Carmen Judith Vanegas y Wolfgang Tutschke

Objectivo: Fixed-point methods can be applied not only to ordinary differential equations but also to partial differential equations. Using Scl1auder's Fixed-Point Theorem instead of the contraction mapping principle, one gets statements under weaker assumptions.

The course is aimed at

• Formulation and proof of Brouwer's and Scl1auder's Fixed-Point Theorems
• Application of Schauder's Fixed-Point Theorem to partial differential equations
• Since functional-analytic methods for partial differential equations have a growing meaning for applications, some parts of the course can contribute to an updating of basic courses on mathematical analysis

Prerrequisitos: Basic knowledge on ( ordinary and partial) differential equations and on Complex Analysis

Programa

• Brouwer's Fixed-Point Theorem
• Two versions of Schauder's Fixed-Point Theorem. Lemma of Mazur
• Proof of Peano's Theorem on ordinary differential equations
• Distributional solutions of partial differential equations. Fundamental solutions of elliptic equations
• Inversion of partial differential operators by singular integrals
• Integral representations
• Reduction of boundary value problems for non-linear partial differential equations to fixed-point problems.
• Solution of boundary value problems
• Comparison with the contraction-mapping principle

Bibliografia

R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II, Partial differential equations (reprint of the 1962 original). New York 1989.
G. F. Manjavidze and W. Tutschke, Some boundary value problems for first-order non-linear partial differential systems in the plane (in Russian). Boundary value problems of the theory of generalized analytic functions and their applications, 79-124, Tbilis. Gos. Univ., Tbi1isi 1983. Engl. translation in preparation.
J. Naas and W. Tutschke, Great theorems and beautiful proofs in Mathematics (in German). 2-ed. Verlag Harri Deutsch 1997.
W. Pogorzelski, Integral Equations and their Applications. Pergamon Press, Warsaw 1966.
W. Tutschke, Partial differential equations. Classical, functional-analytic, and complex methods (in German). Teubner- Text zur Mathematik, vol. 27, Leipzig 1983.
W. Tutschke, Real and complex fundamental solutions - a way for unifying mathematical analysis. Bol. Asoc. Mat. Venez., vol. 9, No.2, 141-179.
C. J. Vanegas, Nonlinear perturbations of systems of partial differential equations with constant coefficients. Electron. J. Differential Equations No.05, 10 pp., 2000.
I. N. Vekua, Generalized Analytic Functions. Pergamon Press 1962.