Nonlinear Potential Theory of Degenerate Elliptic Equations

Aself-containedtreatmentappropriateforadvancedundergraduatesandgraduatestudents,thistextoffersadetaileddevelopmentofthenecessarybackgroundforitssurveyofthenonlinearpotentialtheoryofsuperharmonicfunctions.StartingwiththetheoryofweightedSobolevspaces,thistreatmentadvancestothetheoryofweightedvariationalcapacity.Succeedingchaptersinvestigatesolutionsandsupersolutionsofequations,withemphasisonrefinedSobolevspaces,variationalintegrals,andharmonicfunctions.Chapter7definessuperharmonicfunctionsviathecomparisonprinciple,andchapters8through14formthecoreofthenonlinearpotentialtheoryofsuperharmonicfunctions.Topicsincludebalayage;Perron'smethod,barriers,andresolutivity;polarsets;harmonicmeasure;finetopology;harmonicmorphisms;andquasiregularmappings.Thetextconcludeswithexplorationsofaxiomaticnonlinearpotentialtheoryandhelpfulappendixes.