Second Order Elliptic Equations with Discontinuous

—65— adapt toa ij in (4) some results contained in [9] and make use of the smallness hy- potheses ong ij. On the rest ofV we use some imbedding and co...

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Rendiconti Accademia Nazionale delle Scienze detta dei XL Memorie di Matematica e Applicazioni 118o (2000), Vol. XXIV, fasc. 1, pagg. 63-79

ANNA CANALE - LOREDANA CASO - MARIA TRANSIRICO (*)

Second Order Elliptic Equations with Discontinuous Coefficients in Irregular Domains (**) ABSTRACT. — In this paper we study the Dirichlet problem for a class of linear second order elliptic equations in non divergence form in an open subset V of R n with ¯V irregular and coefficients discontinuous in V and singular on a subset of ¯V .

Equazioni ellittiche del secondo ordine con coefficienti discontinui in domini non regolari SUNTO. — In questo lavoro si studia il problema di Dirichlet per una classe di equazioni ellittiche del secondo ordine in forma non variazionale in un aperto V di R n a coefficienti discontinui in V e singolari su un sottoinsieme di ¯V .

INTRODUCTION Let V be an open subset of R n , n F 2 . Let us consider a weight function r in the class A(V) (see Section 1 for the definition) and denote by Sr the subset of ¯V where r goes to zero. We observe (see (1.1)) that, if Sr c ¯ , then r is related to the distance function to Sr . Let L be the uniformly elliptic differential operator n

n

(1)

Lu 4 2

!

i, j41

aij uxi xj 1

! a u 1a u

i41

i

xi

(*) Indirizzo degli Autori: DIIMA, Facoltà di Scienze, Università di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy. E-mail: canaleHdiima.unisa.it; casoHdiima.unisa.it; transiriHdiima.unisa.it (**) Memoria presentata il 15 dicembre 1999 da Mario Troisi, uno dei XL. Mathematics Subject Classification: 35 J 25, 46 E 35.

— 64 — with coefficients aij 4 aji  L Q (V), i , j 4 1 , R , n . We study the following Dirichlet problem (2)

i

u  Ws2 (V) O W1s 2 1 (V) ,

Lu 1 lbu 4 f ,

f  Ls2 (V) ,

i

where b is a positive function, l , s  R , Ws2 (V), W1s 2 1 (V) and Ls2 (V) are some weighted Sobolev spaces (see Section 1 for definitions) and the weight functions are suitable powers of r . In a recent paper (see [9]) problem (2) has been studied under the following hypotheses on coefficients aij of L: (3)

q

(aij )xk  Lloc (V 0Sr ) ,

sup V(aij )xk VL q (V O B(x , r(x) ) ) E 1Q ,

i , j , k41 , R , n ,

xV

where q D 2 if n 4 2 , q 4 n if n F 3 and B(x , r(x) ) is the open ball centered at x of radius r(x). In such a paper similar hypotheses are made on ai , a and b . If ¯V is singular, further conditions on r , aij , ai , a , b and l are given in order to problem (2) be uniquely solvable. We observe that if, in particular, r is positive constant and V is a bounded open subset of R n , n F 3 , condition (3) becomes (aij )xk  L n (V) ,

i, j, k41, R, n ,

that is the classical hypothesis of C. Miranda (see [16]). This means that the result in [9] extends that one contained in [16] to the case V unbounded open set with singular boundary and (aij )xk , ai and a singular functions near to Sr . Further generalizations of Miranda’s result can be found in literature. For example in [2], [10], [11], [12] the coefficients aij of the operator L belong to wider functional spaces and in [19] V is an unbounded open set. In this paper we study problem (2) with coefficients aij which satisfy a condition more general than (3). In fact we suppose (see conditon a) in Section 3) that aij do not necessarily satisfy the last requirement in (3) and can be split in the following way (4)

aij 4 a ij 1 g ij ,

i, j41, R, n ,

where a ij are bounded simmetric functions which verify an uniformly elliptic condition and hypothesis (3), g ij are sufficiently small near to Sr and at infinity. In Section 6 we give an example of a function with a behaviour similar to g ij , but which does not satisfy the last requirement in (3). We emphasize that in these weaker hypotheses on aij it is not possible to use, as in [9], some results of variational type contained in [5]. In this paper we are able to get a priori bounds for solutions of problem (2) (see sections 3 and 4) as follows. On the part of V close to Sr or to infinity, we suitably

— 65 — adapt to a ij in (4) some results contained in [9] and make use of the smallness hypotheses on g ij . On the rest of V we use some imbedding and compactness theorems contained in [8]. From a priori bounds we deduce some existence and uniqueness results in Section 5.

1. - NOTATIONS

AND FUNCTION CLASSES

Let E be a Lebesgue measurable subset of R n and S(E) the s-algebra of Lebesgue measurable subsets of E . For any A  S(E), NAN is the Lebesgue measure of A , D(A) is the class of restricp tions to A of functions z  CoQ (R n ) such that supp z O A % A , Lloc (A) is the class of functions f : A K C such that zf  L p (A) for any z  D(A). We set NfNp , A 4 V f VL p (A) ,

1 G p G 1Q .

We put B(x , r) 4 ]y  R n : Ny 2 xN E r( ,

Br 4 B( 0 , r) (x  R n , (r  R1 .

Let V be an open subset of R n . We set V(x , r) 4 V O B(x , r)

(x  V , (r  R1 .

We call A(V) the class of functions r : V K R1 satisfying sup x, yV Nx 2 yN E r(y)

N log r(x) N E 1Q . r(y)

It is easy to see that r  A(V) if and only if r : V K R1 and there exists a constant g  R1 such that g 21 r(y) G r(x) G gr(y)

(x  V , (y  V(x , r(x) ) .

Some examples of functions r  A(V) are given in [22] where it is also observed that A(V) contains the class of positive Lipschitz functions with Lipschitz constant less than 1. For any r  A(V) we set Sr 4 ]y  ¯V : lim r(x) 4 0 ( . xKy

As shown in [8], Sr is a closed subset in ¯V . Moreover if Sr c ¯ it results (see [22]) (1.1)

r(x) G dist (x , Sr )

(x  V .

It is well-known (see, e.g., Theor. 2, Chap. VI in [18] and Lemma 3.6.1 in [24]) that

— 66 — there exist a  C Q (V) O C 0 , 1 (V), c1 , c2  R1 such that c1 dist (x , Sr ) G a(x) G c2 dist (x , Sr )

(x  V .

V k 4 ]x  V : NxN E k , a(x) D 1 /k(

(k  N .

We put

If f  D(R1 ) is a fixed function such that 0 GfG1,

f(t) 4 1 if t G 1 /2 ,

f(t) 4 0 if t F 1 ,

we define the functions c k : x  V K (1 2 f(ka(x) ) ) f(NxN/2 k)

(k  N .

We remark that, for any k  N , c k belongs to D(V 0 Sr ) and the following conditions hold 0 GckG1 ,

c kNVk 4 1 ,

supp c k % V2 k .

Let Ao (V) be the class of measurable functions r  A(V). If r  Ao (V), then (see [8]) (1.2)

Q (V ) , r  Lloc

Q r 21  Lloc (V 0 Sr ) .

Further examples and properties of functions of A(V) can be found in [22], [20], [4], [8], [7]. If r  N , 1 G p G 1Q , s  R and r  Ao (V), we denote by Wsr , p (V) the space of distributions u on V such that r s 1 NaN 2 r ¯a u  L p (V) for NaN G r endowed with the norm (1.3)

VuVWsr , p (V) 4

!

NaN G r

Nr s 1 NaN 2 r ¯a uNp , V .

i

Moreover we denote by Wrs , p (V) the closure of CoQ (V) in Wsr , p (V). We put Ws0 , p (V) 4 Lsp (V) ,

Wsr , 2 (V) 4 Wsr (V) ,

i

i

Wsr , 2 (V) 4 Wrs (V) .

For some properties of weighted Sobolev spaces, where the weight functions are powers of a function r  A(V), see, e.g., [3], [14], [17], [15], [21], [4], [8], [7]. If 1 G p E 1Q , s  R and r  Ao (V), we set (1.4)

V(x) 4 V(x , r(x) )

(x  V ,

i A and consider the spaces Ksp (V), Kps (V), Kps (V) defined in [4] in correspondence of the family of open sets defined by (1.4). Let us recall that:

— 67 — p

Ksp (V) is the space of functions g  Lloc (V 0 Sr ) such that V gVKsp (V) 4 sup (r s 2 n/p (x) NgNp , V(x) ) E 1Q ,

(1.5)

xV

endowed with the norm defined by (1.5), A Kps (V) is the closure of LsQ (V) in Ksp (V), i

Kps (V) is the closure of CoQ (V) in Ksp (V). i A For some properties of the spaces Ksp (V), Kps (V) and Kps (V) we refer to [4], [8], [7]. REMARK 1.1: Let us fix r  Ao (V), 1 G p E 1Q , s  R . We observe that if i p g  Lloc (V 0 Sr ), then, for any z  D(V 0Sr ), we have zg  Kps (V). Now, if we fix z  D(V 0Sr ), then zg  L p (V) and so there exists a sequence of functions (gn )n  N , gn  CoQ (V), such that gn K zg

in L p (V) .

For every fixed c  D(V 0Sr ) with c Nsupp z 4 1 , evidently we have (1.6)

cgn K zg

in L p (V) .

From (1.2) and (1.6) we obtain that zg  Ksp (V) and cgn K zg

in Ksp (V) ,

i

so zg  Kps (V).

r 2. - A

PRELIMINAR LEMMA

Let us suppose n F 2 and fix r  A(V) O L Q (V) such that S 4 Sr c ¯ . Set B1 4 ]x  B1 : xn D 0 ( ,

Bo 4 ]x  B1 : xn 4 0 ( ,

we suppose that there exists an open subset V * of R n such that h1 ) there are a d  R1 , an open cover ]Ui (i  I of ¯V * and, for any i  I , a C 2-diffeomorphism c i : Ui K B1 such that (2.1)

c i (Ui O V * ) 4 B1 ,

c i (Ui O ¯V * ) 4 Bo ;

(2.2)

the components of c i and c 21 and of their first and second derivatives are i bounded by a constant independent of i;

(2.3)

for any x  V *d there exists an i  I such that B(x , d) % Ui and, for any x  V * 0 V *d , B(x , d) % V * , where V *d 4 ]x  V * : dist (x , ¯V * ) E d(;

— 68 — V%V*,

(2.4)

¯V0S % ¯V * .

REMARK 2.1: It is easy to prove that (2.1), (2.2) and (2.3) hold when V * has the uniform C 2-regularity property defined in Section 4.6 in [1]. r Let us consider in V the second order linear differential operator n

Lu 4 2

(2.5)

!

i, j41

n

aij uxi xj 1

! a u 1 au

i41

i

xi

with the following conditions on the coefficients: aij  L Q (V) ,

aij 4 aji ,

h2 )

n

!

i, j41

aij j i j j F nNjN2

i, j41, R, n ,

(j  R n , a.e. in V ,

where n is a positive constant independent of x and j; q

(aij )xk  Lloc (V 0S),

h3 )

i, j, k41, R n ,

where qD2

if

n42 ,

Aq ai  K1 (V) ,

h4 )

q4n

if

nF3 ;

A a  Kt2 (V) ,

i41, R, n ,

where t42

if

2 GnE4 ,

tD2

if

n44,

t4

n 2

if

nD4 .

In the sequel

g! h n

ux 4

i41

1 /2

ux2i

g! h n

,

uxx 4

i, j41

ux2i xj

1 /2

.

We consider a function b : V K R1 such that the following hypothesis holds: h5 )

A b  Kt2 (V) ,

A )d  Kq1 (V)

such that b x G bd .

An example of function b which satisfies the hypothesis h5 ) can be given in the following way. From Remark 3.1 in [8] and Theorem 3.2 in [22], there exist s  A(V) O C Q (V) O C 0 , 1 (V), c0 , c1 , c2  R1 such that (2.6)

c1 r(x) G s(x) G c2 r(x)

(2.7)

s x G c0

(x  V ,

(x  V .

— 69 — So, the function (2.8)

1

b4

s2

satisfies the hypothesis h5 ). A 2s Indeed, b  L2Q (V) and then b  Kt2 (V). If we put d 4 x , d  L1Q (V) and so s A d  Kq1 (V). Furthermore we have b x 4 bd . Another example of function which satisfies the hypothesis h5 ) is the function (2.9)

b(x) 4

1 ( 1 1 NxN2 )t

,

xV , tD0 .

A 2 tNxN Q Indeed, b  L2Q (V) and then b  Kt2 (V). If we set d 4 , d  L1 (V) and then 1 1 NxN2 Aq d  K1 (V). Moreover we have b x 4 bd . i We also observe that, from Lemma 2.1 in [8], b  Kt2 (V). REMARK 2.2: One can show that under hypotheses h1 )-h5 ), from Remark 3.1 and Theorem 3.1 in [8] (see also [7]), it follows that for any s , l  R the operator u  Ws2 (V) K Lu 1 lbu  Ls2 (V) is bounded.

r

We put n

Lo u 4 2 gA 4

!

i, j41

aij uxi xj ,

n

!

i, j41

((aij )x 1 Naij Nd) ,

where d is the function defined in h5 ). Let us fix a bounded open subset V of R n such that (2.10)

V%V

or

V O ¯V c ¯

and

V % Ui 0 S

for some i  I .

LEMMA 2.1: If the hypotheses h1 )-h3 ) and h5 ) hold, then, for any l F 0 and for any function v satisfying i

v  W 2 (V) O W1 (V) ,

supp v % V ,

— 70 — we have the bound Nvxx N2 , V G c(NLo v 1 lbvN2 , V 1 N gA vx N2 , V 1 NvN2 , V ) ,

(2.11)

where c  R1 is independent of v and l . PROOF: Proceeding as in the proof of Lemma 5.1 in [7] (see also Section 7 in [6]), we obtain Nvxx N22 , V G c1 (NLo vN22 , V 1 Ngvx N22 , V 1 NvN22 , V ) ,

(2.12) n

where g 4

! (a )

i, j41

and c1  R1 is independent of v and l .

ij x

By means of known techniques (see, e.g., [13], [6], [7]) we have

(L v 1 lbv) dx F (L v) dx 1 l  b

(2.13)

2

o

2

o

V

V

2

2

v 2 dx 1

V





12 ln bvx2 dx 2 2 l b gA NvNvx dx . V

V

Moreover we have (2.14)

 b gA NvNv dx G l2  b v dx 1 21l  N gA v N dx . 2

x

V

2

x

V

2

V

From (2.12), (2.13) and (2.14) we deduce the result.

3. - A

r

PRIORI BOUNDS

Let us suppose that aij satisfy the following further condition: a) there exist functions a ij such that a ij 4 a ji  L Q (V) , n

!

i, j41

Aq (a ij )xk  K1 (V) ,

a ij j i j j F n o NjN2

(j  R n ,

i, j, k41, R, n , a.e. in V ,

where n o is a positive constant independent of x and j and for any e  R1 there exists ke  N such that: n

ess sup V0V ke

!

i, j41

Na ij 2 aij N E e ,

where V k , k  N , are the sets defined in Section 1.

— 71 — LEMMA 3.1: If the conditions h1 )-h5 ), a) hold and l 1 is a real number, then there exists a constant c  R1 such that Nvxx N2 , V G c(NLv 1 lbvN2 , V 1 Nr 21 vx N2 , V 1 Nr 22 vN2 , V ) ,

(3.1)

for any l  [l 1 , 1Q[ and for any function v satisfying i

v  W 2 (V) O W1 (V) ,

supp v % V .

PROOF: Let us suppose l F 0 and consider the functions c k , k  N , introduced in Section 1. Applying Lemma 2.1 in [9] to the function ( 1 2 c k ) v in the case L 4 n

42 (3.2)

!

i, j41

a ij

¯2 ¯xi ¯xj

, we get

N( ( 1 2 c k ) v)xx N2 , V G

uN 2 ! a (( 1 2 c ) v) n

G c1

i, j41

ij

k

xi xj 1 lb( 1 2 c k )

v

N

2, V

1

h

1Nr 21 (( 1 2 c k ) v)x N2 , V 1 Nr 22 ( 1 2 c k ) vN2 , V , where c1  R1 is independent of l , v and k . Moreover we have n

(3.3)

N ! 2

i, j41

a ij (( 1 2 c k ) v)xi xj 1 lb( 1 2 c k ) v

N

2, V G

G NLo (( 1 2 c k ) v) 1 lb( 1 2 c k ) vN2 , V 1 n

1

N ! (a 2 a ) (( 1 2 c ) v) N i, j41

ij

ij

k

xi xj

2, V

G

g

G c2 N( 1 2 c k )(Lo v 1 lbv) N2 , V 1 N( 1 2 c k )x vx N2 , V 1

N

1N( 1 2 c k )xx vN2 , V 1 (( 1 2 c k ) v)xx

n

! (a 2 a ) N

i, j41

ij

ij

2, V

h

,

where c2  R1 is independent of l , v and k . From (3.2), (3.3) and a) we easily deduce that there exists ko  N such that (3.4) N( ( 1 2 c ko ) v)xx N2 , V G c3 (NLo v 1 lbvN2 , V 1 Nr 21 vx N2 , V 1 Nr 22 vN2 , V ) , where c3  R1 is independent of l and v . From now on, we denote by cj , j 4 4 , 5 , 6 , positive constants independent of l and v .

— 72 — Applying Lemma 2.1 to the function c ko v , we get (3.5)

N(c ko v)xx N2 , V G c4 (NLo (c ko v) 1 lbc ko vN2 , V 1 1N gA(c ko v)x N2 , V 1 Nc ko vN2 , V ) .

Let x  D(V 0 S) be a function such that x Nsupp c k 4 1 . Since the function x gA belongs to o Aq the space K1 (V) (see Remark 1.1), from the estimate (3.6) in [8] we deduce (3.6)

2 Nx gA(c ko v)x N2 , V G eV(c ko v)x VWo1 (V) 1 c1 (e) V(c ko v)x VL21 (V) G

G eN(c ko v)xx N2 , V 1 c2 (e) Nr 21 (c ko v)x N2 , V . Then from (3.5) and (3.6) we have (3.7)

N(c ko v)xx N2 , V G c5 (NLo v 1 lbvN2 , V 1 Nr 21 vx N2 , V 1 Nr 22 vN2 , V ) .

From (3.4) and (3.7) we get (3.8)

Nvxx N2 , V G c6 (NLo v 1 lbvN2 , V 1 Nr 21 vx N2 , V 1 Nr 22 vN2 , V ) .

If l 1 E 0 , we fix l  [l 1 , 0[. From (3.6) in [8] we get (3.9)

NlbvN2 , V G Nl 1 N(eNvxx N2 , V 1 c3 (e) Nr 21 vx N2 , V 1 c4 (e) Nr 22 vN2 , V ) .

From (3.8) for l 4 0 and (3.9) we deduce (3.1) with Lo instead of L . Finally, again by (3.6) in [8] we can obtain the result. r i

2 We denote by Wloc (V 0S) (respectively W1loc (V 0 S)) the space of all functions i u : V K R such that zu  W 2 (V) (respectively W1 (V)) for any z  D(V 0 S).

THEOREM 3.1: In the same hypotheses of Lemma 3.1, for any u : V K R such that i

(3.10)

2 (V 0S) O W1loc (V 0S) O Ls22 2 (V) , u  Wloc

Lu 1 l 8 bu  Ls2 (V),

for some s , l 8  R ,

we have u  Ws2 (V). Moreover, for any l 1  R , we have the bound (3.11)

VuVWs2 (V) G c(VLu 1 lbuVLs2 (V) 1 VuVLs22 2 (V) )

(l F l 1 ,

where the constant c  R1 is independent of u and l . PROOF: The result follows from Lemma 3.1 applying arguments similar to those one used in [9] in order to get Theorem 2.1 from Lemma 2.1. r

— 73 — COROLLARY 3.1: In the same hypotheses of Theorem 3.1 and if )m  R1 : b F mr 22

(3.12)

a.e. in V ,

then for any s  R there exist c , l o  R1 such that VuVWs2 (V) G cVLu 1 lbuVLs2 (V)

(3.13)

i

(u  Ws2 (V) O W1s 2 1 (V) ,

(l F l o .

PROOF: By means of known techniques (see, e.g., [23], [19]) and Theorem 3.1, we have lVuVLs22 2 (V) G m 21 VlbuVLs2 (V) G m 21 VLu 1 lbuVLs2 (V) 1 1m 21 VLuVLs2 (V) G m 21 VLu 1 lbuVLs2 (V) 1 c1 VuVWs2 (V) G

(3.14)

G c2 VLu 1 lbuVLs2 (V) 1 c3 VuVLs22 2 (V) i

(u  Ws2 (V) O W1s 2 1 (V) ,

(l  R1 ,

where c1 , c2 , c3  R1 are independent of u and l . From (3.11) and (3.14) we deduce the result.

4. - FURTHER

r

A PRIORI BOUNDS

We assume that the following further hypotheses hold: h6 ) the function s which satisfies (2.6) and (2.7) is such that i

s x  Kqo (V) , where q is the number defined in the hypothesis h3 ); h7 )

i

q

(a ij )xk , ai  K1 (V) , a4a 81a 9, a 9 F m o r 22

i, j, k41, R, n , i

a 8  Kt2 (V) , a.e. in V ,

where t is the number defined in the hypothesis h4 ) and m o R1 is independent of x. An example of function r Ao (V) satisfying the condition h6 ) can be found in [9].

— 74 — THEOREM 4.1: If the hypotheses h1 )-h7 ) and a) hold, then for any s  R there exist c  R1 and an open set V o %% V 0S such that VuVWs2 (V) G c(VLu 1 lbuVLs2 (V) 1 NuN2 , V o )

(4.1)

i

(u  Ws2 (V) O W1s 2 1 (V) ,

(l F 0 .

PROOF: Proceeding as in the proof of Lemma 3.1, let us consider the functions c k , k  N , defined in Section 1. From Theorem 3.1 in [9] we have (4.2)

V( 1 2 c k ) uVWs2 (V) G n

V ! a (( 1 2 c ) u)

G c1 2

G c1

uV

i, j41

ij

k

xi xj 1 a 9 ( 1 2 c k )

Lo (( 1 2 c k ) u) 1 a 9 ( 1 2 c k ) u 1 lb( 1 2 c k ) u

n

1

u 1 lb( 1 2 c k ) u

V ! (a 2 a ) (( 1 2 c ) u) V i, j41

ij

ij

k

xi xj

Ls2 (V)

h

V

Ls2 (V)

V

Ls2 (V) G

1

G

u

G c2 V( 1 2 c k )(Lo u 1 a 9 u 1 lbu) VLs2 (V) 1 V( 1 2 c k )x ux VLs2 (V) 1

V

1V( 1 2 c k )xx uVLs2 (V) 1 (( 1 2 c k ) u)xx

n

! (a 2 a ) V

i, j41

ij

ij

Ls2 (V)

h

,

where c1 , c2  R1 are independent of l , u and k . From hypothesis a) and from (4.2) we deduce that there exists ko  N such that (4.3)

V( 1 2 c ko ) uVWs2 (V) G c3 (VLo u 1 a 9 u 1 lbuVLs2 (V) 1 1V( 1 2 c ko )x ux VLs2 (V) 1 V( 1 2 c ko )xx uVLs2 (V) ) ,

where c3  R1 is independent of l and u , as all the positive constants appearing in the rest of the proof. On the other hand, from Theorem 3.1 we get (4.4)

Vc ko uVWs2 (V) G c4 (VLo (c ko u) 1 a 9 c ko u 1 lbc ko uVLs2 (V) 1 1Vc ko uVLs22 2 (V) ) G c5 (Vc ko (Lo u 1 a 9 u 1 lbu) VLs2 (V) 1 1V(c ko )x ux VLs2 (V) 1 V(c ko )xx uVLs2 (V) 1 Vc ko uVLs22 2 (V) ) .

— 75 — From (4.3) and (4.4) it follows (4.5)

VuVWs2 (V) G c6 (VLo u 1 a 9 u 1 lbuVLs2 (V) 1 V(c ko )x ux VLs2 (V) 1 1V(c ko )xx uVLs2 (V) 1 Vc ko uVLs22 2 (V) ) .

We remark that, by (1.2), we have (4.6)

V(c ko )x ux VLs2 (V) G c7 Nux N2 , supp c ko ,

(4.7)

V(c ko )xx uVLs2 (V) 1 Vc ko uVLs22 2 (V) G c8 NuN2 , supp c ko .

Moreover, from (3.7) in [8], for any e  R1 there exist c(e)  R1 and an open set V e %% V such that n

(4.8)

! Va u

i41

i

xi VLs2 (V) 1 Va 8 uVLs2 (V) G

G eVuVWs2 (V) 1 c(e)(Nux N2 , V e 1 NuN2 , V e ) . From (4.5)-(4.8), taking in mind (1.2), we deduce the assertion.

r

COROLLARY 4.1: If the hypotheses of Theorem 4.1 are satisfied and if (4.9)

Q b 21  Lloc (V 0S) ,

then for any s  R there exist c , l o  R1 such that (4.10)

VuVWs2 (V) G cVLu 1 lbuVLs2 (V) i

(u  Ws2 (V) O W1s 2 1 (V) ,

(l F l o .

PROOF: From (4.9) and (1.2) it follows that (4.11)

lNuN2 , V o G c1 Nr s lbuN2 , V o G c1 VlbuVLs2 (V) i

(u  Ws2 (V) O W1s 2 1 (V) ,

(l  R1 ,

where V o is the open set of Theorem 4.1 and c1 R1 is independent of u and l. Proceeding as in the proof of Corollary 3.1, using Theorem 4.1 instead of Theorem 3.1, we obtain the result. r 5. - EXISTENCE

THEOREMS

THEOREM 5.1: If either the hypotheses h1 )-h6 ), a) and (3.12) or the hypotheses h1 ) 2 h7 ), a) and (4.9) hold, then for any s  R there exists l o  R1 such that for any

— 76 — l F l o the problem (5.1)

i

u  Ws2 (V) O W1s 2 1 (V) ,

f  Ls2 (V)

Lu 1 lbu 4 f ,

is uniquely solvable. PROOF: We denote by A the operator 2D if h1 )-h6 ), a) and (3.12) hold, the operator 2D 1 a 9 in the other case. For any t  [ 0 , 1 ] we set Lt 4 ( 1 2 t) A 1 tL . Using Corollary 3.1 when h1 )-h6 ), a) and (3.12) hold, Corollary 4.1 in the other case, we deduce that there exist c , l o  R1 such that VuVWs2 (V) G cVLt u 1 lbuVLs2 (V) i

(u  Ws2 (V) O W1s 2 1 (V) ,

(l F l o ,

(t  [ 0 , 1 ] .

If we fix l F l o , by Theorem 3.2 in [9], the problem i

u  Ws2 (V) O W1s 2 1 (V) ,

f  Ls2 (V)

Au 1 lbu 4 f ,

is uniquely solvable. Observing that Lt 1 lb 4 ( 1 2 t)(A 1 lb) 1 t(L 1 lb) , we obtain the result by means of the method of continuity.

r

THEOREM 5.2: If the hypotheses h1 )-h4 ), h6 ), h7 ) and a) are satisfied, then for any s  R the problem (5.2)

i

u  Ws2 (V) O W1s 2 1 (V) ,

Lu 4 f ,

f  Ls2 (V)

is an index problem with index equal to zero. PROOF: We consider the function b defined from (2.9) with t 4 1 . Since b satisfies the hypotheses h5 ) and (4.9), by Theorem 5.1 the problem (5.1) is uniquely solvable for l large enough. i

On the other hand, since b  Kt2 (V), from Lemma 2 in [4] and (3.8) in [8] the operator u  Ws2 (V) K bu  Ls2 (V) is compact. So, from well-known results, we deduce the assertion.

r

— 77 — 6. - APPENDIX Let us give an example of function f with a behaviour similar to g ij 4 aij 2 a ij (see q Section 3) and such that fxk  K1 (V) for some k  ] 1 , R , n(. Let us set V 4 R1 3]0 , 2[ and r(x) 4

g h x1

t

2 1 x1

with t  [ 1 , 2[.

We fix d  R1 and consider the function f : V K R defined by

y 2 n11 1 , 21n k 3]0 , 2[ 1 1 , x y k 3]0 , 2[ 2n 2n21

. x [ ( 4 n 2 1 ) x 2 2 n 1 1 ] arctan 1 n ` f(x) 4 / 1 x [ 2 n 2 1 2 ( 2 n 2 1 ) x ] arctan n ` ´0 2

2

1

2

2

Evidently f  L Q (V) and

x

d

lim NxN K 1Q

1

d

x  [ 1 , 1Q[3]0 , 2[ .

f(x) 4 0 . 1

On the other hand we have 0 G f(x) E 2 arctan d and then lim f (x) 4 0 x K xo n (xo  S 4 ]x  ¯V : x1 4 0 (. Q (V 0S) from which we deduce that Moreover we note that fx1 , fx2  Lloc q fx1 , fx2  Lloc (V 0 S). Now we fix a sequence (x n )n  N , x n  V , such that x1n 4

1

x2n 4 1 .

2n21

For n large enough we have

y 21n , 2 n12 1 z 3 y1 , 1 1 12 r(x )z % V(x ) 4 B(x , r(x ) ) . n

n

n

n

Then we get [r(x n ) ]q 2 2

 Nf

x1 N

q

dx F

V(x n )

F

4

1 2 1 2

1 /( 2 n 2 1 )

[r(x n ) ]q 2 1

 y( 2 n 2 1 ) arctan n1 z dx 4 q

2

1

d

1 /2 n

u

1 4n21

v

t(q 2 1 )

( 2 n 2 1 )2 q 2 1 2n

u

arctan

1 nd

v

q

.

— 78 — Finally for d E

( 2 2 t)(q 2 1 ) q

we obtain

lim [r(x n ) ]q 2 2

n K 1Q



Nfx1 Nq dx 4 1Q ,

V(x n )

q

hence fx1  K1 (V).

REFERENCES [1] R. A. ADAMS, Sobolev Spaces, Academic Press, 1975. [2] A. ALVINO - G. TROMBETTI, Second order elliptic equations whose coefficients have their first derivatives weakly-L n, Ann. Mat. Pura Appl. (4), 138 (1985), 331-340. [3] V. BENCI - D. FORTUNATO, Weighted Sobolev spaces and the nonlinear Dirichlet problem in unbounded domains, Ann. Mat. Pura Appl. (4), 121 (1979), 319-336. [4] A. CANALE - L. CASO - P. DI GIRONIMO, Weighted norm inequalities on irregular domains, Rend. Accad. Naz. Sci. XL Mem. Mat., (11) 16 (1992), 193-209. [5] A. CANALE - L. CASO - P. DI GIRONIMO, Variational second order elliptic equations with singular coefficients, Rend. Accad. Naz. Sci. XL Mem. Mat., (1) 17 (1993), 113-128. [6] A. CANALE - M. LONGOBARDI - G. MANZO, Second order elliptic equations with discontinuous coefficients in unbounded domains, Rend. Accad. Naz. Sci. XL Mem. Mat., (1) 18 (1994), 41-56. [7] L. CASO, Spazi con peso ed equazioni ellittiche del secondo ordine con dati singolari, Tesi di Dottorato di Ricerca in Matematica del consorzio Napoli-Salerno, VII ciclo. [8] L. CASO - M. TRANSIRICO, Some remarks on a class of weight functions, Comment. Math. Univ. Carolin., 37 (1996), 469-477. [9] L. CASO - M. TRANSIRICO, The Dirichlet problem for second order elliptic equations with singular data, Acta Math. Hungar., 76 (1-2) (1997), 1-16. [10] F. CHIARENZA - M. FRANCIOSI, A generalization of a theorem by C. Miranda, Ann. Mat. Pura Appl. (4), 161 (1992), 285-297. [11] F. CHIARENZA - M. FRASCA - P. LONGO, Interior W 2 , p estimates for non divergence elliptic equations with discontinuous coefficients, Ricerche Mat., 40 (1991), 149-168. [12] F. CHIARENZA - M. FRASCA - P. LONGO, W 2 , p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853. [13] M. CHICCO, Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients, Ann. Mat. Pura Appl. (4), 92 (1972), 13-22. [14] D. FORTUNATO, Spazi di Sobolev con peso ed applicazioni ai problemi ellittici, Rend. Accad. Sc. Fis. Mat. di Napoli (4), 41 (1974), 245-289. [15] S. MATARASSO - M. TROISI, Teoremi di compattezza in domini non limitati, Boll. Un. Mat. Ital. (5), 18-B (1981), 517-537. [16] C. MIRANDA, Sulle equazioni ellittiche di tipo non variazionale a coefficienti discontinui, Ann. Mat. Pura Appl. (4), 63 (1963), 353-386. [17] R. SCHIANCHI, Spazi di Sobolev dissimmetrici e con peso, Rend. Accad. Sc. Fis. Mat. di Napoli (4), 42 (1975), 349-388. [18] E. M. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, New Jersey, 1970. [19] M. TRANSIRICO - M. TROISI, Equazioni ellittiche del secondo ordine di tipo non variazionale in aperti non limitati, Ann. Mat. Pura Appl. (4), 152 (1988), 209-226.

— 79 — [20] M. TRANSIRICO - M. TROISI, Limitazioni a priori per una classe di operatori differenziali lineari ellittici del secondo ordine in aperti non limitati, Boll. Un. Mat. Ital. (7), 5-B (1991), 757-771. [21] M. TROISI, Teoremi di inclusione negli spazi di Sobolev con peso, Ricerche Mat., 18 (1969), 49-74. [22] M. TROISI, Su una classe di funzioni peso, Rend. Accad. Naz. Sci. XL Mem. Mat., (11) 10 (1986), 141-152. [23] G. VIOLA, Sulle equazioni ellittiche del secondo ordine a coefficienti non regolari, Rend. Mat., 4 (1984), 617-632. [24] W. P. ZIEMER, Weakly Differentiable Functions, Springer-Verlag, 1989.

— 80 —

Direttore responsabile: Prof. A. BALLIO - Autorizz. Trib. di Roma n. 7269 dell’8-12-1959 « Monograf » - Via Collamarini, 5 - Bologna