Volpert V. Elliptic Partial Differential Equations. Volume 1: Fredholm Theory of Elliptic Problems in Unbounded Domains

Springer Basel AG 2011. — 658 p. — (Monographs in Mathematics 101). — ISBN: 978-3-0346-0536-6.The theory of elliptic partial differential equations has undergone an important development over the last two centuries. Together with electrostatics, heat and mass diffusion, hydrodynamics and many other applications, it has become one of the most richly enhanced fields of mathematics. This monograph undertakes a systematic presentation of the theory of general elliptic operators. The author discusses a priori estimates, normal solvability, the Fredholm property, the index of an elliptic operator, operators with a parameter, and nonlinear Fredholm operators. Particular attention is paid to elliptic problems in unbounded domains which have not yet been sufficiently treated in the literature and which require some special approaches. The book also contains an analysis of non-Fredholm operators and discrete operators as well as extensive historical and bibliographical comments.Function Spaces and Operators
A Priori Estimates
Normal Solvability
Fredholm Property
Formally Adjoint Problems
Elliptic Problems with a Parameter
Index of Elliptic Operators
Problems in Cylinders
Non-Fredholm Operators
Nonlinear Fredholm Operators