Fanghua Lin & Qin Han Elliptic Partial differential Equations

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1、INTRODUCTIONInFall1992,thesecondauthorgaveacoursecalled"intermediateP.D.E"attheCourantInstitute.ThepurposeofthatcoursewastopresentsomebasicmethodsforobtainingvariousAPrioriestimatesforthesecondorderpartialdi®erentialequationsofelliptictypewiththeparticularemphasisonthemaxima

2、lprinciples,Harnackinequalitiesandtheirapplications.Theequationsonedealswitharealwayslinearthough,obviously,theyapplyalsotononlinearproblems.ForstudentswithsomeknowledgeofrealvariablesandSobolevfunctions,theyshouldbeabletofollowthecoursewithoutmuchdi±cul-ties.Thelecturenotes

3、wasthentakenbythe¯rstauthor.In1995attheuniversityofNotre-Dame,the¯rstauthorgaveasimilarcourse.Theoriginalnoteswasthenmuchcompleted,anditresultedinthepresentform.Wehavenointentiontogiveacompleteaccountoftherelatedtheory.Ourgoalissimplythatthenotesmayserveasbridgebe-tweeneleme

4、ntarybookofF.John[J]whichstudiesequationsofothertypetoo,andsome-whatadvancedbookofD.GilbargandN.Trudinger[GT]whichgivesrelativelycompleteaccountofthetheoryofellipticequationsofsecondorder.WealsohopeournotescanserveasabridgebetweentherecentelementarybookofN.Krylov[K]onclassic

5、altheoryofellipticequationsdevelopedbeforeoraround1960's,andthebookbyCa®arelliandXivier[CX]whichstudiesfullynonlinearellipticequations,thetheoryobtainedin1980's.CHAPTER1HARMONICFUNCTIONSGUIDEThe¯rstchapterisratherelementary,butitcontainsseveralimportantideasofthewholesubject

6、.Thusitshouldbecoveredthroughly.Whiledoingthesections1.1-1.2,theclassicalbookofT.Rado[R]onsubharmonicfunctionsmaybeaverygoodreference.Alsowhenonereadssection1.3,somestatementsconcerningHopfmaximalprincipleinsec-tion2.1canbeselectedasexercises.Theinteriorgradientestimatesofse

7、ction2.3followsfromthesameargumentsasintheproofoftheProposition3.2ofsection1.3.Inthischapterwewillusevariousmethodstostudyharmonicfunctions.Theseincludemeanvalueproperties,fundamentalsolutions,maximumprinciplesandenergymethod.Foursectionsinthischapterarerelativelyindependent

8、ofeachother.x1.MeanValuePropertyWebeginthissectionwiththede¯nitionofmeanval