Hueber, H.

Sieveking, M.

Annales de l’institut Fourier (AIF)

ANNALES DE L’INSTITUT FOURIERH. HUEBERM. SIEVEKINGUniform bounds for quotients of Greenfunctions on C1,1-domainsAnnales de l’institut Fourier, tome 32, no 1 (1982), p. 105-117<http://www.numdam.org/item?id=AIF_1982__32_1_105_0>© Annales de l’institut Fourier, 1982, tous droits réservés.L’accès aux archives de la revue « Annales de l’institut Fourier »(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-tion commerciale ou impression systématique est constitutive d’une in-fraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Ann. Inst. Fourier. Grenoble32, 1 (1982). 105-117UNIFORM BOUNDS FOR QUOTIENTSOF GREEN FUNCTIONS ON C1 '-DOMAINSby H. HUEBER and M. SIEVEKINGConsider a partial differential operator L on R" which hasthe formn nLU = ^ ^j^n11 + S A'^f^ + cu '( , /=1 /= !We assume that L is strictly elliptic and has H61der continuous coef-ficients. We will also assume c < 0. Hence every bounded domainft has a Green function which we will denote by G^ . As usualA denotes the Laplace operator. The aim of our paper is to provethe following theorem:THEOREM. — For every bounded C1'1-domain ft there existsa constant C such that we haveC-^ <G" <CG"A L Aon ft x ft. The constant C may be chosen depending only onthe ellipticity constant of L, on the Holder norms of the coeffi-cients of L, on the diameter of f t , and on the curvature of 3ft.We proceed as follows: In § 1 we introduce notations concern-ing the geometry of ft and give two elementary lemmas. In § 2we introduce the harmonic space and the adjoint harmonic spaceassociated to L and quote some results which we will use in ourproof. The proof is given in § 3. We finish our paper with someremarks which include an application of our result to the Dirichletproblem for Lu = f.106 H. HUEBER AND M. SIEVEKING1. The geometrical situation.For r > 0 and x G R" let ByQc) denote the open ball withradius r and center x and let Sy(x) := By(x)\Bi (x). Throughout3^this paper ft will denote a bounded domain in R" which is ofclass C1'1 . The distance from x E R" to 9ft will be denoted by d^ .For every XQ G 8ft there exists a unique inner normal n(,Xo)at 3ft with \n(Xo)\ = 1 . By definition the mapping XQ——> n(Xo)fulfills a local Lipschitz condition, and a simple compactness argu-ment shows that there exists a constant C^ > 1 such that we have\n(Xo) — n(yo)\ < C^ |Xo —^ | for all Xo ,^e3 f t . Furthermorethere exists r^ G ]0,1 [ such that we have By (^g 4- r^n^x^)) c ftand By (XQ — r^n(Xo)) ̂  R" \ft for all Xg € 8ft . We may assumeC^n^ l -For every x G ft with fi?^ < r^ there exists exactly one pointXQ G 3ft with x = JCo + d^(Xo). Hence for such a point ^ G ftthe following definitions make sense:x^ := XQ 4- an(xQ) a€Rn(x) := n(Xo)Z^(x) := { ^ E f t | (^ -x^ < 2~12a for some j3G ]—a, a[} AG R^..LEMMA l.l.-Z^ ^ , z E f t wzT/z d x ^ d z < r s ^ f Let J 3 , a , X e R +^c/z ^ar 0 < j 3 < a < y ^ a^d 0 < X < — • Assume further\ZQ— x^\< ̂  -\- 2\a . Then one has \ZQ - XQ \ < 4a^X.Proof. — Regarding the situation in a plane which contains XQ ,Xy and ZQ the problem becomes a two-dimensional one. The state-ment then follows from the theorem of Pythagoras and elementaryestimates which we will leave to the reader, nLEMMA 1.2. - a) Let x € ft wrt d(x , 3ft) < r^ , leta G j 0. — r^ | and let z G Z^QcT. Then we have \ ZQ —XQ \ < y. a.b) Let x , ̂  € ft mrt d^ < r^ fl^d dy <r^ . AssumeZ^)nzjy)^(J)for some o i E ] 0 , r ^ [ . Hence we have \x^—y^\< . a. for all^[0,^].QUOTIENTS OF GREEN FUNCTIONS 107Proof. — a) Since z G Z ^ ( x ) there exists j 3 € ] — a , a [ such that\x. — z | < 2~12a . Hence we have\ZQ ~x^ < ^o ~z\ + \z ~x^< \XQ— z \ + \z —x^\< \xo~x(3\ + 2 I 2 "~^1 < l ^ l + 2•2-12a.Let X = 2~12 . Since ZQ f. By (Xy ) the assertion follows fromLemma 1.1 at least for j3 > 0 . For j3 < 0 the assertion is obviouslytrue.b) Let z e Z^(x) H Z^CY") . By a) we have1^0 ~XO\ ̂  \YO ~ zol + \ZQ ~XO\ < g- a 'Hence for j3 ̂  [0 , r^] we get1^ ~~^1 == l^o + ̂ (^o)"-3;o "iS^^o)!< l ^ o - ^ o l +^ l^o ) -^o ) l < l ^o -> ; o l (1 +<3C^)< - i ^ ( l + ^ C ^ ) < ^ a .2. Potential theory of L.Following [2] we say that the operatorL = S ^^ + S 6 , D , + ct,;=i < = ibelongs to the class A^, ̂ ) with X > l a n d a o G ] 0 , l [ i f i t fulfillsthe following properties:(i) For all x G R" and all { G R" we have£ ^W^^-1!^2' ,7=1(ii) For all x , y € R» we have£ |a,,(x)-^OQ| + ^ \b,(x)-b,(y)\ij=l i=l a^+ \c(x)-c(y)\<\\x-y\ °108 H. HUEBER AND M. SIEVEKING(iii) For all x E R" we have£ 1^0c)l+ 1 \b,(x)\ + |c0c) |<\ .ij=l i=lFrom now on we will always assume L € ff(\, o^), and furthermorewe will assume c < 0 .The sheaf 9^ °^ solutions of Lu = 0 is a harmonic sheafwhich gives rise to a Brelot space ([3], [4], [6]). Every bounded domainV has a Green function G^ which may be characterized by thefollowing properties:(i) G ^ ( - , y ) is a potential on V for every j ^ G V . Its supportis W.(ii) For all y € V we havelim G ^ ( x , y ) _ ^^det(^))(1).—y (V(Jc-^)B(^)(^-^))2-" n v f,Here B(jQ is the inverse of (ci^(y)\. and o?^ denotes as usualthe volume of the unit ball in R".R.M. Herve [6] has shown that there exists a unique Brelotspace on R" such that the extrbmal potentials on V for this adjointharmonic space are just the functions G^ (x , • ) with x G ^2 . Weuse the symbol L* to indicate this adjoint space. If the adjoint ope-rator of L exists as a differential operator and is denoted by L*the notation will stay consistent.For every L-regular bounded open set V C R" and every/E (3(8 V) we denote by H^/ the unique continuous functionon V which is equal to / on 3V and L-harmonic in V. ForL*-regular V the meaning of H^ / is obvious. For our purposesit is enough to know that Lipschitz domains are L-regular and Ir-regular (cf. [1] Theoreme 4.1), and we will use this without furthercomment.It is clear that A G K(n, a) for every a G ]0,1 [. To be ableto work with the same constants for L and A we will henceforthassume X > n.(1) For n < 2 this formula has to be modified in the usual way. The sameremark applies to the statement of Lemma 2.4.QUOTIENTS OF GREEN FUNCTIONS 109LEMMA 2.1. — There exists a constant C^ = Ci(X, o^, n) suchthat for every x E R" , r G ]0,1 [ and /€ e^(3B/^)) rt^ /o/tow-ing inequalities hold on B := By.(x):C^H^H^ <C,H^/C^H^Hf/^C^/.Proo/ — The first inequality is due to Serrin and the secondone is due to Ancona. They can be found in [2] page 13 and page 20(for r < 1 the constant C in Proposition 10 on page 20 does notdepend on r ) . DLEMMA 2.2. — There exists a constant C^ := C^(\, c^, n) suchthat for every x E R " , r € ]0,1 [, and every positive L-harmonicor L*-harmonic function h on B/x) the following inequality holdsforall y , , y ^ B , (x): h(y,) < C, h(y^).2 ^Proof. — With the use of the ordinary Harnack inequality thisstatement follows immediately from Lemma 2.1. DLEMMA 2.3. - Let x G ft mth d^ < r^ and let a E j 0, -. r^ |.Let h be a positive ^.-harmonic or \^-harmonic function in Z^(x)which tends to zero at each y £ Z^(x) 0 Q^l. Then we haveC^1 ^ h(xj < h(x^) < €3 ^ h(x^)for all 7 E [0, a] with a constant €3 depending only on \, OQ , nand r^.Proof. — In the following the symbol L has to be replaced byL* if h is L*-harmonic.Let j3 := S-^-^a and letr := {z | \z-x^\ = P ' , \z-x^\ <ft}.By Lemma 2.2 we have h (x^) < C^ inf h(T) .By [1] Theoreme 2.2 and Thtorfeme 4.1 there exists a constantC = C(X, OQ , n , r^) such thatsupA(B^Oco)nn)<CA(^).110 H.HUEBER AND M. SIEVEKINGUsing the maximum principle we getC2-1A(^)HW ^(^X^^XC^X^H^^ Ic^Oc,/^)for all 7 G [0, j3]. Using Lemma 2.1 we get^(^ 1^^7)>C71H^(^) ̂ (^^ ^ r '• ^ , /^ 'Y> F-1 /?-! | Y — Y | = C~1 -—^ ^1 P 1^-y ^Ol ^1 nwith C^ = C ^ ( X , OQ , n). A standard transportation argument toge-ther with Lemma 2.1 showsT ^ T^^.-^ ^Ba^x,^)^) ^C^ ^-with C^ = C^(\, OQ , n) (cf. [2^ Theoreme 2).Thus we haveC,1 C,-1 1 h(x^) < h(x^) < C C - h(x^)for aU 7 G [0 , j3].Lemma 2.2 implies that there exists a constant €3 such thatwe have h(x^ ) < €3 h(x^^) for all 7^, ̂  G [j3, a]. Thus for7 E [0 ? j3] we have5.212 C,-1 C,-1 C3-1 ^ A(xJ </z(^) < S^^CC, €3 ^ /z(x,),and for 7 G [j3, a] we have70-1 C^1 A(xJ < C^1 h(x^) < h(x^) < €3 h(x^)< T^-1^-1)^ A^) = ja-1 5-212 €3 /z(^).This shows that the assertion is thie with €3 := 5 -l^C C^ C^ C^ €3. nLEMMA 2.4. — There exists a constant €4 = C4(X, OQ , n) suchthat we haveC^jx-^ l2-" < G^(;c,^)<C4|x-^|2-"for all x , ^ C n wfr/z | x -^ |<^-^ or | x - ^ | < — d^.Proo/; —This Lemma is essentially due to Gilbarg/Serrin [5].It can be found in [2] on page 14.QUOTIENTS OF GREEN FUNCTIONS 11 ]3. Proof of the theorem.Let K :=} x € S2 | d^ > — r^ . Using Lemma 2.4 one cansee that C ^ G ^ ^ G ^ ^ C^ G^ is true at least in a neighbourhoodof the diagonal of K x K. Since G^ and G^ are strictly positiveand continuous off the diagonal there exists a constant Cs suchthat we have(*) C,-1 G^ (x , y ) < G?(x , y) < C, G^(x , y)for all x , y G K . Looking very carefully at Lemma 2.2 and Lemma2.4 it is obvious that €5 may be chosen depending only onX , UQ , n, r^ and the diameter of Sl.Now let x , y € S2. Since all arguments which we will use inthe following proof are also true if one interchanges the role of 9^and S^,* we may assume d^ <d . The proof is divided into threecases:l d x > 16"̂  ; dy> '4 rnn ̂  < ]jr" ; dy> ̂ r ^III ̂  < ^ ^.Case L - In this case we have x , y G K and our statementfollows from (*).Ca^ //. - In this case we have y ^ Z ^ (x). With j3 := d^we get rnG^Oc^)<C3^G^Oc, ,y)<C,C, ^G^x, , y)rS^ '8^ r^ "8^< C^ Cs G^(x , ̂ ) < C^Cs ^tL G^(x, , y )ra ^n<^C| ^G^(x, ^XC^C^G^,^).''n 8'''"Here we have used (*), and we have applied Lemma 2.3 to the L-harmonic function G^(* , j0 and the A-harmonic function G^(- , y )on Z^ (^).'̂'n112 H. HUEBER AND M. SIEVEKINGCase HI. — Let j3 := dy apd 7 := d . By assumption we have0 < j3 < 7 < -j r^ . By a we will denote the minimum of all posi-tive numbers 5 for which we have Zg (x) H Zg (y) ^= 0 . We willconsider four subcases.Subcase 1: a > -. r^ =: jn.We have ^ ,x^eK and Z^(x) H Z^(^) = 0. With (*) andLemma 2.3 we getG^(x,y)<C^ ^ G^(x^y)<Cl^G^,^)<C^C, ^G^x^y,)<CiCs ^G^,^)<C^C5 G^(^,^)<C^CJG^(x,^),where the dots indicate that we must repeat our arguments in theinverse order.Subcase 2: . r^ > a > 7.Obviously we have Z^(x) H Z^(^) ̂  0, and hence Lemma 1.2implies \ x ^ — y ^ \ < - . a < — d . We apply Lemma 2.3 andLemma 2.4 and getG^x,y)<C2, ^G^(x,,^)<C^ ^G?(^,^)<C^C^G^^)< p6 p2 P ^ r'^^v i, ^3 4 a a ^L^^)QUOTIENTS OF GREEN FUNCTIONS 113<W ^G^y^)a a<C^C^G^(x,y).Subcase 3: j3 > a.If | x — y | < — j3 the desired inequalities follow from Lemma2 12.4. Hence we may assume \x — y | > ^- j3. Since j3 < 7 we have2 1^^Z.(^). From Lemma 1.2 we get \x. — y . \ < ~ 7 P , and1 . i^ 2 ^ •I ̂  "~" ̂  i I ^ TT j3 is clear. Hence we get2^ 2G^(jc,3/)<C3 2G^(x^ ,>/)y^<C^C,2G^(y,_ y)<C2CJ2r-^G^ ^)2 7<C,C^C^27-^G^(^^)<C,(%2G^ ^)y^<C^C|C^2G^(x^,^)<C^C^C^ G^(x,y)<C^C^ G^(x,y).To derive the first of these inequalities we applied Lemma 2.3 to theA-harmonic function G ^ ( ' , y ) on ZQ(X). In the second line weapplied Lemma 2.2 to the same function in B^ (y ^ ) . In the third2^ 2^inequality we used Lemma 2.3 on Z, (y) observing that y f. Z < ( y ) .y7 y7The fourth inequality followed from Lemma 2.4. Then we appliedagain Lemma 2.3, then Lemma 2.2, then Lemma 2.3 once more, andthe dots indicate that we must repeat all our arguments in theinverse order.Subcase 4: 7 > a > j 3 .114 H.HUEBER AND M.SIEVEKINGWe apply subcase 3 to the points x^ , y and getC^-4 C,-2 G^,, jQ < G^(x^y) < C^C^ G^(^,^).Since y f. Z^(x) we may apply Lemma 2.3 to Z^(x), and we getG^(x,y)<C, ̂  G^(x^y)< C^CiC2^ G^ y)<C^C^G?(x^)<C^C^G^^)< r4^1^4 -^ 0,"̂  ^^^2^3 ^4 ^A^a'^)^C^C^G^x^).Now we see that with C := C^C^C^Cg the statement of the theoremis true. a4. Remarks.4.1. Obviously our theorem is true if the Laplace operator is replacedby any L' G &(\, c^) with c < 0 . HenceC(L, L') := inf {c > 0 I c-1 G^ < G^ < c G^}is a well defined real number > 1. We can show that for sufficientlysmooth domains C(L, L') depdnds continuously on the coefficientsof L, L' with respect to uniform convergence. The proof is by amodification of a method presented in [7]. n4.2. For Lipschitz domains our theorem is not true:Let n := {x = ( x ^ x ^ G R 2 ) 0<x, < 1; 0 < X 2 <x,}; fix^ G f t and let V denote the intersection of ft and a small ballaround zero whose closure does not contain y . Hence there existsa constant C^ such thatC^^x^-x^XG^x^yXC^x^-x^)for all x = (jc^jc^ev.QUOTIENTS OF GREEN FUNCTIONS 115a2 a a2Let A := —— + 2 ;—-—4- 3 ——. Then there exists a9x^ Qx^Qx^ 9x^constant C^ such thatCA^I - ̂ 2) < G^ . ̂ ) < C^Oc2 - x , x ^ ) .This shows that G^( • , ^)/G^( • , y ) is unbounded near zero. n4.3. Let X be a locally compact space with countable base, andassume that we have two harmonic sheafs 9€ 1,3^2 on ^ suc!1that (X.3^.) i •= 1,2 is a y-Brelot space. Assume that S2 isa bounded domain in X which is regular for both structures andhappens to have Green functions G^ 0' = 1,2). Let / := BS2 —> Rbe continuous and positive, and let u^ denote the solution of theDirichlet problem for / with respect to the sheaf 96,. We knowthat u^ is the limite of an increasing sequence (pn\^ °f 3^i-potentials, and we may even assume that the potential theoreticcarrier of ?„ is contained in ?2\U^ where (U^)^^N is an exhaus-tion of t2.Hence /,PnW-J G^(x,y)^(dy)with ^ (U^)=0 .Let /*^nW- JG^(x,y)^(dy).Assume now that there exists a constant C such thatC—l r^S^l ^ (-^^i ^ f-\ r^^l(Jr^ -s (J.. ^ L Lr^ •d- •j j(. f\ <f\-1HenceC^q^p^Cq^.By Azela-Ascoli a subsequence of (q^ )^^ is locally uniformlyconvergent and therefore we may assume that (qn\ itself convergeslocally uniformly to an 3€ ̂ -harmonic function q .Obviously we have C~1 q < u^ < Cq and from this we deriveC-1 q < u^ < C-1 q . Thus we have C~2 u^ < u^ < C2 u^ .This shows that our theorem implies the statement of Lemma2.1. In a similar way we may also derive the other statements of § 2from the theorem. This shows that our paper characterizes in a116 H.HUEBER AND M. SIEVEKINGcertain way all Brelot spaces on R" for which the comparabilitystatement of our theorem is true with a constant C which dependsonly on the diameter of ft and the curvature of 3ft .4.4. For L^L^ Gj?(X, a^) the following comparability statementis an easy conclusion of 4.1 and 4.3:For any ^, u^ G (°(ft) 0 (^(ft) with u^ = u^ > 0 on 3ftand Li^ = L^ < 0 in ft we have u^ <Cu^ with a constantC = C ( X , a o , ^ , f t ) . DBIBLIOGRAPHY[1] A. ANCONA. Principe de HArnack a la fronti^re et thtoreme deFatou pour un operateur elliptique dans un domaine Lipschitzien,Ann. Inst. Fourier, 28, 4 (1978), 169-213.[2] A. ANCONA. Principe de Harnack a la frontiere et problemes defronti^re de Martin, Lecture Notes in Mathematics, 787 (1980),9-28.[3] N. BOBOC. P. MUSTATA, Espaces harmoniques associes aux ope-rateurs differentiels lineaires du second ordre de type elliptique,Lecture Notes in Mathematics, 68 (1968).[4] C. CONSTANTINESCU, A. CORNEA, Potential theory on harmonicspaces, Berlin-Heidelberg-New York, 1972.[5] D. GILBARG, J. SERRIN. On isolated singularities of solutions ofsecond order elliptic differential equations, J. d'Anai Math.,4 (1954-1956), 309-340.[6] R.M. HERVE, Recherches ^xiomatiques sur la theorie des fonc-tions surhamoniques et du potentiel, Ann. Inst. Fourier, 12(1962), 415-571.[7] H. HUEBER. M. SffiVEKiNG, On the quotients of Green functions(preliminary version), Bielefeld, September 1980 (unpublished).[8] J. SERRIN, On the Hamack inequality for linear elliptic equations,J. d'Anal. Math., 4 (1956), 292-308.QUOTIENTS OF GREEN FUNCTIONS 117[9] J.-C. TAYLOR, On the Martin compactification of a boundedLipschitz domain in a Riemannian manifold, Ann. Inst. Fourier,28, 2 (1977), 25-52.Manuscrit re^u Ie 13 avril 1981.H. HUEBER,Fakultat fur MathematikUniversitat Bielefeld4800 Bielefeld 1 (R.F.A.)&M. SlEVEKING,Angewandte MathematikRobert-Mayer-Str. 106000 Frankfurt/Main (R.F.A.).

Uniform bounds for quotients of Green functions on $C^{1,1}$-domains

Numérisation de Documents Anciens Mathématiques

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Uniform bounds for quotients of Green functions on $C^{1,1}$ -domains

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Uniform bounds for quotients of Green functions on C1,1C^{1,1}C1,1-domains

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Uniform bounds for quotients of Green functions on $C^{1,1}$ -domains