Solution of Continuous Nonlinear PDEs through Order Completion


  • M.B. Oberguggenberger, Institut für Mathematik und Geometrie, Universität Innsbruck, Innsbruck, Austria
  • E.E. Rosinger, Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

This work inaugurates a new and general solution method for arbitrary continuous nonlinear PDEs. The solution method is based on Dedekind order completion of usual spaces of smooth functions defined on domains in Euclidean spaces. However, the nonlinear PDEs dealt with need not satisfy any kind of monotonicity properties. Moreover, the solution method is completely type independent. In other words, it does not assume anything about the nonlinear PDEs, except for the continuity of their left hand term, which includes the unkown function. Furthermore the right hand term of such nonlinear PDEs can in fact be given any discontinuous and measurable function.
View full description


Book information

  • Published: July 1994
  • Imprint: NORTH-HOLLAND
  • ISBN: 978-0-444-82035-8

Table of Contents

Part I: General Existence of Solutions Theory. 1. Introduction. 2. Approximation of Solutions of Continuous Nonlinear PDEs. 3. Spaces of Generalized Functions. 4. Extending T(x,D) to the Order Completion of Spaces of Smooth Functions. 5. Existence of Generalized Solutions. 6. A Few First Examples. 7. Generalized Solutions as Measurable Functions. Part II: Applications to Specific Classes of Linear and Nonlinear PDEs. 8. The Cauchy Problem for Nonlinear First Order Systems. 9. An Abstract Existence Result. 10. PDEs with Sufficiently Many Smooth Solutions. 11. Nonlinear Systems with Measures as Initial Data. 12. Solution of PDEs and the Completion of Uniform Spaces. 13. Partial Orders Compatible with a Nonlinear Partial Differential Operator. 14. Miscellaneous Results. Part III: Group Invariance of Global Generalized Solutions of Nonlinear PDEs. 15. Introduction. 16. Group Invariance of Global Generalized Solutions of Nonlinear PDEs Obtained Through the Algebraic Method. 17. Group Invariance of Generalized Solutions Obtained Through the Algebraic Method: An Alternative Approach. 18. Group Invariance of Global Generalized Solutions Obtained Through the Order Completion Method. Appendix. References. Index.