Post on 01-Feb-2023
The Dirichlet Problem for Degenerate EllipticDarboux Equation
Saoussen Kallel-Jallouli*
Faculte des Sciences, Campus Universitaire, Tunis, Tunisie
ABSTRACT
We study the Dirichlet problem
det@2u
@xi@xjþ aij
� �¼ K xð Þf x; u;Huð Þ in O � Rn
u j @O ¼ j
8><>:where K vanishes on a compact set of O. We prove under some hypothesis on
K; f , j and if jdetðjij þ aijÞ � KfðjÞjCs� is sufficiently small, that we can find aunique sufficiently smooth solution u to the Dirichlet problem such that
ðuij þ aijÞ � 0:
Key Words: Darboux equations; Degenerate elliptic.
Mathematics Subject Classification: 35J70; 35Mxx.
*Correspondence: Saoussen Kallel-Jallouli, Faculte des Sciences, Campus Universitaire, Tunis1060, Tunisie; Fax: (216)-71-885-350; E-mail: saoussen.kallel@fst.rnu.tn.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
Vol. 29, Nos. 7 & 8, pp. 1097–1125, 2004
1097
DOI: 10.1081/PDE-200033756 0360-5302 (Print); 1532-4133 (Online)
Copyright # 2004 by Marcel Dekker, Inc. www.dekker.com
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I. INTRODUCTION
Let O be a bounded domain in Rn with smooth boundary. We shall consider theDarboux equation
det@2u
@xi@xjþ aij
� �¼ K xð Þf x; u;Huð Þ in O � Rn
f j @O ¼ j:
8><>:where aij ¼ aji are sufficiently smooth real functions satisfying
ðiÞ @iakp ¼ @ka
ip; 8i; k;pðiiÞ ðaijÞ � 0 in O:
ðI:2Þ
We assume that j is a real function defined on O and
gðx;fÞ ¼ KðxÞfðx;f;HfÞ ¼ KðxÞ½f1ðx;fÞ þ f2ðHfÞ�:Let S be a compact C1 submanifold of O such that OnS is connected. We need
the following assumptions:
ðiÞ K � 0 in O
ðiiÞ K�1ð0Þ ¼ S
(ðI:3Þ
ðiÞ fðx; u;pÞ > 0 in O�R�Rn
ðiiÞ f2 is concave:
ðiiiÞ 2K@f1@u
�Xk
@K
@xk
@f2@pk
in O�R�Rn
8>>><>>>: ðI:4Þ
ðiÞ ðjij þ aijÞ jOnS is strictly positive
ðiiÞ ðjij þ aijÞ jS is of rankðn� 1Þ
ðiiiÞ The eigenvalues of ðjij þ aijÞ on S are distincts:
8>>><>>>: ðI:5Þ
Our main result is the following
Theorem A. For any integer s� � 7þ n; a 2 �0; 1½, G > 1, aij 2 Cs�;aðOÞ satisfying(I.2) and j 2 Cs�þ2;aðOÞ satisfying the condition (I.5), there exists a constante0 > 0 such that for any g ¼ Kf ; Cs� of its arguments satisfying (I.3) (I.4), if
jdetðjij þ aijÞ � gðjÞjCs� � e0 ðI:6Þ
and
jHðu;pÞgjCs�þa � G:
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Then the problem (I.1) admits a unique solution f 2 C½s��3�n2�ðOÞ satisfying
ðfij þ aijÞ � 0 in O: ðI:7Þ
In the case when aij ¼ 0, f � 1, related results were found by Amano (1988) forthe dimension n ¼ 2, then generalized, in the special case when S is an isolated pointin O, by Atallah (2000) for higher dimension.
Equations of this type appear in many geometrical (see Pogorelov, 1978) andphysical problems (see Oliker and Waltman).
Their strong nonlinearity makes their study rather difficult, and this difficulty isgreater when the equation imposes the vanishing of the Monge–Ampere determi-nant, since then the equation is degenerate.
Among the constructive existence proofs in analysis, the Implicit FunctionTheorem is well known; it furnishes the construction of the solutions of manynonlinear problems in the theory of ordinary and partial differential equations (seeAlinhac and Gerard, 1991).
In our case this method fails, however, because of a phenomenon which may bedescribed as ‘‘loss of derivatives’’. This loss will be circumvented by an appropriatesmoothing process and the use of a new type of Implicit Function Theorem as a basicmethod in the proof of our Theorem.
In the first step of the proof, we shall introduce the basic norms and state somesimple calculus lemmas which play important roles in the proof of convergence ofour iteration scheme.
Section 3 is devoted to the study of a priori estimates, for modified linearizedoperators, in suitable functional spaces.
In fact, although we have a strong estimate in an elliptic region (Lemma III.6), itis quite hard to show certain a priori estimates (III.15–III.17) in a neighborhood ofzero points of K since the linearized operators are degenerating. The compactness offK ¼ 0g, Lemma III.7 and the hypothesis on g and j will play an important role inthe proof of Theorem III.4.
Finally, in Sec. 4, one must use an iteration algorithm of Nash–Moser type(see Moser,1961) (a new type of the inverse function theorem) and takes into accountthe fact that one does not solve the linear problem exactly. We construct a sequence ofgood approximating solutions that converges, du to the key estimates (III.15–III.17)established previously and the hypothesis on g and j, to a solution to our problem.
II. SOME TECHNICAL LEMMAS
We use three kinds of norms
j jk ¼ k kCkðOÞ; k kk ¼ k kHkðOÞ and j jk;t ¼ k k
Ck;tðOÞ
where k 2 N and t 2 �0; a½.Let us denote n� ¼ n
2 þ t, and recall some technical lemmas.
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1099
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Lemma II.1 (see Alinhac and Gerard, 1991). Let s� be an integer �7þ n. We canfind a constant b � 2 such that for any 0 � i; j; k � s� þ 2; we have
juji;t � bkukiþn� ðII:1Þ
kukj � bkukk�j
k�i
i kukj�i
k�i
k ; i < j < k ðII:2Þ
kuks� � bjujs� ðII:3Þ
For any l � 1, we can find a family of smoothing operators Sl: HiðOÞ ! HjðOÞ
satisfying
kSluki � bkukj; if i � j ðII:4Þ
kSluki � bli�jkukj; if i � j ðII:5Þ
kSlu� uki � bli�jkukj; if i � j ðII:6Þ
((II.1) and (II.2) are Sobolev and Gagliardo–Nirenberg interpolation inequal-ities, respectively.)
Lemma II.2 (see Alinhac and Gerard, 1991; Hormander, 1984).
(1) If u; v 2 L1 \ Ht (t > 0) then uv 21 \Ht and
kuvkt � K1ðjuj0kvkt þ kuktjvj0Þ ðII:7Þ
K1 is a constant � 1 independent of u, v.(2) Let H : Rm ! C be a function C1 of its arguments.
– If o 2 ðL1 \ HsÞm (s > 0) and joj0 � M, then
kHðoÞks � K2ðs;H;MÞðkoks þ 1Þ ðII:8Þ
K2 is a constant �1 independent of o.– If o 2 ðCi;mÞm, m 2 �0; 1½, i 2 N then HðoÞ 2 Ci;m and if we suppose that
joj0 � M, then we can find a constant K3 ¼ K3ðm; i;H;MÞ � 1 such that
jHðoÞji;m � K3ðjoji;m þ 1Þ: ðII:9Þ
Following Zuily (1988, Lemma 1.6), we get
Lemma II.3. We denote by F the determinant function.
FðuijÞ ¼ detðuij þ aijÞ
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For 1 � i; j; a; b � n, we have
F@2F
@uij@upq¼ @F
@upq
@F
@uij� @F
@uip
@F
@uqjðII:10Þ
Proposition II.4. Under the hypothesis (I.2) the operator P ¼Pni;j¼1
@F@uij
uij� �
@i@jwhere u 2 C3ðOÞ is a formally self-adjoint operator.
Proof. Let us set G ¼ fx 2 O=FðuijÞðxÞ ¼ 0g: Since
P ¼Xni¼1
@i
Xnj¼1
@F
@uijðuijÞ@j
!�Xni;j¼1
@i
�@F
@uij
�ðuijÞ@j
it is sufficient to prove that for j ¼ 1; . . . ; n and x 2 O
AjðxÞ ¼Xni¼1
@i
�@F
@uij
�ðuijÞðxÞ
¼Xn
i;p;q¼1
@2F
@uij@upqðuijÞ½uipqðxÞ þ @ia
pqðxÞ� ¼ 0
Using the identity (II.10), and the hypothesis (I.2) we get AjðxÞ ¼ 0 for anyx 62 G. The continuity of the determinant function allow as to conclude either whenx 2 G.
III. A PRIORI ESTIMATES FOR LINEAR OPERATORS
Following Atallah (2000), by means of the change of unknown f ¼ jþ ew wecan reduce (I.1) to the following equation
detðfijÞ ¼ detðjij þ aij þ ewijÞ ¼ g ðIII:1Þ
We shall consider
GðwÞ ¼ 1
e½detF� g� ðIII:2Þ
The linearized operator of G at w is given by
LGðwÞ ¼Xni;j¼1
fij@xi@xj þXni¼1
bi@xi þ b ðIII:3Þ
where eFF ¼ ðfijÞ is the matrix of cofactors of F ¼ ðfijðx; e;wÞÞ, bi ¼ � @g@pi
andb ¼ � @g
@u.
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1101
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Now, for any smooth real valued function w, the matrix ðfijÞ is symmetric andwe can find a unitary matrix Tðx; eÞ satisfying
Tðx; eÞðfijÞtTðx; eÞ ¼ diagðl1; . . . ; lnÞ ðIII:4Þ
Let us consider a point x0 2 S that can be supposed to be the origin. By means ofchange of variables, we may assume using (I.5) that
jijð0Þ þ aijð0Þ ¼ sidji ; i; j ¼ 1; . . . ; n ðIII:5Þ
where dji is the Kronecker symbol. The constants si are chosen such that
si > 0 for i ¼ 1; . . . ; n� 1; sn ¼ 0 and si 6¼ sj for i 6¼ j ðIII:6ÞLet 0 < t � a
4 : We have
Lemma III.1. We can find three constants e1ðx0Þ > 0, d0ðx0Þ > 0 and M > 0depending only on j; aij; g, n, O such that when
Vx0 ¼ fðx; e;wÞ=jx� x0j � d0; 0 � e � e1; w 2 C3;tðOÞ; jwj3;t � 1g;
we have
(i) The eigenvalues li, i ¼ 1; . . . ; n of F are distinct on Vx0 and of class C1 in�VVx0 . Moreover, li > 0 in Vx0 , for i ¼ 1; . . . ; n� 1:
(ii) For ðx; e;wÞ 2 Vx0
Xni¼1
jsi � liðx; e;wÞj þ����Fnnðx; e;wÞ �
Yn�1
i¼1
si
���� � Mðeþ jxjÞ ðIII:7Þ
(iii) For ðx; e;wÞ 2 Vx0 and i ¼ 1; . . . ; n� 1
li � inf1�i�n�1
si �Md0 � ðM þ 1Þe1 > 0 and
Fnn �Yn�1
i¼1
si �Md0 �Me1 > 0 ðIII:8Þ
Proof. Let us consider the function
Hðx; e;w; lÞ ¼ detðjij þ aij þ ewij � ldji Þ
H is C1 of its arguments, and by (III.5) (III.6) we have for i 2 f1; . . . ; ng
Hð0; 0; 0; siÞ ¼ 0 and@H
@lð0; 0; 0; siÞ 6¼ 0
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By means of the implicit mapping theorem, there exists three constants e1 > 0,d0 > 0 and M > 0 such that we have (i). Moreover by (III.5), we have
@F
@unnðjijÞð0Þ ¼ Fnnð0; 0;wÞ ¼
Yn�1
i¼1
si > 0
which gives (ii) and (iii).Moreover, we have
Lemma III.2. One can find a positive constant e2 such that for 0 < e < e2,w 2 C3;tðOÞ a real valued function satisfying jwj3;t � 1 and y ¼ max
x2O jGðwÞj,the operator
L ¼ �LGðwÞ � y4 ðIII:9Þis a degenerate elliptic operator
�here ¼ 4 ¼Pn
i¼1@2
@xi2
�:
Proof. We have to prove
A ¼ yjxj2 þXni;j¼1
fijxixj � 0; 8ðz; xÞ 2 O�Rn ðIII:10Þ
(�) If x 2 OnS, by (I.5.i), we can deduce that A > 0 for all x 2 Rnnf0g.(�) If x 2 S, we can suppose x ¼ x0 ¼ 0:
For x 2 Rn, let us set x ¼ tTðt; eÞexx. We have
A ¼ yjxj2 þt xeFFx ¼ yjxj2 þt exxT eFFtTexxbut FeFF ¼ detF Id so by (III.4),
detF Id ¼ TFtTT eFFtT ¼ diagðliÞT eFFtT
T eFFtT ¼ detF diag�
1
li
�¼Yni¼1
li diag�
1
li
�¼ ðeGþ gÞdiag
�1
li
�Then,
A ¼ yjexxj2 þ detF Xni¼1
exx2ili
¼ yjexxj2 þXn�1
i¼1
detFjexxij2li
þYn�1
i¼1
liexx2n¼�yþ
Yn�1
i¼1
li
�exx2n þXn�1
i¼1
eGþ g þ ylili
exx2i
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1103
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by (III.8), for i ¼ 1; . . . ; n� 1, e � e1ðx0Þ, jwj3;t � 1 eGþ yli � yðsi �Md0�ðM þ 1Þe1Þ � 0 and A � 0.
Since S is compact, then S � SLi¼0 Bðxi; diÞ, where diðxiÞ is constructed in the
same way us d0ðx0Þ given by Lemma III.1. It suffices then to takee2 ¼ min0�i�L e1ðxiÞ.
We shall also need the result
Proposition III.3. There exists a constant K0 � 1 such that for any functionwi 2 Cs�þ2;tðOÞ, jwij2 � 1, i ¼ 1; 2; 3 and any e � 1 we have
jGðw1Þ �Gðw2Þj0 � K0jw1 �w2j2�kjk2þn� þ kw1k2þn� þ kw2k2þn� þ 1
�ðIII:11Þ
and for t 2 ½0; 1�, s 2 ½0; s����� d
dt½LGðw1 þ tw2Þw3�
���s� eK0½ðkjk2þs þ ekw1k2þs þ ekw2k2þs þ 1Þjw2j2jw3j2þ ðkjk2þn� þ ekw1k2þn� þ ekw2k2þn� þ 1Þ� ðjw2j2kw3k2þs þ jw3j2kw2k2þsÞ� ðIII:12Þ
Proof. Just write
Gðw1Þ �Gðw2Þ
¼ 1
e½detðjij þ ew1
ij þ aijÞ � detðjij þ ew2ij þ aijÞ þ gðw1Þ � gðw2Þ�
¼Z 1
0
Xni;j¼1
@F
@uijðjij þ ew2
ij þ teðw1ij � w2
ijÞÞðw1ij � w2
ijÞdt
þ þZ 1
0
@g
@uðjþ ew2 þ teðw1 � w2ÞÞðw1 � w2Þ d
dt½LGðw1 þ tw2Þw3�
¼ d
dt
"Xni;j¼1
@F
@uijðjij þ ew1
ij þ tew2ijÞw3
ij þ@g
@uðjþ ew1 þ tew2Þw3 þ
#
¼ eXn
i;j;p;q¼1
@2F
@uij@upqðjij þ ew1
ij þ tew2ijÞw2
pqw3ij þ
then combining (II.1) (II.7) (II.8) and (II.9) with the inequalities
jjij þ ew2ij þ teðw1
ij � w2ijÞj0 � jjj2 þ 2jw2j2 þ jw1j2 � 3þ jjj2
and
jjij þ ew1ij þ tew2
ijj0 � jjj2 þ ejw1j2 þ tejw2j2 � 2þ jjj2;
we can deduce (III.11) (III.12).
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Now, we will study a boundary value problem for the degenerate ellipticoperator
L ¼ �LGðwÞ � y4 ¼Xni;j¼1
bijDxiDxj þXni¼1
bi@xi þ b
bij ¼ @F
@uij
ðjij þ ewijÞ þ ydji ¼ Fij þ ydji
bi ¼ K@f
@pi
b ¼ K@f
@u
For the sake of simplicity, we put
AðkÞ ¼ max�1; max
1�i;j�njbijjk; max
1�i�njbijk; jbjk
�Ls ¼ fði; jÞ; 0 � i; j � s; iþ j � s and iþ 2 � maxðs; 2Þg:
8<: ðIII:13Þ
for k; s 2 NThe main result of this section is the following
Theorem III.4. Suppose that y � 1 and for some constant M0 > 0, we haveAð2Þ � M0: One can find e3 > 0 such that for any e 2 �0; e3� any w 2 Cs�þ2;tðOÞ a realvalued function satisfying the inequality jwj3;t � 1 and any real valued functionh 2 Hs� , the problem
Lu ¼ h in O
u j @O ¼ 0:
ðIII:14Þ
possess a unique solution u 2 Hs� : Moreover for 0 � s � s�, the inequalities
kuk0 � C0khk0 ðIII:15Þ
kuk1 � C1ðkhk1 þ kuk0Þ ðIII:16Þ
kuks � Cs
(khks þ
Xði;jÞ2Ls
j�s�1
ð1þ jjþ ewjiþ4;tÞkukj); s � 2 ðIII:17Þ
for some constant Cs ¼ Csðj; aij; g; s;O;M0; e3Þ, independent of w and e holds.
In order to solve the Dirichlet problem (III.14), we need the following. Forn 2 �0; 1½; we denote Ln ¼ L� n4.
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1105
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Proposition III.5. Suppose that y � 1 and for some constant M0 > 0, we haveAð2Þ � M0: One can find e3 > 0 such that for any e 2 �0; e3� any real valued functionw 2 Cs�þ2;tðOÞ satisfying the inequality jwj3;t � 1 and any real valued functionh 2 Hs� ðOÞ the regularized problem
Lnu ¼ h in O
u j @O ¼ 0:
ðIII:140Þ
admits a unique solution u 2 Hs�þ1ðOÞ.
Proof. First of all, we recall that LGðwÞ is a real second order operator with realcoefficients. From the Lemma III:2, Ln is uniformly elliptic with coefficients inCs�;tðOÞ then we conclude using Gilbarg and Trudinger (1983) (Theorems 6.14 and8.13).
Now, suppose (III.15) to (III.17) are valid for the regularized problem (III.140)with an uniform constantCs for n 2 �0; 1�, then by letting n go to zero we shall get a solu-tion u 2 Hs� ðOÞ to the original problem which of course will satisfy (III.15) to (III.17).
In the rest of this paragraph, we shall prove the estimates (III.15) to (III.17) forthe modified linearized operators Ln .
Unless otherwise specified, Q denotes a degenerate elliptic operator of the form
Q ¼Xni;j¼1
bijDiDj þXni¼1
bi@i þ b
with Cs�;t real coefficients b, bi, bij ¼ bji defined in O.Assume that there is a continuous function lðxÞ � 0 defined in O such that
Xni;j¼1
bijxixj � lðxÞ
III.1. Estimates in the Elliptic Zone of L
Lemma III.6. Assume that Q is uniformly elliptic in O, that is
Xni;j¼1
bjiðxÞxixj � l0jxj2; l0 is a constant >0
Then for any integer 1 � s � s�, there exists a constant C 0s depending only on
s; l0 and Að0Þ such that for any real function u 2 Cs�;tðOÞ \ H10ðOÞ
kuk1 � C 01ðkQuk0 þ Að2Þkuk0Þ ðIII:18Þ
kuks � C 0s
kQuks�1 þ
Xiþj�s�1i�s�2
Aðiþ 2Þkukj!; s � 2 ðIII:19Þ
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It is not difficult to prove (III.18). In fact, we need only to apply well-knownstandard techniques to the linear elliptic operator Q and to calculate several con-stants precisely. By induction with respect to s and patient calculation, (III.19)follows from (III.18).
Let S be a subset of O satisfying
fx 2 O; lðxÞ ¼ 0g � S
and for d > 0; we define a set Sd by
Sd ¼ fx 2 O;dðx; SÞ < dg
Lemma III.7. Assume that S is a compact C1 submanifold of O and OnS is con-nected. Then there exists a function m 2 L1ðOÞ and a constant C > 0 such thatm ¼ 0 on S, md ¼ infOnSd m > 0 for any sufficiently small d andZ
Omu2 dx � C
kQuk0kuk0 þ
1
2sup½bijij þ bii � 2b�kuk20
ðIII:20Þ
for u 2 Cs�;tðOÞ \ H10ðOÞ.
Proof. Stand techniques of elliptic operators giveZljDuj2dx � C
kQuk0kuk0 þ
1
2sup½bijij þ bii � 2b�kuk20
Hence, using the inequalityZmu2 dx �
ZljDuj2 dx
established by (see Amano, 1988), we get (III.20)
Lemma III.8. We have for u 2 C10ðOÞX
k
k½@k;Q�uk20 � CðAð2ÞkQuk1kuk1 þ Að2Þ2kuk21Þ ðIII:21ÞXk
k½@k;Q�uk2s � C
�Að2ÞkQuksþ11kuksþ1 þ
Xði;jÞ2Lsþ1
Aðiþ 2Þ2kuk2j�; s � 1
ðIII:22Þ
Proof. ð�Þ Lemma 1.7.1 of Oleınik and Radkevic shows that
ðbijk uijÞ2 � CAð2Þbijuliulj
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1107
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which implies
Xk
k½@k;Q�uk20 � CXk
Z nðbijk uijÞ2 þ ðbkuÞ2
o� CAð2Þ
Xk
Zbijuliulj þ CAð1Þ2kuk20
By integrating by parts, we haveZbijuliulj ¼ �hðQuÞl; uli þ h½@l;Q�u; uli þ 1
2hðbijij � 2bÞul; uli
which meansZbijuliulj � C
�kQuk1kuk1 þ
Xk
k½@k;Q�uk0kuk1 þ Að2Þkuk21�
From these inequalities, it follows that
Xk
k½@k;Q�uk0 � CðAð2ÞkQuk1kuk1 þ Að2Þ2kuk21Þ
(�) For s � 1, (III.22) is proved by recursion on s using (III.21).
Lemma III.9. Let w 2 C1ðOÞ satisfying supp Hw � O. For any integer 0 � s � s�,there exists a constant Cs > 0 such that 8u 2 Cs�;tðOÞ
k½w;Q�uk2s � Cs
�Að2ÞkQukskuks þ
Xði;jÞ2Ls
Aðiþ 2Þ2kuk2j�
ðIII:23Þ
Proof. Let us consider a cut-off function eww 2 C10 ðOÞ satisfying 0 � eww � 1 and eww ¼ 1
onS
i supp @iw, and define an operator eQQ ¼ ~bbijDxiDxj þ ~bb by eQQ ¼ ewwQ. Since
½w; eQQ�u ¼ ½w;Q�u and keQQuks � CkQuks, it will suffice to prove ðIII:23Þ for eQQ.
(�) For s ¼ 0, Corollary of Lemma 1.7.1 of Oleınik and Radkevic showsthat
�Xj
~bbijuj
�2
� 2Að0Þ~bbijuiuj
which gives
k½w; eQQ�uk20 � CAð0ÞZ
~bbijuiuj þ CAð0Þ2kuk20
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by integrating by parts, we haveZ~bbijuiuj ¼ heQQu; ui þ 1
2
D�~bbijij � 2~bb
�u; uE
� keQQuk0kuk0 þ CAð2Þkuk20
which implies (III.23)0.
(�) For s � 1, we prove (III.23)s using (III.23)0 and an induction withrespect to s.
III.2. Estimates Near the Degenerate Points of L
For t � 1, we define
Ut ¼(x 2 O; jxnj < 1
t
)
Lemma III.10. We can find two constants C0 > 0 and C � 0 independent of Q andt such that; 8u 2 C2ðUtÞ, u j @Ut
¼ 0
C0t2 inf
Ut
ðbnnÞkuk20 þ t infUt
ðbnÞkuk20 � C
(kQuk0kuk0 þ
1
2supUt
½bijij þ bii � 2b�kuk20)
ðIII:24ÞProof. For a real-valued function u 2 C2ðUtÞ satisfying u j @Ut
¼ 0, we put
v ¼ ðT � etxnÞ�1u; T = constant >5e
Direct computation gives
Qu ¼ ðT � etxnÞQv� tetxnf2bnjvj þ bnvþ tbnnvgZðT � etxnÞ�1
Qu:v ¼ �Iþ II� III� IV� V
with
I ¼Z
bijvivj
II ¼ 1
2
Zfbijij � bii þ 2bgv2
III ¼ t2ZetxnbnnðT � etxnÞ�1
v2
IV ¼ t
ZetxnbnðT � etxnÞ�1
v2
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1109
ORDER REPRINTS
V ¼ 2t
ZetxnðT � etxnÞ�1
vbnjvj
Using the Cauchy–Schwartz inequality, we get
jVj �Zbijvivj þ 4t2
Ze2txnðT � etxnÞ�2
bnnv2
Since
etxnðT � etxnÞ�1 � 4e2txnðT � etxnÞ�2 ¼ etxnðT � etxnÞ�2ðT � 5etxnÞThen
t2Ze2txnðT � etxnÞ�4ðT � 5etxnÞbnnu2 � �
ZðT � etxnÞ�2
Qu u� II� IV
but, e�1 � etxn � e, ðT � e�1Þ�1 � ðT � etxnÞ�1 � ðT � eÞ�1. So, we obtain (III.24).Let x be any point of S and t � tx � 1. The main result of this section is
Proposition III.11. We can find a neighborhood VtðxÞ of x, such that: For anyinteger s 2 ½0; s�� and any function u 2 C
s�;t0 ðVtðxÞÞ, there exists a constant
C00s ¼ C00
s ðn;O;j; dxÞ > 0 such that
kuk0 � C000t
�1kLnuk0 ðIII:25Þ
kuks � C00s t
�1
�kLnuks þ
Xði;jÞ2Ls
Aðiþ 2Þkukj�; s � 1 ðIII:26Þ
(where dx is given in the same way us d0ðx0Þ of Lemma III.1).
Proof. We can suppose x ¼ x0 ¼ 0. Let us set Vtðx0Þ ¼ Ut \ Bðx0; d0Þ.ð�Þ To get (III.25), we apply Lemma III.10, so we can write for
u 2 Cs�;t0 ðVtðx0ÞÞ,
t tC0 infUðtÞ
ðbnnÞ � C
2supUðtÞ
jbijij þ bii þ 2bj( )
kuk20 � CkLnuk0kuk0
but
bnn ¼ ðFnn þ 4ðyþ nÞÞ
If jwj3;t � 1, jxj � d0 and e � e1ðx0Þ, we have
Fnn �Yn�1
i¼1
si �Md0 �Me1 ¼ a > 0
1110 Kallel-Jallouli
ORDER REPRINTS
We have only to take
t � t0 ¼ 1þ 2CAð2ÞaC0
and (III.25) is proved.
ð�Þ To prove (III.26), we use (III.25) and a recursion on s.
III.3. Proof of the Estimates (III.15) to (III.17) for Ln
Proposition III.12. For any point x 2 S, we can find a neighborhood V ðxÞ such that:for any cut-off function w 2 C1
0 ðV ðxÞÞ, u 2 Cs�;tðOÞ \H10ðOÞ and 1 � s � s�
kwuks � 2C00s
�kLnuks þ k½w;Ln �uks þ
Xði;jÞ2Ls
j<s
ðjjþ ewjiþ4;t þ 1Þkukj�
ðIII:27Þ
Proof. It suffices to consider the point x0. We shall take the neighborhood Vtðx0Þconstructed in the proof of Proposition III.11.
Let us consider a cut-off function w 2 C10 ðVtðx0ÞÞ,
For u 2 Cs�;t \H10 ðOÞ. Since supp w � Vtðx0Þ, we have by (III.26) for any
1 � s � s�
kwuks � C00s t
�1
�kwLnuks þ k½w;Ln �uks þ
Xði;jÞ2Ls
j<s
Aðiþ 2Þkukj�
þ C00s t
�1Að2Þkwukswe have Að2Þ � M0. We fix t � t0 such that for 1 � s � s�
C00s t
�1Að2Þ � 1
2
In the other hand,
Aðiþ 2Þ
¼ max1�p;q�n
�1;
���� @g@u ðjþ ewÞ����iþ2
;
���� @g@pp
ðjþ ewÞ����iþ2
;
���� @F@upqðjkl þ ewklÞ
����iþ2
þ y�;
but for k 2 f0; 1; 2g, j@kjþ e@kwj0 � jjj2 þ 1, then by (II.9), since y � 1, we get for0 � i � s� � 2;
Aðiþ 2Þ � CðjÞðjjþ ewjiþ4;t þ 1Þ ðIII:28Þ
and we deduce (III.27).
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1111
ORDER REPRINTS
Proof of Theorem III.4. Since S is compact, then
S �[
0�i�N
VðxiÞ ¼ V
(where VðxiÞ is defined in Proposition III.12). Later we shall denotee1 ¼ inf0�i�N e1ðxiÞ, where e1ðxiÞ is constructed in the same way as in Lemma III.1.
Using partition of unity, there exists a family of regular functions wi such that,for each i 2 f1; . . . ;Ng
supp wi � VðxiÞ; 0 � wi � 1 andXi
wi ¼ w � 1 in a neighborhood of S:
Since kuks � kð1� wÞuks þP
i kwiuks; it will suffice to estimate kð1� wÞuks andkwiuks, respectively.
Proof of III.15. Since w ¼ 1 in a neighborhood of S in V , then, Suppð1� wÞ � OnS.Let us consider the following cut-off functions: eww, eewweww 2 C1
0 ðOnSÞ, 0 � eww, eewweww;� 1 andsuch that; eww ¼ 1 on
Si;j supp @iwj and
eewweww ¼ 1 on supp eww. We consider now the functionm given by Lemma III.7 (md depends only on j;O; n)
By III.20, there exists C0 ¼ C0ðj;O; nÞ > 0 such that
kð1� wÞuk20 ¼ZOnS
u2dx � 1
md
Zmu2dx � C0ðkuk0kLuk0 þ Bkuk20Þ
where B ¼ 12 sup½bijij þ bii � 2b�:
By Proposition II.4,P
ij bijij ¼ 0, and the hypothesis (I.4) imply that
Xi
bii � 2b ¼Xi
@K
@xi
@f2@pk
þ K
Xij
@2f2
@pi@pj
ðjij þ ewijÞ � 2@f1@u
� eXij
���� @2f2
@pi@pj
����so B � en2kf2kC2 ¼ r. Hence, we have
kð1� wÞuk20 � C0ðkuk0kLuk0 þ rkuk20ÞSince Suppeewweww � OnS, we also have by the same way,
keewwewwuk20 � C0ðkuk0kLuk0 þ rkuk20ÞOn the other hand by (III.25) and fixing t � max0�i�N txi , we have
kwiuk20 � C2kLnwiuk20 � C2ðkLnuk20 þ k½wi;Ln �uk20Þ
but ewwLneewwewwu ¼ ewwLnu and ½wi;Ln �u ¼ ½wi;ewwLn �eewwewwu, so since Að2Þ � M0 and n � 1 we get
1112 Kallel-Jallouli
ORDER REPRINTS
using Lemma III.8,
k½wi;Ln �uk20 ¼ k½wi;ewwLn �eewwewwuk20 � C½kewwLnw2uk20 þ ðM0 þ 1Þ2keewwewwuk20�� C0ðkLnuk20 þ keewwewwuk20Þ
Combining these inequalities we get (III.15), for e small enough.
Proof of ðIII.16Þ. Since Suppð1� wÞ � OnS. then by (I.5.i), for e � e4 smallenough, L is uniformly elliptic on E: Using (III.18) and the estimation Að2Þ � M0,we have
kð1� wÞuk1 � C01ðkLnuk0 þ ðM0 þ 1Þkuk0 þ k½w;Ln �uk0Þ
Applying Lemma III.8, we get
k½w;Ln �uk0 � C0ðkLnuk0 þ ðM0 þ 1Þkuk0Þ
so,
kð1� wÞuk1 � C1ðM0ÞðkLnuk0 þ kuk0Þ
On the other hand, since Að2Þ � M0, we get using (III.27)
kwiuk1 � C1ðM0ÞðkLnuk1 þ k½wi;Ln �uk1 þ kuk0ÞBut ewwLn
eewwewwu ¼ ewwLnu and ½wi;Ln �u ¼ ½wi;ewwLn �eewwewwu, so since Að2Þ � M0, Lemma III.8gives
k½wi;Ln �uk1 � C1ðkewwLneewwewwuk1 þ ð1þM0Þkeewwewwuk1Þ
� C1ðkLnuk1 þ ð1þM0Þkeewwewwuk1ÞSince L is uniformly elliptic on Suppeewweww and Að2Þ � M0, then we have by (III.18)
keewwewwuk1 � C01ðkLnuk0 þ ðM0 þ 1Þkuk0 þ k½eewweww;Ln �uk0Þ
that gives using (III.23)
keewwewwuk1 � C1ðM0ÞðkLnuk1 þ kuk0Þ
The proof of (III.17) is identical to that of (III.16) using the inequalities (III.18)(III.19) (III.23) and (III.27).
IV. PROOF OF THEOREM A
Now we shall construct a sequence which converges to a solution of ourproblem.
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1113
ORDER REPRINTS
We begin by fixing the constants. We choose first
M0 ¼ 1þmaxH2F
K3ð2; t;H; ð1þ jjj2ÞÞð1þ jjj4;tÞ ðIV:1Þ
where
F ¼
@F
@uij;@g
@u;@g
@pi
�1 � i; j � n
and K3 is the constant introduced in (II.9) (i.e., jHðuÞjj;m � K3ðj; m;H;MÞjujj;m).Then
D ¼ max�max0�s�s�
Cs; 1�
ðIV:2Þ
Cs being the constant (depending only on s;j; g; aij;O;M0) given by theTheorem III.4
m ¼ maxðb; 3Ds2�ð1þ jjjs�þ2;tÞ; 21tÞ and ~mm ¼ b2ms� ðIV:3Þ
a1 ¼ 9K0m5
a2 ¼ 5a1ms�þ1
a3 ¼ 7K0m5:
8><>: ðIV:4Þ
where K0 is the constant determined by Proposition III.3.Finally, we fix e0 satisfying
ffiffiffiffie0
p � minh
inf1�i�4
ð1; eiÞ; ð3D2a2 þ 6~mmD2Þ�2i
ðIV:5Þ
where ei are given in the proof of Theorem III.4, Lemma III.2, Theorem III.4 and theproof of (III.16).
After the constants being fixed, let g be Cs� of its arguments and satisfy
jdetðjij þ aijÞ � gðjÞjs� � e0
Let denote by Sn ¼ Smn the family of operators given by Lemma II.1, whenmn ¼ mn (m is given by IV.3).
We will construct wn, n ¼ 0; 1; . . . ; by induction on n as follows; starting with u0,w0 ¼ 0, suppose w0, w1; . . . ;wn have been chosen and define wnþ1 as follows
wnþ1 ¼ wn þ unþ1 ðIV:6Þ
where unþ1 is the solution of the Dirichlet problem
LGð~wwnÞunþ1 þ yn4unþ1 ¼ gn; in Ounþ1 j @O ¼ 0 ðIV:7Þ
1114 Kallel-Jallouli
ORDER REPRINTS
satisfying
~wwn ¼ Snwn ðIV:8Þyn ¼ jGð~wwnÞj0 ðIV:9Þg0 ¼ �S0Gð0Þ; gn ¼ Sn�1Rn�1 � SnRn þ Sn�1Gð0Þ � SnGð0Þ ðIV:10Þ
R0 ¼ 0; Rn ¼Xnj¼1
rj ðIV:11Þ
r0 ¼ 0; rj ¼ ½LGðwj�1Þ � LGð~wwj�1Þ�uj þQj � yj�14uj; 1 � j � n ðIV:12ÞQj ¼ GðwjÞ �Gðwj�1Þ � LGðwj�1Þuj; 1 � j � n ðIV:13ÞThe wk are well defined. We give first the following result.Let us denote ~ee ¼ ffiffiffiffi
e0p
:
Proposition IV.1. Under assumptions
s� � 7þ n
4þ nþ 2t � s < s� � 2
then we have for any s 2 N,
kujks �ffiffi~ee
p½maxðm; mj�1Þ�s�s; j 2 N�; 0 � s � s� ðIV:14Þ
kwjks �2ffiffi~ee
p; for s � s� tffiffi
~eep
ms�sj ; for s� t � s � s�; j 2 N�
(ðIV:15Þ
j~wwjj4;t � 1; j 2 N� ðIV:16Þ
kwj � ~wwjks � 2bffiffi~ee
pms�sj ; 0 � s � s�; j 2 N� ðIV:17Þ
krjks � ~eea1½maxðm; mj�1Þ�s�s; 0 � s � s� � 2; j 2 N� ðIV:18Þkgjks � ~eea2ms�s
j ; 0 � s � s�; j 2 N ðIV:19Þ
yj � a3ffiffi~ee
pm�2j � 1; j 2 N ðIV:20Þ
Ajð2Þ � M0; j 2 N ðIV:21Þ
Proof. We have u0 ¼ 0: Let begin by proving (IV.19)0 to (IV.21)0.
(a) (IV.19)0: Using (III.2) and (IV.10), we have
g0 ¼ �S0Gð0Þ and Gð0Þ ¼ 1
e½detðjij þ aijÞ � gðjÞ�
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1115
ORDER REPRINTS
But j 2 Cs�þ2;aðOÞ, g is Cs� of its arguments and Sn are smoothing operators, sog0 2 Hs� ðOÞ. (II.3) (II.4) (III.2) and (I.6) show that
kg0ks � bkGð0Þks �b~eek detðjij þ aijÞ � gðjÞks�
� b2
~eej detðjij þ aijÞ � gðjÞjs� � b2~ee
using (IV.4) and b � m, we get
kg0ks � m2~ee � a2~ee
(b) (IV.20)0: (III.2) (IV.4) (IV.5) and (I.6) give
y0 ¼ jGð0Þj0 �1
~eej detðjij þ aijÞ � gðjÞjs� � ~ee �
ffiffi~ee
pa2 � 1
(c) (IV.21)0: We have
A0ð2Þ ¼ maxi;j
1;
���� @g@pi
ðjÞ����2
;
���� @g@u ðjÞ����2
;
���� @F@jij
ðjlqÞ����2
þ y0
!
Then, by (II.9) (IV.1) and (IV.20)0, A0ð2Þ � M0.
Assume that we have constructed u0, u1; . . . ; un�1 2 Hs� ðOÞ satisfying (III.15) to(III.17) and we have proved (IV.14) to (IV.21) for j � n� 1. We shall constructun 2 Hs� ðOÞ satisfying (III.15) to (III.17) and prove that (IV.14) to (IV.21) are satis-fied for j ¼ n. Combining (IV.16)n�1 to (IV.21)n�1, we have j~wwn�1j4;k � 1, yn�1 � 1,An�1ð2Þ � M0 and gn�1 2 Hs� ðOÞ. We can then apply Theorem III.4 to get a solutionun 2 Hs� ðOÞ to the problem (IV.7)n satisfying (III.15) to (III.17). Then:
(a) (IV.14)n: ð�Þ For n ¼ 1, using (I.6) (II.3) (III.2) (III.15) and (IV.2), weget
ku1k0 � Dkg0k0 � DbkGð0Þk0 � Db2
~eejdetðjij þ aijÞ � gðjÞjs� � Db2~ee
(IV.3) (IV.5) and s� � s give
ku1k0 �ffiffi~ee
pm�s ðIV:22Þ
We have by (III.16)
ku1k1 � Dðkg0k1 þ ku1k0Þ
so, using (I.6) (II.3) (IV.22) and s� � s, we get
ku1k1 � Dðb2~eeþffiffi~ee
pm�sÞ �
ffiffi~ee
pm1�s
1116 Kallel-Jallouli
ORDER REPRINTS
Suppose that we have for 0 � l � s; s � 2 :
ku1kl �ffiffi~ee
pml�s ðIV:23Þ
Using (III.17), we have for s � 2,
ku1ks � D
kg0ks þ
Xði;lÞ2Ls
l�s�1
ð1þ jjjiþ4;tÞku1kl!
ðIV:24Þ
(I.6) (II.3) (II.4) (IV.3) (IV.10) and s� � s imply
kg0ks � bkGð0Þks � b2jGð0Þjs � b2~ee � ~mm~eems�s
that gives by (IV.23) (IV.24)
ku1ks � D
~mm~eems�s þ
Xði;lÞ2Ls
l�s�1
ð1þ jjjiþ4;tÞffiffi~ee
pml�s
!
� D
�~mm~eems�s þ s2�ð1þ jjjiþ4;tÞm�1
ffiffi~ee
pms�s
�which shows by (IV.3) (IV.5) that
ku1ks �ffiffi~ee
pms�s
ð�Þ For n � 2, (III.15) (IV.2) (IV.5) (IV.19)n�1 imply that
kunk0 � Dkgn�1k0 � D~eea2m�sn�1 �
ffiffi~ee
pm�sn�1
ðIV:25Þ
By the same way, (III.16) (IV.2) (IV.5) ((IV.19)n�1) and (IV.25) give
kunk1 �ffiffi~ee
pm1�sn�1
Suppose that for 0 � l < s, s � 2
kunkl �ffiffi~ee
pml�sn�1
By (III.17), we have
kunks � D
kgn�1ks þ
Xði;lÞ2Ls
l�s�1
ð1þ jjþ ~ee~wwn�1jiþ4;tÞkunkl!
But, (II.1) (II.5) (IV.15)n�1 and 4þ n� � s� t imply that for 0 � i � s� 2
j~wwn�1jiþ4;t � bk~wwn�1k4þn�þi � b2min�1k~wwn�1k4þn� � 2b2ffiffi~ee
pmin�1
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1117
ORDER REPRINTS
so, using ðIV:19Þn�1
kunks � D
�~eea2ms�s
n�1 þX�
1þ jjjs�þ2;t þ 2b2ffiffi~ee
pmin�1
� ffiffi~ee
pml�sn�1
�� D
�~eea2ms�s
n�1 þ 2b2s2�~eems�sn�1 þ
�1þ jjjs�þ2;t
�s2�
ffiffi~ee
pms�1�sn�1
�combined with (IV.4) (IV.5), that gives
kunks �ffiffi~ee
pms�sn�1
(b) (IV.15)n: (IV.6) shows that wn ¼Pn
j¼1 uj. By (IV.14)j, 1 � j � n we have
kwnks �Xnj¼1
kujks �ffiffi~ee
pms�s þ
Xnj¼2
ffiffi~ee
pms�sj�1 �
ffiffi~ee
pms�s þ
Xn�1
j¼2
ffiffi~ee
pms�sj
For s � s� t, since m � 21t � 2 then ms�s
j � m�tj � 1
2j and
kwnks �Xn�1
j¼0
ffiffi~ee
pms�sj �
ffiffi~ee
p Xn�1
j¼0
1
2j� 2
ffiffi~ee
p
For s � s� t, we have
kwnks �ffiffi~ee
pms�s þ
ffiffi~ee
p mnðs�sÞ � ms�s
ms�s � 1
Since m � 21t then ms�s � mt � 2 so,
kwnks �ffiffi~ee
pms�sn
(c) (IV.16)n: Combining (II.1) (II.4) (IV.5) (IV.15)n and 4þ n� � s� t, weobtain
j~wwnj4;t � bk~wwnk4þn� � b2kwnk4þn� � 2b2ffiffi~ee
p� 1
(d) (IV.17)n:
In the case s � s� t, we obtain using (II.6) and (IV.15)n
kwn � ~wwnks � bms�½sþt��1n kwnk½sþt�þ1
� bms�½sþt��1n
ffiffi~ee
pm½sþt�þ1�sn � b
ffiffi~ee
pms�sn
1118 Kallel-Jallouli
ORDER REPRINTS
In the case s > s� t, (II.6) (IV.15n) and b � 1 give
kwn � ~wwnks � bkwnks � bffiffi~ee
pms�sn
(e) (IV.18)n: We have by (IV.12)
rn ¼ ½LGðwn�1Þ � LGð~wwn�1Þ�un|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ð1Þ
� yn�14un|fflfflfflfflffl{zfflfflfflfflffl}ð2Þ
þ Qn|{z}ð3Þ
When n ¼ 1, ð1Þ ¼ 0. In the case n � 2, since
ð1Þ ¼Z 1
0
d
dt½LGð~wwn�1 þ tðwn�1 � ~wwn�1ÞÞun�dt
We have by (II.1) and (IV.17)n�1
jwn�1 � ~wwn�1j2 � bkwn�1 � ~wwn�1k2þn� � 2b2ffiffi~ee
pm3þn��sn�1
But 2b2ffiffi~ee
p � 1 and 3þ n� � 4þ 2n� � s, so jwn�1 � ~wwn�1j2 � 1. By the sameway, (II.1) (IV.5) and (IV.14)n give
junj2 � bkunk2þn� � bffiffi~ee
pm3þn��sn�1 � 1
We also have by (IV.16)n�1, j~wwn�1j2 � 1:Hence, we can apply Proposition III.3 to get
kð1Þks � ~eeK0f½kjksþ2 þ k~wwn�1ksþ2 þ kwn�1ksþ2 þ 1�jwn�1 � ~wwn�1j2junj2þ ðkjk2þn� þ k~wwn�1k2þn� þ kwn�1k2þn� þ 1Þ� ðjwn�1 � ~wwn�1j2kunksþ2 þ kwn�1 � ~wwn�1ksþ2junj2Þg
Using (II.3) (IV.3), we get for 0 � s � s�
kjksþ2 � bjjjs�þ2 � bm � m2
By (II.2), it suffices to prove (IV.18)n for s ¼ 0 and s ¼ s� � 2:
s ¼ 0: Combining (II.1) (IV.14)n(IV.15)n�1 and (IV.17)n�1, we have
kð1Þk0 � ~eeK0
h�m2 þ 2b
ffiffi~ee
pþ 2
ffiffi~ee
pþ 1�2b3~eem4þ2n��2s
n�1
þ�m2 þ 2bffiffi~ee
pþ 2
ffiffi~ee
pþ 1�4b3~eem4þn��2s
n�1
ithat gives using (IV.5) and s � 4þ 2n� � 4þ n�
kð1Þk0 � ~eeK0m�sn�1:
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1119
ORDER REPRINTS
s ¼ s� � 2: (IV.5) and s� � sþ t imply as in the previous case
kð1Þks��2 � ~eeK0ms��2�sn�1
By (II.2), we obtain for 0 � s � s� � 2
kð1Þks � b~eeK0ms�sn�1
Next,
kð2Þks � yn�1kunksþ2
In the case n ¼ 1: combining (IV.5) (IV.9) and (IV.14)n; we obtain
kð2Þks � jGð0Þj0ku1ksþ2 � ~eeffiffi~ee
pmsþ2�s � ~eems�s
In the case n � 2: (IV.14)n and (IV.20)n�1 imply
kð2Þks � a3~eem�2n�1m
sþ2�sn�1 ¼ a3~eems�s
n�1
Finally, since by (IV.13)
ð3Þ ¼ Qn ¼ Gðwn�1 þ unÞ �Gðwn�1Þ � LGðwn�1Þun
¼Z 1
0
Z t
0
d
dh½LGðwn�1 þ hunÞun�dh
!dt
Then using (II.1) (IV.5) and (IV.15)n�1, we obtain
jwn�1j2 � bkwn�1k2þn� � 2bffiffi~ee
p� 1
Since we proved that junj2 � 1, we can apply Proposition III.3 to have
kð3Þks � ~eeK0½ðkjksþ2 þ kunksþ2 þ kwn�1ksþ2 þ 1Þjunj22þ 2junj2kunksþ2ðkjk2þn� þ kunk2þn� þ kwn�1k2þn� þ 1Þ�
By combining (II.1) (IV.14)n(IV.15)n�1, we get For s ¼ 0:
kð3Þk0 � ~eeK0
��m2 þ
ffiffi~ee
p½maxðm;mn�1Þ�2�s þ 2
ffiffi~ee
pþ 1�b2~ee½maxðm;mn�1Þ�4þ2n��2s
þ 8�m2 þ
ffiffi~ee
pb½maxðm;mn�1Þ�2þn��s þ 2
ffiffi~ee
pþ 1�~eeb½maxðm;mn�1Þ�4þn��2s�
which gives combined with (IV.5) and s � 4þ 2n�
kð3Þk0 � ~eeK0½maxðm; mn�1Þ��s:
1120 Kallel-Jallouli
ORDER REPRINTS
For s ¼ s� � 2: we get also since s � 4þ 2n�
kð3Þks��2 � ~eeK0½maxðm; mn�1Þ�s��2�s
Then (II.2) shows that for 0 � s � s� � 2
kð3Þks � b~eeK0½maxðm; mn�1Þ�s�s
and we conclude that
krnks � ð2bK0 þ a3Þ~ee½maxðm; mn�1Þ�s�s
� 9K0m5~ee½maxðm; mn�1Þ�s�s ¼ a1~ee½maxðm; mn�1Þ�s�s
(f) (IV.19n): By (IV.10) and (IV.11)
gn ¼ Sn�1Rn�1 � SnRn þ ðSn�1 � SnÞGð0Þ¼ ðSn�1Rn�1 � SnRn�1Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð4Þ
� Snrn|{z}ð5Þ
þ ðSn�1 � SnÞGð0Þ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}ð6Þ
The case s ¼ 0: (II.6) (IV.11) and (IV.18)j, j � n� 1 imply
kð4Þk0 � kðI � Sn�1ÞRn�1k0 þ kðI � SnÞRn�1k0� bkRn�1ks��2m
2�s�n�1 þ bm2�s�
n kRn�1ks��2
� ðba1~eem2�s�n�1 þ ba1~eem2�s�
n Þ ms��2�s þ
Xn�1
j¼2
ms��2�sj�1
!
Since s� � 2 > s and b � m, then
kð4Þk0 � ba1~eeðm2�s�n�1 þ m2�s�
n Þms��2�sn�1
� 2a1m2~eem�sn
In the other hand, combining (II.4) (IV.18)n, s < s� � 2 and b � m we obtain
kð5Þk0 � bkrnk0 � ba1~ee½maxðm; mn�1Þ��s � a1m2~eem�sn
We also have by (I.6) (II.3) (II.6) and s < s� � 2
kð6Þk0 � kðI � Sn�1ÞGð0Þk0 þ kðI � SnÞGð0Þk0� bm�s
n�1kGð0Þks þ bm�sn kGð0Þks
� b2m�sn�1jGð0Þjs� þ b2m�s
n jGð0Þjs�� b2~eem�s
n ðms þ 1Þ � 2ms�~eem�sn
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1121
ORDER REPRINTS
We finally get
kgnk0 � ð2þ 3a1Þms�~eem�sn
The case s ¼ s�: (II.5) (IV.11) (IV.18)j, 1 � j � n and s < s� � 2 show that
kð4Þ þ ð5Þks� � kSn�1Rn�1ks� þ kSnRnks�� bm2n�1kRn�1ks��2 þ bm2nkRnks��2
� bm2n�1a1~ee
ms��2�s þ
Xn�1
j¼2
ms��2�sj�1
!
þ bm2na1~ee
ms��2�s þ
Xnj¼2
ms��2�sj�1
!� ba1~eeðm2n�1m
s��2�sn�1 þ m2nm
s��2�sn Þ
� 2ba1~eems��sn � 2ma1~eems��s
n
Next, we have by (I.5) (II.5) (II.3) and b � m
kð6Þks� � kSnGð0Þks� þ kSn�1Gð0Þks�� bms��s
n kGð0Þks þ bms��sn�1 kGð0Þks
� 2b2~eems��sn � 2m2~eems��s
n
so,
kfnks� � 2mða1 þ mÞ~eems��sn
We can finally conclude using (IV.4) and m � a1
kfnks� � 4a1m2~eems��sn � a2~eems��s
n
(g) (IV.20)n: We have by (IV.9)
yn ¼ jGð~wwnÞj0 � jGðwnÞ �Gð~wwnÞj0 þ jGðwnÞj0
Combining (IV.7) to (IV.13), we have
GðwnÞ ¼ ðI � Sn�1ÞRn�1 þ ðI � Sn�1ÞGð0Þ þ rn
Then
yn � jGðwnÞ �Gð~wwnÞj0|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ð7Þ
þ jðI � Sn�1ÞRn�1j0|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}ð8Þ
þ jðI � Sn�1ÞGð0Þj0|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ð9Þ
þ jrnj0|{z}ð10Þ
1122 Kallel-Jallouli
ORDER REPRINTS
Since we proved that jwnj2 � 1 and j~wwnj2 � 1, we can apply Proposition III.3to get
ð7Þ � bK0kwn � ~wwnk2þn� ðkjk2þn� þ kwnk2þn� þ k~wwnk2þn� þ 1Þ
(II.4) (IV.15)n(IV.17)n and 3þ n� � 4þ 2n� � t � s� t give
ð7Þ � 2b2K0
ffiffi~ee
pm2þn��sn ðm2 þ 2
ffiffi~ee
pþ 2b
ffiffi~ee
pþ 1Þ
Since ~ee � 1ð6b2Þ2, b � m and 4þ n� � s � 4þ 2n� � s � 0 then
ð7Þ � 4m5K0
ffiffi~ee
pm�2n
In the case n ¼ 1, ð8Þ ¼ 0. For n � 2, since b � m, n� � s � �2 andm4a1
ffiffi~ee
p � a2ffiffi~ee
p � 1, combining (II.1) (II.6) (IV.11) and ðIV:18Þj, j � n� 1 we obtain
ð8Þ_� bkðI � Sn�1ÞRn�1kn�� b2mn��s�þ2
n�1 a1~ee
ms��2�s þ
Xn�1
j¼2
ms��2�sj�1
!� b2a1~eem
s��sn�1 �
ffiffi~ee
pm�2n
(I.6) (II.1) (II.3) (II.6) (IV.5) and b � m imply
ð9Þ � bkðI � Sn�1ÞGð0Þkn� � b2mn��s�n�1 kGð0Þks�
� b3m�2n�1~ee � b3m2~eem�2
n �ffiffi~ee
pm�2n
Finally, we get by (II.1) and (IV.18)n
ð10Þ � bkrnkn� � ba1~ee½maxðm; mn�1Þ�n��s
� ma1~ee½maxðm; mn�1Þ��2 �ffiffi~ee
pm�2n
Thus, we conclude that
yn � 7K0m5ffiffi~ee
pm�2n ¼ a3
ffiffi~ee
pm�2n � 1
(h) (IV.21): We have
Anð2Þ � max1�i;j�n
�1;
���� @g@pi
ðj þ ~ee~wwnÞ����2
;
���� @g@u ðj þ ~ee~wwnÞ����2
;
����� @F@uij
ðjkl þ ~eeð~wwnÞklÞ����2
þ yn
�Using (II.9) (IV.1) (IV.16)n and (IV.20)n, we get Anð2Þ � M0:
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1123
ORDER REPRINTS
Proof of Theorem A. We shall prove using Proposition IV.1 the convergence of thesequence ðwnÞ.
Set s ¼ s� � 2� t and s ¼ s� t. By (IV.6) (IV.14), for any i; k 2 N� i > k
kwi � wkks �Xij¼kþ1
kujks � bffiffi~ee
p Xij¼kþ1
m�tj�1 ¼ b
ffiffi~ee
p Xij¼kþ1
ðm�tÞj�1
Since m � 2, t > 0 then kwi � wkks ! 0 as i; j ! 1. Hence, there is a functionw 2 Hs��2�2tðOÞ satisfying wn ! w in Hs��2�2tðOÞ. Since Hs��2�2tðOÞ �Cs��2�n�3tðOÞ then w 2 Cs��3�nðOÞ.
On the other hand,
GðwnÞ ¼ ðI � Sn�1ÞRn�1 þ ðI � Sn�1ÞGð0Þ þ rn
For n � 2, we have using (II.2) and (IV.18)
krnks��2�2t � a1b~eems��2�2t�sn�1 ¼ a1b~eem�t
n�1
Combining (II.3) with (II.6) and (I.6), we get
kðI � Sn�1ÞGð0Þks��2�2t � bm�2�2tn�1 kGð0Þks� � b2m�2�2t
n�1 ~ee
Combining (II.6) (IV.11) and (IV.18), we can write
kðI � Sn�1ÞRn�1ks��2�2t � bm�2tn�1kRn�1ks��2
� bm�2tn�1
Xn�1
j¼1
krjks��2
� bm�2tn�1~eea1
ms��2�s þ
Xn�1
j¼2
ms��2�sj�1
!� ~eeba1m�2t
n�1ms��2�sn�1
� ba1~eem�tn�1
These inequalities imply GðwnÞ ! 0 in Hs��2�2tðOÞ as n ! 1. SinceHs��2�2tðOÞ � C2ðOÞ and wn j @O ¼ 0, we conclude that GðwÞ ¼ 0 and w j @O ¼ 0. Thatis u ¼ jþ ew is a solution to the original Monge–Ampere equation which satisfies byLemma III.1, (I.7) since g is nonnegative. The uniqueness of the solution followsimmediately from Caffarelli et al. (1984).
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1124 Kallel-Jallouli
ORDER REPRINTS
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Received April 2003Accepted April 2004
Dirichlet Problem for Degenerate Elliptic Darboux Equation 1125
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