Giaquinta_Elliptic_Systems_Harmonic_Map_Minimal_Graphics

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1、11APPUNTILECTURENOTESMarianoGiaquintaScuolaNormaleSuperiorePiazzadeiCavalieri,756126Pisa,ItalyLucaMartinazziRutgersUniversity110FrelinghuysenRoadPiscatawayNJ08854-8019,USAAnIntroductiontotheRegularityTheoryforEllipticSystems,HarmonicMapsandMinimalGraphsMarian

2、oGiaquintaandLucaMartinazziAnIntroductiontotheRegularityTheoryforEllipticSystems,HarmonicMapsandMinimalGraphsc2012ScuolaNormaleSuperiorePisaSecondaedizionePrimaedizione:2005ISBN978-88-7642-442-7ISBN978-88-7642-443-4(eBook)ContentsPrefacetothefirsteditionxiPre

3、facetothesecondeditionxiii1Harmonicfunctions11.1Introduction..........................11.2Thevariationalmethod...................21.2.1Non-existenceofminimizersofvariationalintegrals........................31.2.2Non-finitenessoftheDirichletintegral.......41.3S

4、omepropertiesofharmonicfunctions...........51.4Existenceingeneralboundeddomains...........111.4.1SolvabilityoftheDirichletproblemonballs:Pois-son’sformula.....................121.4.2Perron’smethod...................131.4.3Poincar´e’smethod..................15

5、2Directmethods172.1LowersemicontinuityinclassesofLipschitzfunctions.....................182.2Existenceofminimizers...................192.2.1MinimizersinLipk(Ω)................192.2.2Apriorigradientestimates.............202.2.3Constructingbarriers:thedistancef

6、unction....232.3Non-existenceofminimizers.................252.3.1AnexampleofBernstein...............252.3.2Sharpnessofthemeancurvaturecondition....272.4Areaofgraphswithzeromeancurvature.........302.5TherelaxedareafunctionalinBV.............322.5.1BVminimizers

7、fortheareafunctional.......33viContents3Hilbertspacemethods373.1TheDirichletprinciple....................373.2Sobolevspaces........................393.2.1Strongandweakderivatives.............393.2.2Poincar´einequalities.................413.2.3Rellich’stheorem

8、...................433.2.4ThechainruleinSobolevspaces..........463.2.5TheSobolevembeddingtheorem..........483.2.6TheSobolev-Poincar´einequality..........493.3Ellipticequations:existenceof