Partial differential equations 2. Functional analytic methods. With consideration of lectures by E. Heinz (Q5891515)

The second volume of the revised edition of this book presents functional analytic methods and applications to problems in differential geometry. The author is mainly interested in the following topics: (i) solvability of operator equations in Banach spaces; (ii) linear operators in Hilbert spaces and spectral theory; (iii) Schauder's theory of linear elliptic differential equations; (iv) weak solutions of differential equations; (v) nonlinear partial differential equations and characteristics; (vi) nonlinear elliptic systems; and (vii) boundary value problems from differential geometry. In Chapter 7 nonlinear operators in Banach spaces are considered. With the aid of Brouwer's degree of mapping Schauder's fixed point theorem and Banach's fixed point theorem are proved. Chapter 8 deals with the basic methods in the linear theory of operators in Hilbert spaces, such as eigenvalue problems, Sturm-Liouville theory, or Hilbert-Schmidt operators. In Chaper 9 the Riemann-Hilbert boundary value problem is solved by the integral-equation method. The next chapter is concerned with the theory of weak solutions in Sobolev spaces. Chaper 11 deals with some fundamental methods and tools in differential geometry, in relationship with some classes of partial differential equations. Nonlinear elliptic systems are carefully studied in Chapter 12 of this volume. In Chapter 13 the theory of nonlinear elliptic systems is applied to boundary value problems from differential geometry. The book will be a useful addition to the libraries of all those interested in the theory and applications of partial differential equations.