Heinonen J., Kilpeläinen T., Martio O. Nonlinear Potential Theory of Degenerate Elliptic Equations

Mineola: Dover Publications, 2006. - 412 p.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.Preface to the Dover Edition
CorrigendaWeighted Sobolev spaces
Capacity
Supersolutions and the obstacle problem
Refined Sobolev spaces
Variational integrals
A-harmonic functions
A-superharmonic functions
Balayage
Perron’s method, barriers, and resolutivity
Polar sets
A-harmonic measure
Fine topology
Harmonic morphisms
Quasiregular mappings
Ap-weights and Jacobians of quasiconformal mappings
Axiomatic nonlinear potential theory
Appendix I: The existence of solutions
Appendix II: The John-Nirenberg lemmaList of symbolsEpilogue
The John-Nirenberg lemma
Admissible Weights
The Riesz measure of an A-superharmonic function
Generalizations
New Bibliography