; TeX output 1996.03.04:1612 hGS K`y UU cmr10AnnalesAcademiScienĽtiarumFN8ennic MathematicaVN8olumen21,1996,179{187 h֍!"V cmbx10ANINHOMOGENEOUSDIRICHLETPROBLEMFORANON-HYPOELLIPTICLINEARPٚARTIAL^DIFFERENTIALOPERAٚTORN @"V cmbx10ReinhardTHoQc9hmuth^SnFN8reieUnivĽersitUatBerlin,DepartmentofMathematics ?aArnimallee3,D-14195Berlin,GermanĽy;ho;Bchmuth@math.fu-berlin.deN#"V UU cmbx10Abstract.Inthispap;BerwĽestateaninhomogeneousDirichletproblemforaclassoflin- earppartialdierenĽtialop;Beratorswhicharenon-hyp;Boelliptic. WN8epproveuniqueness, DexistenceandregularitĽyresultsforitssolutions.4k 1.pIn9troQduction5 K`y cmr10Inv[1]K.DoppGelandthepresentauthorstatedahomogeneousDirichletprob-lem#fornon-hypGoelliptic#linearpartialdierentialopGerators,-especially#foraproGd-uctofuniformlyellipticdierentialopGeratorswithsmoothcoecientfunctions.HerewestateaninhomogeneousDirichletproblemforthesameclassofpar-tial4dierentialopGerators.cTherelatedhomogeneousproblemturnsouttobeanequivqalent3reformulationofthehomogeneousDirichletproblemin[1]. Butthisnewzformulationyieldsmoregeneralresultsthantheformerone.F*orfurtherde-tailsMofthisproblemandforsomeliteraturerelevqantinthiscontextwerefertotheUUintroGductionin[1].n!2.pTheTinhomogeneousproblem5Most;ofthedenitionsusedherearetakenfrom[1].%)W*erecapitulateonlysomeUUnotationnecessaryforunderstandingthesubsequenttheorems.Let5: ٓR cmr71 and 2bGe5:boundeddomainsin5:R^0er cmmi7n Z cmr51ثorR^n 2( b> cmmi10n1|s;n2!", cmsy10<>2,m3respGec-tively)"withbGoundaries"@ 8 ("m=1;2)of"classC ^O! cmsy71 #(cf.e.g.Grisvqard[2,Deni-tion&1.2.1.1]).;Thusthedomains& 0satisfytheuniformconeconditionandtheproGductdomain := 1_ 2(R^n Ce( n=n1+n2|s)hasthesamepropGerty(cf.HoGchmuth[3,Satz3.1]).wF*urthermore, isaLipschitzdomain(cf.Grisvqard[2,TheoremUU1.2.2.2]).On feachofthedomains f }'weconsiderauniformlyellipticdierentialopGer-atorUUP\(;Dx O \ cmmi5 ț)ofUUsecondorder;P\(;Dx ț):= jWn YA u cmex10Xti;jg=1`KDj6b a:()Zijë()DiTLb "+j*n Y8Xti=1UQb:()Zi()Di,+8c()()ö ff xsUU1991MathematicsSubvjectClassication:%Primary35B30;Secondary35A05. * 180}Ep0J cmsl10ReinhardUUHoGchmuthhG with+SDx =(D1|s;:::;Dn 15)andDjī=@ 8=@xj6,3where+Sa:()Zijî;b:()Zi;c^()c۱2C ^1 0(} fe 8 ㌟@M)+SareˍgivenXreal-vqaluedfunctionswithXa:()Zij4=Gqa:()Zjgië.XNotethatthenthereareconstants %#ٱ2R^+ withj!H:n YXtWi;jg=11Pa:()Zijë(x\)iTLjı%j gn YiXt&i=1#ڮuǬ2፯iGforUUall9y\x#ٱ2 and#(1|s;:::;n 15)2Rn :#0<Byx䍑KeUUP(;Dx ț)UUweUUdenotetheformaladjointopGerators Y x䍑7Ҵe5 PAZX(;Dx ț):= jWn YAXti;jg=1`KDj6b a:()Zjgië()DiTLb " j*n Y8Xti=1UQDiTLb 飮b:()Zi()b7+8c()(): bW*eUUconsidertheclassicalhomogeneousellipticDirichletproblems:Problem@( \)( =1;2).ЉF*or@f涱2C ^0;5X( \)ұ\C (} fe 8 ㌟@M)@(2(0;1])nd@afunctionUUu#ٱ2C ^23( \)8\C (} fe 8 ㌟@M)suchUUthat#Ǎ UP\(x;Dx ț)u(x) =f\(x)UWfor&x#ٱ2 ; u\(x) =0.for=x#ٱ2@ 8 \:GW Now3weformulatetheinhomogeneousDirichletproblemofclassicaltypGefortheUUnon-hypGoellipticproductoperatorUUPc(;Dx)denedUUby)Pc(x;Dx):=P1|s(x1;Dx 1)P2(x2;Dx 2)UWfor&x=(x1|s;x2)2 1S8 2|s:WF*orUUPc(;Dx)theUUformaladjointopGeratorisdenotedbyx䍑KeUUP$r(;Dx) .Problemb#( ) .1F*orf2oC ( )andg8H2C (@ 8 )ndb#afunctionb#u2C ^2;2( )Ai\C (} fe 8 8)UUsuchUUthat k