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Nun-linearEl'l'ipticEguationsWithoutNun-rmearEatireSolutionsImtituteofMathematicalScienceaNewPorkUnivereity,MewYorkCity$1.htroduction.A)Theequationofminimalsurfaces(1+(o:)cpzz-2rpzP,(o*+(1+P:)PvY=0occupiesaspecialpositioninthetheoryofquasi-linearellipticequationssince,1justlikeitslinearanalogue,theLaplaceequation,itcanbeinvestigatedbymethodsofcomplexfunctiontheory.Inparticular,thefol1011-ingpropertiesofsolutionsofthisequationareknown.a.Everyentiresolutionisalinearjunction.(Byanentiresolutionwemeanhereandhereafteronewhichisdefinedforallvaluesofxandy.)b.-4single-valuedsolutiondefinedina?~ezghborhoodofthepointatiltfinityisalwayssuchthatitsgradientattainsa$finitelimitatinfinity.Thesameistruejoramultiple-valuedsolution,providedthatitsgradientisfinitelymany-valued.c.Atafinitepointasingle-vak7iedsolutioncanhulleatmostcrrcazovableisolatedsingularity.B)TheoremaisduetoS.Bernstein(21,~whoobtaineditasacorollaryofhishelebratedgeometrictheorem.Thelattertheoremimpliesthatej7eryellipticzquationoftheformpossessesonlyconstantentireboundedsolutions.OtherproofsofBernstein'stheoremonminimalsurfacesweregivenbyT.Rado1191,myself131,andE.Reinz[13].IprovedTheoremsbandcbythesamemethodasBernsteb's'Sponsoredhyt,heOfficeofOrdnanceResearch,U.S.Army,undercontrnct,sDA-30-069-ORn-4822ndDA-3'J-Og$)-QRD-835.'sumbersinbracketsrefertothebibliography.Cj.alsoHOP^[14],MICKLE[lsl-767LIPMANBERStheorem.Theothermethodsdonotyieldtheseresults.AdifferentproofofTheoremcwasgivenlaterbyFinn[ll].Iconjectured[4]thatTheoremsaandcremaintrueforellipticequationsoftheformwherep(q)isagivenfunctionsuchthatqp(q)isincreasingandbounded.ForTheoremcthiswasverifiedbyFinn[ll],who,infact,provedastrongerresultonremovableirregularities.Asfarasaisconcerned,theconjectureisalmostcertainlywrong.SimpleexamplesshowthatneitheroftheTheoremsaandcimpliestheother.Ontheotherhand,itiseasytoseethatTheorembimpliesTheorema,atleastforquasi-linearellipticequat.ionsoftheform@)Themainpurposeofthispaperistoindicatetwoclassesofequationsof$heformstatedaboveforwhichTheoremaholds.(Anothersuchclasswasdis-covered,independently,byFinn[12].)Oneoftheseclasses(calledequationsoffiniteconformalradiusandslowlygrowingcharacteristic)containstheequationofminimalsurfaces.TheproofinthiscaseisverysimilartomyproofofBernstein'stheorem,exceptthatpseudo-analyticfunctionsandquasi-conformalmappingsareusedinsteadofanalyticfunctions.Inparticular,arecentresultofAgmononquasi-conformalmappingsisusedatanessentialpoint.IbelievethatTheorembremainstrueinthiscase,butIwasunabletoproveit.Theokherclassofe~trations(equationsoffiniteconformalradiusandslowlygrowir~geccentricity)isofinterestprimarilybecauseitcontainsthepotentialequationofasubsonicadiabaticgasflow.ForthisclassofequationsTheoremsaandbcanbeprovedbyarelativelysimplequasi-conformalityargument.ThesameargumentalsoyieldsastrongerformofTheoremc:anisolatedsingularityofasolutionisremovableifnotthesolutionitselfbutmerelyitsfirstderivativesareassumedtobesingle-valued.Forminimalsurfacessuchastatementisfalse,6sisshownbytheexample=arctan(ylx).D)In$2wecollectinformationonquasi-conformalmappingswhichareusedintheproofs.Thedefinitionsoftheconcepts:conformalradius,eccentricity,characteristic,aregivenin$53,4.In$5wegiveaversionofthefamiliarhodo-graphlinearizationofquasi-linearequationswhichisconvenientforourpurposes.Theproofsoftheresultsdescribedabovewillbefoundin§$5,6.Thelastsectioncontainssomeexamples.(2.2)S=r(z)=Hx,Y)t-idx,?/IdsedomainDinthez-plane(z=z+iy)willbecalledWO&8thederivativesPITON-LINEARELLIPTICEQUATIONS769/fz,tg,q2,qgexistandarecontinuous,andf,q,-5,q,9.TheeccentricityofasmoothhomeomorphismatapointofDisdefinedasthevalueofatthispoint.Itisalsotheeccentricityofthehomeomorphisminverseto(2.1)(ofcourse,atthecorrespondir,apointinthe{-plane),anditiss?sotheeccen-tricityofanyhomeomorphismobtainedfrom(2.1)byprecedingorfollowingitbyaconformalmapping.Ifsomerestrictionisimposed011thevaltr_eofE,themappifigiscalledquasi-conformal.Itiscalledofboundedeccentricilgiftheratio(2.2)isuniformly'rt~unded.~B)Asmoothhomeomorphism(2.1)ismidtobeconformalwithrespecttoaRiemannmetric(whereg11g22-g:2,gll0)ifA=X(x,y)beingapositivefunction.TheeccentricityofsuchamappingiseasilyseentoequalInorderthatthemapping(2.1)beconformalwithrespectto(2.3)itisneces-saryandsufficientthat[(x,p)andq(x,y)satisfythesystemofBeltramiequa-tionsdeterminedbythemetric.ICarn1153andLichtenstein116,173provedthattheBeltramiequgtionsaresolvableinthesmallifthecoeilicientsofthemetric(2.3)areHijldercontinuouslydifferentiableorevenonlyM6'tdercontinuous.Inthiscasethesolutionsaretwice(once)Holdercontinuouslydifferentiable.(HereandhereafterwecallafunctionHoldercontinuousonasetifitsatisfiesauniformHijlderconditiononeverycompactsubset.)Combiningthisresultwiththegeneraluniforxnizationtheoremoneobtainsthefollowingwell-knownstatement.Ifthecoefficientasofthemetric(2.3)aredefinedandHoldercontinuousinaplanedomainD(or,moregenerally,onzRiemannsurfaceofgenuszero),thenthereexistsasmoothhomeomorphism5;=~(2)ofD(ofb)ontoaplanedomainAwhichisconformalwithrespectto(2.