Nonlinear Potential Theory of Degenerate Elliptic Equations

This volume provides a detailed introduction to the nonlinear potential theory based on supersolutions to certain degenerate elliptic equations of the p-Laplacian type. Recent research has shown that such classical notions as blayage, polar sets, Perron's method and fine topology have their proper analogues in a nonlinear setting, and a coherent exposition of this natural extension of classical potential theory is presented. Yet fundamental differences to classical potential theory exist, and in many places a new approach is mandatory. Sometimes new, or long forgotten methods emerge that are applicable also to problems in classical potential theory. Quasiregular mappings constitute a natural field of applications and a careful study of the potential theoretic aspects of these mappings is included. The aim of the book is to explore the ground where partial differential equations, harmonic analysis and function theory meet. The quasilinear equations considered in this book involve a degeneracy condition given in terms of a weight function and, therefore, most results appear here for the first time in print. However, the reader interested exlusively in the unweighted theory will find new results, new proofs and a reorganization of the material as compared to the existing literature. No previous knowledge of the subject is required.Table of ContentIntroduction. 1: Weighted Sobolev spaces. 2: Capacity. 3: Supersolutions and the obstacle problem. 4: Refined Sobolev spaces. 5: Variational integrals. 6: A-harmonic functions. 7: A superharmonic functions. 8: Balayage. 9: Perron's method, barriers, and resolutivity. 10: Polar sets. 11: A-harmonic measure. 12: Fine topology. 13: Harmonic morphisms. 14: Quasiregular mappings. 15: Ap-weights and Jacobians of quasiconformal mappings. 16: Axiomatic nonlinear potential theory. Appendix I: The existence of solutions. Appendix II: The John-Nirenberg lemma. Bibliography. List of symbols. Index