- 11. Complete Asymptotic Expansions Under Exponential and Power Perturbations of A(Dt).- 12. Reduction to a First Order System.- 13. General Asymptotic Representation.- 14. Power-Exponential Asymptotics.- 15. The Case of One Simple Eigenvalue on the Line.- 16. Several Simple Eigenvalues on the Line.- 17. The Case of a Single Multiple Eigenvalue.- A. Holomorphic Operator Functions.- A.1 Introduction.- A.2 Prerequisites on Fredholm Operators.- A.3 Basic Notions of the Spectral Theory of Holomorphic Operator Functions.- A.5 The Local Equivalence of Holomorphic Operator Functions.- A.6 The Smith Form of a Holomorphic Matrix Function.- A.7 The Resolvent of a Holomorphic Matrix Function.- A.8 Fredholm Holomorphic Operator Functions.- A.9 The Adjoint Holomorphic Operator Function.- References.- Index of Notation.- Index of Names.

From the reviews of the first edition:

"The book under review is the first systematic and self-contained presentation of a theory of arbitrary order ordinary differential equations with unbounded operator coefficients in a Hilbert or Banach space ... . this is an excellent book, that contains recent results of the topic, deep theoretical results and various applications to PDE-s. It is warmly recommended to specialists in ODE-s, PDE-s, functional analysis." (Jeno Hegedus, Acta Scientiarum Mathematicarum, Vol. 72, 2006)

Erscheint lt. Verlag	7.12.2010
Reihe/Serie	Springer Monographs in Mathematics
Zusatzinfo	XX, 444 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	697 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Schlagworte	asymptotics of solutions • Banach space • Boundary value problem • differential equation • Differential operator • eigenvalue • ordinary differential equation • Ordinary differential equations • Ordinary differential equations with operator coef • Ordinary differential equations with operator coefficients • partial di • partial differential equation • Partial differential equations
ISBN-10	3-642-08453-2 / 3642084532
ISBN-13	978-3-642-08453-9 / 9783642084539
Zustand	Neuware