A self-contained treatment appropriate for advanced undergraduates and graduate students, this text 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, this treatment 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 text concludes with explorations of axiomatic nonlinear potential theory and helpful appendixes.

Introduction1. Weighted Sobolev spaces2. Capacity3. Supersolutions and the obstacle problem4. Refined Sobolev spaces5. Variational integrals6. Harmonic functions7. Superharmonic functions8. Balayage9. Perron's method, barriers, and resolutivity10. Polar sets11. Harmonic measure12. Fine topology13. Harmonic morphisms14. Quasiregular mappings15. Ap-weights and Jacobians of quasiconformal mappings16. Axiomatic nonlinear potential theory17. Appendix I: The existence of solutions18. Appendix II: The John-Nirenberg lemmaBibliographyList of symbolsIndex