Juha Heinonen, Tero Kilpeläinen and Olli Martio

Introduction
1. Weighted Sobolev spaces
2. Capacity
3. Supersolutions and the obstacle problem
4. Refined Sobolev spaces
5. Variational integrals
6. Harmonic functions
7. Superharmonic functions
8. Balayage
9. Perron's method, barriers, and resolutivity
10. Polar sets
11. Harmonic measure
12. Fine topology
13. Harmonic morphisms
14. Quasiregular mappings
15. Ap-weights and Jacobians of quasiconformal mappings
16. Axiomatic nonlinear potential theory
17. Appendix I: The existence of solutions
18. Appendix II: The John-Nirenberg lemma
Bibliography
List of symbols
Index

A self-contained treatment appropriate for advanced undergraduate and graduate students, this volume offers a detailed development of the necessary background for its survey of the nonlinear potential theory of superharmonic functions.
Starting with the theory of weighted Sobolev spaces, the text advances to the theory of weighted variational capacity. Succeeding chapters investigate solutions and supersolutions of equations, with emphasis on refined Sobolev spaces, variational integrals, and harmonic functions. Chapter 7 defines superharmonic functions via the comparison principle, and chapters 8 through 14 form the core of the nonlinear potential theory of superharmonic functions. Topics include balayage; Perron's method, barriers, and resolutivity; polar sets; harmonic measure; fine topology; harmonic morphisms; and quasiregular mappings. The book concludes with explorations of axiomatic nonlinear potential theory and helpful appendixes.