A NON HOMOGENEOUS EXTRA TERM FOR THE LIMIT OF DIRICHLET PROBLEMS IN PERFORATED DOMAINS

Transcription

1 J. Casado Diaz & A. Garroni A NON HOMOGENEOUS EXTRA TERM FOR THE LIMIT OF DIRICHLET PROBLEMS IN PERFORATED DOMAINS J. CASADO DIAZ A. GARRONI Abstract: We study te asymptotic beaviour of Diriclet problems on varying domains wit a nonlinear elliptic differential operator. In te limit problem it appears a nonlinear extra term wose properties are strictly connected wit te form of te differential operator. We investigate tis connection wen te domains ave a periodic structure.. Introduction to te problem. Te contents of te present paper is related to te study of te asymptotic beaviour of te following problems (.) div a(x, Du ) = f in, u W,p 0 ( ), were is a sequence of open subsets of a fixed bounded open set R n, a(x, ξ) is a Caratéodory function wic satisfies standard conditions of monotonicity and of p order growt ( < p < + ), and f belongs to W,p (). For every N we extend te function u to a function in te space W,p 0 () by setting u = 0 on \. By te monotonicity of a it is easy to prove tat te sequence (u ) is bounded in W,p 0 () and ten, up to a subsequence, converges weakly in W,p 0 () to some function u W,p 0 (). Te problem is to caracterize u as te solution of some limit problem. Tis problem as been studied by many autors in te literature under different assumptions on te differential operator and wit different approaces (see for instance [0], [4], [], [7], [5], [6], [2],...) In te case of monotone operators G. Dal Maso and F. Murat in [8] (see also [5]) proved te following result: assume tat te function a(x, ξ) satisfies te omogeneity condition (.2) a(x, tξ) = t p 2 t a(x, ξ) ξ R n and t R 8

2 A non omogeneous extra term... (tis assumption is clearly satisfied if a(x, ξ) = ξ p 2 ξ, i.e., wen te operator div a(x, Du) is te p-laplacian operator p ). Ten, tere exists a subsequence of indices, tat we still denote by (), and a non negative Borel measure µ suc tat for every f W,p () te solutions of (.) converge weakly in W,p 0 () to te solution u of te problem (.3) u W,p 0 () L p µ(), a(x, Du)Dv dx + u p 2 uv dµ = f, v v W,p 0 () L p µ(). Te measure µ wic appears in problem (.3) does not carge sets of p-capacity zero (for te definition and te properties of te p-capacity we refer to [2]). Let us remark tat te class of problems of te type (.3) is quite large and includes, wit a suitable coice of µ, te class of Diriclet boundary valued problems on open subset of. Namely if E is a closed subset of and µ is defined as follows 0, if p-cap(b E) = 0, µ(b) = +, oterwise for every Borel set B, ten it is easy to see tat problem (.3) is equivalent to te problem div a(x, Du) = f in \ E, u W,p 0 ( \ E), If we assume tat µ is a Radon measure, te problem (.3) may be written as (.4) div a(x, Du) + u p 2 uµ = f in, u W,p 0 () L p µ(), were te equation above is understood in te sense of distribution. Comparing problems (.) and (.3), we find tat in te limit problem it appears an extra term, u p 2 uµ, wic is (p )-omogeneous. In [3] we generalize te result of G. Dal Maso and F. Murat to te case in wic te function a(x, ξ) does not satisfy te assumption (.2). More precisely we assume tat te function a(x, ξ) satisfies te following conditions if 2 p < + : (i) tere exists a constant α > 0 suc tat (a(x, ξ ) a(x, ξ 2 ))(ξ ξ 2 ) α ξ ξ 2 p for every ξ, ξ 2 R n and for a.e. x ; (ii) tere exist a constant β > 0 and a function L p () suc tat a(x, ξ ) a(x, ξ 2 ) β((x) + ξ + ξ 2 ) p 2 ξ ξ 2 82

3 for every ξ, ξ 2 R n and for a.e. x ; (iii) a(x, 0) = 0 for a.e. x. J. Casado Diaz & A. Garroni We make analogous assumptions in te case < p < 2. Under tese conditions on te operators div a(x, ) te following result olds. Teorem.. Let ( ) be a sequence of arbitrary open subsets of. Tere exist a subsequence of ( ), still denoted by ( ), a non negative Borel measure µ and a function F : R R, suc tat for every f W,p () te sequence (u ) of solutions of te problems (.) converges weakly in W,p 0 () to te solution u, of te following problem u W,p 0 () L p µ(), (.5) a(x, Du)Dv dx + F (x, u)v dµ = f, v v W,p 0 () L p µ(). Moreover te measure µ is zero on sets of p-capacity zero and te function F (x, s) satisfies te following conditions if 2 p < + : (I) for every s, s 2 R and for every x we ave F (x, s ) F (x, s 2 ) L( s + s 2 ) p 2 p s s 2 (II) for every s, s 2 R and for every x we ave (F (x, s ) F (x, s 2 ))(s s 2 ) α s s 2 p ; (III) F (x, 0) = 0 for every x ; Analogous conditions old in te case < p < 2. p ; Tis result for general monotone operator witout any omogeneity condition as been also proved in [] and [2] under additional geometric assumptions on te sequence ( ) wic assure in particular tat te measure µ wic appears in te limit problem is a Radon measure. Remark.2. Te measure µ in problem (.5) may be taken equal to te measure wic defines te extra term wen te operator A is te p-laplacian operator. Remark.3. Te result above as a local caracter. In te sense tat te function F and te measure µ wic appear in te extra term in te limit problem do not depend on te domain and on te fact tat we consider solutions of Diriclet problems wit boundary value zero on. Namely if ( ) is te sequence given by Teorem. and (z ) is a sequence of functions in W,p () satisfying a(x, Dz )Dv dx = f, v for every v W,p 0 ( ) wit compact support in, wic converges to some function z weakly in W,p (), ten z belongs to L p µ( ) for every open set and a(x, Dz)Dv dx + 83 F (x, z)v dµ = f, v

4 A non omogeneous extra term... for every v W,p 0 () L p µ() wit compact support in. Te goal of te present paper is to sow in a particular case tat te function F (x, s) for a non omogeneous operator can be in general non omogeneous wit respect to s (see Section 3). We preliminarily study (in Section 2) te omogenization problem (.) in te simple case in wic a(x, ξ) does not depend on x and te sequence ( ) is given by ( \ E ), were E is te union of closed balls of radius r wose centers are periodically distributed. 2. Te periodic case. In tis section we consider te asymptotic beaviour of te problem (.) in te special case wen te sequence ( ) as a periodic structure. For every 0 < ε < let us consider a partition of R n, wit n > p, composed of semi-open cubes Q i ε, i Z n, of side lengt 2ε and center x i ε = 2εi. Let r = ε n/n p and for every i Z n let B i r be te closed ball in Q i ε of radius r and center x i ε. By Q ε and B r we sall denote Q 0 ε and B 0 r, respectively. Let E ε = i Bi r. It is well known (see [4] and [9]) tat, under tis special geometrical assumption, te limit problem wic describes te asymptotic beaviour of te Diriclet problems in \E ε is determined by te measure C n dx in, were C n is a positive constant wic depends only on n and p. Namely for every f W,p () if w ε is te solution of p w ε = f in \ E ε, w ε W,p 0 ( \ E ε ) ten w ε converge weakly in W,p 0 () to te solution w of te problem (2.) p w + w p 2 w = f in, w W,p 0 () as ε 0. Let us suppose now tat te function a(x, ξ) does not depend on x and let (r ) be a sequence wic converges to zero suc tat Let us define te function a (ξ) by lim (r ) p a( r ξ) ξ Rn. (2.2) a (ξ) = lim (r ) p a( r ξ) ξ Rn. It is easy to see tat a satisfies (i) (iii). In te sequel (ε ) will be te sequence of positive numbers converging to zero defined by (2.3) r = ε n/n p. 84

5 J. Casado Diaz & A. Garroni By Teorem. and by Remark.2 we can suppose tat for every f W,p () te sequence (u ) of te solutions of te problems (2.4) div a(du ) = f in \ E ε, u W,p 0 ( \ E ε ), converges weakly in W,p 0 () to te unique solution u of te problem div a(du) + F (x, u) = f in, (2.5) u W,p 0 (), were F : R R satisfied conditions (I) (III). We sall sow tat since te function a does not depend on x, te function F does not depend on x too (Lemma 2.). Moreover we sall prove tat, in tis case, it is possible to construct F by means of te function a defined in (2.2) (Teorem 2.2). Lemma 2.. Te function F (x, s) in problem (2.5) does not depend on x, i.e., F (x, s) = F (s) for every x R n. Proof: Let us consider an open set. Let ε 0 = dist (, ), ten for every i Z n, wit i =, and for every 0 < ε ε 0 we ave tat = + εi. Let s R, let s k be te solution of problem div a(ds k ) = k( s p 2 s s k p 2 s k ) in \ E ε, s k W,p 0 ( \ E ε ), and let s k be te solution of te analogous problem corresponding to. By te local caracter of Teorem. (see Remarks.3) we ave tat te sequences (s k ) and ( sk ), extended by zero in ( \ ) E ε and in ( \ ) E ε, converge weakly in W,p 0 () to te solutions s k and s k of problems div a(ds k ) + F (x, s k ) = k( s p 2 s s k p 2 s k ) in, (2.6) s k W,p 0 ( ), and (2.7) div a(d s k ) + F (x, s k ) = k( s p 2 s s k p 2 s k ) s k W,p 0 ( ), in, Moreover a penalization result proved in [3] permits us to conclude tat (2.8) lim k ϕ D(s k s) p dx + k ϕ s k s p dx = 0 85

6 for every ϕ C 0 ( ), wit ϕ 0, and A non omogeneous extra term... (2.9) lim ϕ D( s k s) p dx + k ϕ s k s p dx = 0 k for every ϕ C 0 ( ), wit ϕ 0. By (2.7) we ave a(d s k )Dϕ dx + F (x, s k )ϕ dx = k ( s p 2 s s k p 2 s k )ϕ dx for every ϕ C 0 ( ) and ten by canging variables in (2.7) we obtain a(d s k (x + εi))dϕ dx + F (x + εi, s k (x + εi))ϕ dx = = k ( s p 2 s s k (x + εi) p 2 s k (x + εi))ϕ dx for every v C 0 ( ). Tus for every ϕ C 0 ( ) by (2.6) we ave (2.0) F (x + εi, s k (x + εi)) F (x, s k (x)) ϕ dx a(d( s k (x + εi)) a(ds k (x)) Dϕ dx+ +k s k (x + εi) p 2 s k (x + εi) s k (x) p 2 s k (x) ϕ dx. Since by (2.8) and (2.9) we easily obtain tat te rigt and side of (2.0) converges to zero as k. By condition (I) te sequence (F (x + εi, s k (x + εi))) converges to F (x + εi, s) and (F (x, s k (x))) converges to F (x, s) for a.e. x in a compact subset of ; ence, from (2.0), we get F (x + εi, s) = F (x, s) for every x and for every ε ε 0 ; so tat F (x, s) = F (s) for every x and te conclusion follows from te arbitrariness of and s. From now on we sall denote by H p (R n ) te space of all functions belonging to L p (R n ), /p = /p /n, wose first order distribution derivatives belong to L p (R n ). We sall say tat a function u : R n R is Q ε -periodic if u(x + ε i) = u(x) for every x R n and for every i Z n. Te following teorem gives an explicit representation of te function F (s) in terms of te p-capacity of te closed unit ball B in R n relative to te operator div a. Tis result can be obtained as a particular case of te results proved by Skrypnik in a more general context (see []). For te sake of completeness we sall give ere an alternative proof tat olds in te particular case of a periodic structure. 86

7 J. Casado Diaz & A. Garroni We sall denote by C a positive constant wic can cange from line to line and wic depends only on n, α, and β. Teorem 2.2. Let s R and let ζ be te solution of te problem div a (Dζ) = 0 in { x > } (2.) ζ = s in { x } ζ H p (R n ). Ten te function F ( ) in (2.5) is given by te following formula (2.2) F (s) = 2 n a (Dζ)Dv dx, R n were v is an arbitrary function in H p (R n ) suc tat v = on { x }. Proof: Let s W,p loc (Rn ) be te solution of te problem div a(ds ) = F (s) in Q ε \ B r, (2.3) s = 0 on B r, s Q ε -periodic, were Q ε is te cube of center 0 and side 2ε and B r is te closed ball of center 0 and radius r. Let us prove tat te sequence (s ) converges to s weakly in W,p (). For every function v W,p () te following version of te Poincaré inequality olds (2.4) v p dx K p-cap(n(v), ) Dv p dx, were K is a positive constant independent of v and, N(v) = {x : v(x) = 0}, denotes te Lebesgue measure of, and p-cap(n(v), ) is te p-capacity of te set N(v) in (see [2]). Since {x Q ε : s (x) = 0} B r by (2.4) we get (2.5) Q ε s p dx K(2ε ) n p-cap(b r, Q ε ) Q ε Ds p dx for every N. Moreover it is well known tat p-cap(b r, Q ε ) p-cap(b r, B 2ε ) = (r p n (2ε ) p n ) and ence, by (2.3), p-cap(b r, Q ε ) ε n ; so tat by (2.5) we obtain (2.6) s p dx K Ds p dx, Q ε Q ε were K is positive constant independent of. Taking s as a test function in (2.3), by Hölder inequality and (2.6), we ave Ds p dx = F (s) s dx Q ε Q ε F (s)(2ε ) n/p ( Q ε s p dx ) p K p F (s)(2ε ) n/p ( Q ε ) Ds p p dx 87

8 A non omogeneous extra term... and ence (2.7) (2ε ) n Ds p dx CF (s) p. Q ε Since (ε ) tends to zero and (s ) is Q ε -periodic we ave (2.8) were lim o = 0. Tus by (2.6) we get Ds p dx + Ds p dx = + o (2ε ) n Ds p dx, Q ε s p dx = + o (2ε ) n s p dx, Q ε s p dx + o ( + K) Ds (2ε ) n p dx Q ε and ence, by (2.7), we ave tat (s ) is bounded in W,p (). Ten, up to a subsequence, (s ) converges weakly in W,p () to some function v. Since s is Q ε -periodic it is easy to ceck tat v is constant, i.e., v = c R. Moreover for every ϕ C 0 () te function s satisfies (2.9) a(ds )Dϕ dx = F (s) ϕ dx. Indeed if ϕ C 0 (), ten te function ψ(x) = i Z n ϕ(x + ε i) is Q ε -periodic and by (2.3) we ave a(ds )Dϕ dx = a(ds )Dϕ dx = a(ds )Dψ dx = i Z n Q i ε Q ε = F (s)ψ dx = F (s)ϕ dx = F (s)ϕ dx. Q ε Q i ε i Z n Tus by Remark.3 we ave tat F (c)ϕ dx = F (s)ϕ dx and ence by te monotonicity of F (condition (II)) we get c = s. Let us consider now te function z (x) = s (r x) and let us denote by Q te cube of center 0 and side 2ε /r. By canging variables in (2.3) we obtain tat z satisfies (2.20) r Q a(r Dz )Dv dx = F (s) v dx Q 88

9 J. Casado Diaz & A. Garroni for every v Q -periodic and v = 0 on B. By (2.7) and (2.3) we ave (2.2) Q Dz p dx = r n Q ε r p Ds p dx = ε n Q ε Ds p dx 2 n CF (s) p Let us denote by (z ) Q = Q Q z dx te average of z on Q. Since by (2.8) we ave (z ) Q = rn 2 n ε n z dx = Q 2 n ε n s dx = Q ε s dx ( + o ) and (s ) converges to s strongly in L p (), we obtain tat (z ) Q converges to s. Moreover by te Sobolev inequality we ave (2.22) ( Q z (z ) Q p ) /p dx ( ) /p C Dz p dx, Q were te constant C is independent of. Ten by (2.2) and (2.22), and by te fact tat (z ) Q converges to s we ave tat, up to a subsequence, te sequence (z ) converges to some function z W,p loc () weakly in W,p (B) for every bounded open set B R n. Let ϕ C0 (R n ) be wit compact support. Since for large enoug we ave supp ϕ Q we can take (z z)ϕ as test function in (2.20) and by (i) we get (2.23) = r p supp ϕ supp ϕ α D(z z) p ϕ dx supp ϕ (r ) p (a(r Dz ) a(r Dz))D(z z)ϕ dx = F (s)(z z)ϕ dx (r ) p a(r Dz )Dϕ(z z) dx supp ϕ (r ) p a(r Dz)D(z z)ϕ dx. supp ϕ Since, by (ii) and (2.2), te rigt and side of (2.23) tends to zero as we obtain tat (z ) converges to z strongly in W,p loc (Rn ). If we take as test function in (2.20) a function v W,p (R n ) wit compact support and v = 0 in B, ten we can take te limit as and by (2.2) we ave tat (2.24) R n a ( Dz)Dv = 0 for every v W,p (R n ) wit compact support and v = 0 in B. Let now ζ = s z. By (2.2) and te fact tat (z ) converges to z strongly in W,p loc (Rn ) we ave Dζ p dx C B 89.

10 A non omogeneous extra term... for every bounded open set B R n and ence Dζ L p (R n, R n ). Similarly by (2.22) and te fact tat (z ) Q converges to s we get tat ζ L p (R n ) and ence ζ H p (R n ). Finally, since ζ = s on B, by (2.24) we ave tat ζ is te unique solution of problem (2.). Let us prove te representation formula (2.2). Let v H p (R n ) be wit compact support and v = on B. Taking v as test function in problem (2.20) we ave (2.25) Q (r ) p a(r Dz )Dv dx = 2 n F (s) r p F (s) Q v dx. Taking te limit as we obtain (2.2). If v as not compact support, it is enoug to consider a function ṽ H p (R n ) wit compact support and ṽ = on B. Using v ṽ as test function in (2.) we get and tis concludes te proof. R n a (Dζ)Dv = R n a (Dζ)Dṽ = 2 n F (s) Remark 2.3. Te representation formula (2.2) assure tat if te function a is asymptotically (p )-omogeneous, i.e., (2.26) lim r 0 r p a( r ξ) ξ R n, ten te limit problem (2.5) does not depend on te coice of te sequence (ε ). More precisely if (2.26) is satisfied ten te asymptotic beaviour of te solutions u ε of te problems (2.27) diva(du ε ) = f in \ E ε, u ε = 0 on ( \ E ε ), is completely described by te solution of problem (2.5). So tat in tis case we ave te same omogenization penomenon wic is well known in te omogeneous case. 3. Example of non omogeneous extra term. In tis section, under special assumption for te function a, we sall prove tat te function F is (p )-omogeneous if and only if a is (p )-omogeneous. Tis result will permit us to exibit simple examples were te function F is not omogeneous (Remark 3.3) and to sow tat in tis case te omogenization penomenon describe by Remark 2.3 does not occurs (Proposition 3.2). Proposition 3.. Let us suppose tat te function a defined by (2.2) is of te form (3.) a (ξ) = γ( ξ )ξ 90

11 J. Casado Diaz & A. Garroni were γ : [0, + ] [α, β]. Ten te function F (s) given by (2.2) is omogeneous of degree (p ) if and only if a is omogeneous of degree (p ). Proof: If a is omogeneous of degree (p ) ten te result is a direct consequence of formula (2.2) once we note tat if ζ is te solution of problem (2.) at te level s R ten te function tζ, wit t R, is te solution of te same problem at te level ts. Vice versa let F (s) be omogeneous of degree (p ). Let ω n be te (n )-dimensional + r n u p dr + measure of te unit spere in R n and let H p = {u : [0, + ) R : + r n u p dr < + }. By assumption (3.) it is easy to see tat te solution ζ of problem (2.) is radially symmetric, i.e., ζ(x) = z( x ) wit z H p, and (3.2) F (s) = ω n 2 n + r n γ( z )z v dr for every v H p wit v() =. Moreover for every v H p wit v() = 0 we ave + r n γ( z )z v dr = 0. Ten tere exist a constant t(s) depending on s suc tat r n γ( z )z = t(s). By (3.2) we ave tat 2 n F (s)/ω n = t(s) and ence (3.3) ω n r n γ( z )z = 2 n F (s). Let us denote by P (r) te function defined by P (r) = γ( r )r for every r R. By (i) and (ii) we ave tat tere exists te inverse function P of P. Ten by (3.3) we ave (3.4) z (r) = P ( 2 n F (s) ω n r n ) and, since (3.5) + z dr = s and F (s) = s p 2 sf (), we get + ( s p 2 s ) G r n dr = s, were G(t) = P ( 2 n F ()t/ω n ). By canging variables in (3.5), wit ρ = s p 2 s/r n, we obtain p n s p 2 s s n s G(ρ) ρ n n dρ = s n 0 and ence s p 2 s 0 G(ρ) ρ n n dρ = (n ) s n p n. If we derive wit respect to s we get G(s) s n(p ) n γ( r )r = 2 n ω n (n p) F () r p 2 r, 9 = (n p) s p n s/ s and ence

12 wic concludes te proof. A non omogeneous extra term... Proposition 3.2. Under te same assumption of Proposition 3., let us assume tat te function a is not omogeneous. Ten tere exists a sequence (ε ) of positive number converging to zero suc tat te sequence (u ε ) of te solutions of problems (2.26) corresponding to ε = ε,p do not converges weakly in W0 () to te solution of problem (2.5). Proof: Let us prove te result by contradiction. Let u ε be te solution of problem (2.26) and let us suppose tat every subsequence of (u ε ) converges weakly in W,p 0 () to te solution u of problem (2.5). Let (r ) be a sequence suc tat a (ξ) = lim (r ) p a( r ξ) for every ξ Rn, for every t R, t > 0, let us define r t = t r, and let ε t be defined by r t = (ε t ) n/(n p) = t n/(n p) ε n/(n p).,p Since for every t > 0 te sequence (u ε t ) converges weakly in W 0 () to te solution u of problem (2.5), by Proposition 2.2 for every s R and for every v H p (R n ), wit v = in { x }, we ave (3.6) F (s) = 2 n were for every ξ R n R n a t (Dζ s t )Dv dx, (3.7) a t (ξ) = lim (rt ) p a( (r t ) ξ) = t p a (tξ) and ζ s t (3.8) is te solution of te problem div a t (Dζt s ) = 0 in { x > } ζt s = s in { x } ζt s H p (R n ). Moreover, since a (ξ) = γ( ξ )ξ, for every ξ R n we ave tat (3.9) a t (ξ) = t p 2 γ( tξ )ξ = t p 2 t a (tξ) t R, t 0. Now let zt s = t ζ s for every s, t R, wit t 0. By (3.8) zt s is te solution of te problem div a t (Dzt s ) = 0 in { x > } z t s = s/t in { x } zt s H p (R n ), ten, by uniqueness, we deduce tat z s t = ζ s/t t and ence tζ s/t t = ζ s s, t R, t 0. 92

13 J. Casado Diaz & A. Garroni Terefore by (3.6) and (3.9), for every s, t R wit t 0, we ave F (s) = 2 n a (Dζ)Dv s dx = t p 2 t R 2 n a t (Dζ s/t t )Dv dx = t p 2 tf (t s) n R n for every v H p (R n ), wit v = in { x }. Tis implies tat F is (p )-omogeneous and ence, by Proposition 3., a is (p )-omogeneous wic is a contradiction. Example 3.3. Let us consider te operator div (γ( Du )Du) were γ( t ) = (2 + sin(log t )) t p 2, tat can be considered as a non linear perturbation of te p-laplacian operator. It is easy to see tat te function a(ξ) = γ( ξ )ξ satisfies condition (i) (iii). Moreover if we coose r = exp( 2πn), ten γ(r ξ ) = γ( ξ ) for every ξ Rn and ence a (ξ) = a(ξ) for every ξ R n. Since te function a(ξ) is clearly non-omogeneous, by Proposition 3. we obtain tat te function F (s) wic appears in te limit problem (2.5) is non-omogeneous. Moreover wit tis coice of a(ξ), by Proposition 3.2, te limit problem for te sequence (u ε ) depend on te coice of te sequence (ε ), i.e., te omogenization result wic olds for te omogeneous case does not occur. References. [] Attouc H., 984, Variational Convergence for Functions and Operators, Pitman, London. [2] Casado Diaz J., Te omogenization problem in perforated domains for monotone operators, to appear. [3] Casado Diaz J., Garroni A., Asymptotic beaviour of non linear Diriclet systems on varying, to appear. [4] Cioranescu D., Murat F., 982 and 983, Un terme étrange venu d ailleurs, I and II, Nonlinear Partial Differential Equations and Teir Applications, Collège de France Seminar. Vol. II, 98-38, and Vol. III, 54-78, Res. Notes in Mat., 60 and 70, Pitman, London. [5] Dal Maso G., Defrancesci A., 988, Limits of nonlinear Diriclet problems in varying domains, Manuscripta Mat, 6, p [6] Dal Maso G., Garroni A., 994, New results on te asymptotic beaviour of Diriclet problems in perforated domains, Mat. Mod. Met. Appl. Sci., Vol. 4, No. 3, p [7] Dal Maso G., Mosco U., 987, Wiener s criterion and Γ-convergence, Appl. Mat. Optim., 5, p [8] Dal Maso G., Murat F., Asymptotic beaviour and correctors for Diriclet problems in perforated domains wit omogeneous monotone operators, to appear. [9] Labani N., Picard C., 989, Homogenization of nonlinear Diriclet problem in a periodically perforated domain, Recent Advances in Nonlinear Elliptic and Parabolic Problems (Nancy, 988), p , Res. Notes in Mat. 208, Longman, Harlow. [0]Marcenko A. V., Kruslov E. Ya., 978, New results in te teory of boundary value problems for regions wit closed-grained boundaries, Uspeki Mat. Nauk, 33, p. 27. [] Skrypnik I. V., 986, Nonlinear Elliptic Boundary Value Problems, Teubner-Verlag, Leipzig. [2] Ziemer W. P., 989, Weakly Differentiable Functions, Springer-Verlag, Berlin. 93

14 A non omogeneous extra term... Juan Casado Diaz Departamento de Ecuaciones Differenciales y análisis numérico Apartado de Correos Sevilla SPAIN jcasado@numer.us.es Adriana Garroni Dipartimento di Matematica Universitá di Roma La Sapienza Piazzale Aldo Moro Roma ITALY garroni@tsmi9.sissa.it 94