UNIQUE CONTINUATION FOR A QUASILINEAR ELLIPTIC EQUATION IN THE PLANE

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1 UNIQUE CONTINUATION FOR A QUASILINEAR ELLIPTIC EQUATION IN THE PLANE SEPPO GRANLUND AND NIKO MAROLA Abtract. We conider planar olution to certain quailinear elliptic equation ubject to the Dirichlet boundary condition. We how that if the boundary data ha finite number of relative maxima and minima then a olution ha the unique continuation property. Our method i new and applicable in the plane. 1. Introduction and preliminarie In thi paper we conider olution of quailinear econd order elliptic partial differential equation of the form A(x, u) = B(x, u), (1.1) where A: R 2 R 2 R 2 and B : R 2 R 2 R are Carathéodory function under certain tructural condition dicued in Section 1.2. A noteworthy example of uch equation i the p-laplace equation ( u p 2 u) = 0, (1.2) where 1 < p <, which give the Laplace equation when p = 2; we refer to [11]. The reult of thi note i the following. Let G be a bounded implyconnected Jordan domain in R 2. Suppoe u i a olution to (1.1) ubject to the Dirichlet boundary condition u = g on G, where g W 1,p (G) C(G). We aume minimal regularity condition on G o that every boundary point i regular, and hence u C(G). If g G ha finite number of relative maxima and minima, then u atifie the unique continuation principle, i.e. if u vanihe in ome open ubet of G, then u vanihe identically in G. For linear equation the tudy of unique continuation i rather complete [9]. It i known, however, that unique continuation doe not hold in certain cae for olution to the equation in (1.1). We mention a paper by Martio [13] in which he contructed ome counterexample in 2000 Mathematic Subject Claification. Primary: 35J62; Secondary: 35J25, 35J92. Key word and phrae. Dead core, Harnack inequality, maximum principle, nodal domain, p-laplacian, p-harmonic, quailinear elliptic equation, unique continuation. 1

2 the cae of p = n in (1.3), n 3, and B 0. The planar cae for (1.1) till remain an open problem, but in the preent paper we provide a partial olution. A typical two-dimenional phenomenon i the connection between quairegular mapping and linear econd order uniformly elliptic equation of the general form of two variable. Moreover, the unique continuation property can be derived from thi intrinic connection. Unique continuation for nonlinear equation, uch a (1.2), i till, to the bet of our knowledge, an open problem. There are ome reult. The two-dimenional cae of the p-laplace equation wa olved by Aleandrini [1] by conidering the et of critical point of a twodimenional olution to (1.2). We refer alo to Bojarki Iwaniec [7] and Manfredi [12] who oberved that the complex gradient of a olution to (1.2) i quairegular, and hence the unique continuation property follow for olution in two variable. We mention paper [4] and [5] in which the unique continuation property i achieved for olution to certain generalization of the p-laplace equation in the plane a a conequence of tudying the propertie of critical point and level line of uch olution; in [5], for intance, the author conider planar olution to the equation ( A u u (p 2)/2 A u) = 0 with A = A(x) uniformly elliptic and Lipchitz continuou ymmetric matrix. In the preent note, we conider two-dimenional olution to more general nonlinear equation and provide a new approach to thi problem in the plane. Our approach i baed on the analyi of nodal domain, which are maximal connected component of the et and nodal line {x G : u(x) 0}, {x G : u(x) = 0}, which are the boundarie of nodal domain. Our main tool i to couple the trong maximum principle and the Harnack inequality with ome topological argument; thi argument applie in the ituation in which there are finite number of nodal domain. Interetingly, the unique continuation property can be deduced from ome feature of the topology of planar et. The topological approach taken in the paper can be applied alo to the nonlinear eigenvalue problem involving the p-laplacian and to more general eigenvalue problem contituting the Fučik pectrum. We refer to a recent paper [10] for more detail. We want to refer to [2] ince the framework and ome idea there are omewhat related to thoe taken in the preent note. Finally, we refer to three recent paper on unique continuation. Aleandrini [3] 2

3 how the (trong) unique continuation property for olution to linear elliptic equation in two variable in divergence form, poibly non-elfadjoint and with lower order term. Armtron Silvetre [6] how that a vicoity olution of a C 1,1 uniformly elliptic fully nonlinear equation atifie the (trong) unique continuation property. In [8], on the other hand, the author tudy the game p-laplace equation on a tree and provide a characterization of the ubet of the tree that enjoy the unique continuation property Notation. Throughout, G i a bounded imply-connected Jordan domain of R 2. A domain i an open connected et in R 2. We write B r = B r (x) = B(x, r) for concentric open ball of radii r > 0 centered at ome x G. We denote the cloure, interior, exterior, and boundary of E by E, int(e), ext(e), and E, repectively Structural aumption. Let u pecify the tructure of A and B in (1.1); We hall aume that there are contant 0 < a 0 a 1 < and 0 < b 1 < uch that for all vector h in R 2 and almot every x G the following tructural aumption apply A(x, h) h a 0 h p, A(x, h) a 1 h p 1, B(x, h) b 1 h p 1 (1.3) where 1 < p <. We do not aume the monotoneity or the homogeneity of the operator A ince we do not conider exitence or uniquene problem. The tructural condition (1.3) reult in Hölder continuity of a weak olution to (1.1), and moreover in the Harnack inequality and the trong maximum principle, we refer to Serrin [18]. We could alo allow for the following tructural condition A(x, h) h a 0 h p, A(x, h) a 1 h p 1, (1.4) B(x, h) b 0 h p b 1 h p 1 where 0 < b 0 < and 1 < p <. In thi cae local Hölder continuity and the Harnack inequality for a locally bounded weak olution of (1.1) follow from Trudinger [19]. We do not conider the cae in which A and B may depend on u, or, for that matter, aim at the mot general tructure in (1.3) or (1.4). We will only need that olution of (1.1) are continuou, and atify the Harnack inequality and the trong maximum principle Some element of the plane topology. We recall a few fact about the topology of planar et; a good reference i [16]. Let Ω be any domain in R 2. A Jordan arc i a point et which i homeomorphic with [0, 1], whera a Jordan curve i a point et which i homeomorphic with a circle. By Jordan curve theorem a Jordan curve in R 2 ha 3

4 two complementary domain, and the curve i the boundary of each component. One of thee two domain i bounded and thi domain i called the interior of the Jordan curve. A domain whoe boundary i a Jordan curve i called a Jordan domain. A a related note, it i well known that the boundary of a bounded imply-connected domain in the plane i connected. In the plane a imply-connected domain Ω can be defined by the property that all point in the interior of any Jordan curve, which conit of point of Ω, are alo point of Ω [15]. A Jordan arc with one end-point on Ω and all it other point in Ω, i called an end-cut. If both end-point are in Ω, and the ret in Ω, a Jordan arc i aid to be a cro-cut in Ω. A point x Ω i aid to be acceible from Ω if it i an end-point of an end-cut in Ω. Acceible boundary point of a planar domain are aplenty: The acceible point of Ω are dene in Ω [16, p. 162]. We recall a few fact about connected et and ε-chain. If x and y are ditinct point, then an ε-chain of point joining x and y i a finite equence of point x = a 1, a 2,..., a k = y uch that a i a i1 ε, for i = 1,..., k 1. A et of point i ε-connected if every pair of point in it can be joined by an ε-chain of point in the et. A compact et F in R 2 i connected if and only if it i ε-connected for every ε > 0 [16, Theorem 5.1, p. 81]. If a connected et of point in R 2 interect both Ω and R 2 \ Ω it interect Ω [16, Theorem 1.3, p. 73]. Latly, we recall the following topological property [16, p. 159]. A ubet E of R 2 i aid to be locally connected at any x R 2 if for every ε > 0 there exit δ > 0 uch that any two point of B δ (x) E are joined by a connected et in B ε (x) E. A et i uniformly locally connected, if for every ε > 0 there exit δ > 0 uch that all pair of point, x and y, for which x y < δ can be joined by a connected ubet of diameter le than ε. All convex domain and, more generally, Jordan domain are uniformly locally connected [16, Theorem 14.1, p. 161]. However, imply-connected domain are not necearily locally connected. 2. Unique continuation and nodal domain We may interpret equation (1.1) in the weak ene. We recall that it follow from the tructural aumption (1.3) (or (1.4)) that a weak olution to (1.1) i Hölder continuou and atifie the following Harnack inequality. We refer to Serrin [18]. The proof i baed on the Moer iteration method [14]. 4

5 Theorem 2.1 (Harnack inequality). Suppoe u i a non-negative olution to (1.1) in B 3r G. Then where C = C(p, a 0, a 1, b 1 ). up u C inf u, B r B r Having the tructure (1.4) in (1.1), Theorem 2.1 can be found in Trudinger [19]. Moreover, in thi cae we hall aume that a weak olution u i locally bounded. We alo point out the following important property, the trong maximum principle, which can be deduced from the Harnack inequality. We refer to a monograph by Pucci and Serrin [17] on maximum principle. Theorem 2.2 (Strong maximum principle). Suppoe u i a non-contant olution to (1.1) in G. Then u cannot attain it maximum at an interior point of G. We hall make ue of the fact that if u i a olution to (1.1), then u c, c R, i alo a olution to an equation imilar to (1.1). Hence Harnack inequality and the trong maximum principle apply to both u and u c. Our main reult i the following theorem. We aume minimal regularity condition on G o that every boundary point i regular, and hence u C(G). Theorem 2.3. Suppoe u i a olution to the equation (1.1) under tructural condition (1.3) (or (1.4), and u locally bounded) in a bounded imply-connected Jordan domain G of R 2 ubject to the Dirichlet condition u = g on G, where g W 1,p (G) C(G). If g G ha finite number of relative maxima and minima, then u ha the unique continuation property: If u vanihe in ome open ubet of G, then u vanihe identically in G. We could alo tate the reult a follow: If u i a contant in ome open ubet of G, then u i identically contant in G. In what follow, however, we tick to the claical formulation by dealing with a vanihing olution. The crux of the proof i to tudy o-called nodal domain. A maximal connected component, i.e. one that i not a trict ubet of any other connected et, of the et {x G : u(x) 0} i called, in what follow, a nodal domain. We denote thee component by N i = {x G : u(x) > 0}, and N j = {x G : u(x) < 0}, where i, j = 1, 2,.... We remark that if u i, for intance, a olution to the p-laplace equation (1.2) it i not known whether the number of nodal domain i finite. 5

6 In the proof of the following key lemma, we make ue of the fact that G i a Jordan domain, and more preciely, G i uniformly locally connected at every x G, we refer to Section 1.3. Lemma 2.4. Let the hypothei of Theorem 2.3 be atified. Then the number of nodal domain, N i and N j, i finite. Proof. We note firt that u vanihe on all nodal line in G, i.e. on N i G and N j G. Hence by the trong maximum principle each nodal line meet the boundary of G. By the trong maximum principle the et N i G contain a global maximum of u in N i. We then how that uch maximum point x 0 N i i alo a relative maximum of g on G: Let x 0 G be a maximum point of u on ome fixed nodal domain N i. We hall then apply local connedtedne of G at every boundary point x G (Section 1.3). We claim next that there exit δ x0 > 0 uch that B δ (x 0 ) G, for each δ < δ x0, contain only point of N i. But aume that thi i not the cae. Hence for each δ < δ x0 there exit x B δ (x 0 ) G uch that x belong to ome other nodal domain than N i, ay, N j or u( x) = 0. Obviouly, we need to conider only the former cae. We write u(x 0 ) = max x N u(x) = σ > 0. There exit a poitive i δ < δ x0 uch that u(x) > σ 2 for every x N i G B δ(x 0 ) (it can be verified that uch point exit ince G i a Jordan domain). Moreover, ince there exit a point x B δ(x 0 ) G uch that x N j and G i locally connected at every x G, there mut exit alo a point x B δ(x 0 ) G o that u( x) = 0. For mall enough δ thi i not poible ince u i continuou and u(x 0 ) > 0. We have therefore obtained that there exit a poitive δ x0 uch that B δ (x 0 ) G, δ < δ x0, contain only point of N i. It follow that the inequality u(x) u(x 0 ) i valid both for every x B δ (x 0 ) N i and for every x B δ (x 0 ) G (in fact B δ (x 0 ) N i = B δ (x 0 ) G a et). Hence each maximum point x 0 N i contitute a relative maximum of g on G. An analogou reaoning applie for minima and relative minima on N j and G, repectively. Since g i aumed to poe only finite number of relative maxima and minima on G, the number of nodal domain mut be finite. Our idea in the proof of the preceding lemma ha certain imilarity to that of Lemma 1.1 in Aleandrini [2] where the number of interior critical point wa conidered to olution of linear equation. 6

7 The following proof reemble the argument preented in a recent paper by the author, we refer to [10] for more detail. Proof of Theorem 2.3. We aume for contradiction that (A) u vanihe in a maximal open et D G but i not identically zero in G. The maximal open et D i formed a follow: for every x G for which there exit an open neighborhood uch that u 0 on thi neighborhood we denote by B(x, r x ), r x = up { t > 0 : u B(x,t) 0 }, the maximal open neighborhood of x where u vanihe identically. Then the maximal open et D i imply the union of all uch neighborhood. We pick a connected component of D, till denoted by D. It i worth noting that (A) implie that the boundary data function g change ign at leat once on G. By Lemma 2.4, there exit poitive M, M < uch that we may index the nodal domain i = 1,..., M and j = 1,..., M. Each nodal domain i imply-connected which can be een a follow. Suppoe that N i i not imply-connected for ome i {1,..., M }. Then there exit a Jordan curve γ N i with it interior S γ and S γ contain point which do not belong to the fixed nodal domain N i. Moreover, S γ G ince G i aumed to be imply-connected. It follow that the et E = {x S γ \ N i : u(x) 0 or u(x) > 0} i non-empty. If Ẽ = {x S γ : u(x) < 0 or u(x) > 0} wa empty, then u(x) = 0 for all x E, and u(x) > 0 for all x S γ \ E. Thi i impoible by Harnack inequality, Theorem 2.1. Hence N i i implyconnected. We conider next the cae in which Ẽ = {x S γ : u(x) < 0 or u(x) > 0} E i non-empty. It uffice to conider only the point at which u < 0 (the point at which u > 0 are handled in the ame way); thi et i till denoted by Ẽ. The et Ẽ i open and each component of Ẽ i a ubet of ome nodal domain N j. Thi contradict with the fact that each nodal line meet G. Hence N i i imply-connected. An analogou, ymmetric, reaoning applie to N j. Hence N i and N j are connected a the boundarie of imply-connected domain, and thu continua, i.e. compact connected et with at leat two point, for each i and j. Suppoe next there exit a point x D G and it neighborhood B δ (x), δ > 0, uch that B δ G and B δ (x) ext(d) contain only point of either N i or N j for ome i and j, i.e. point at which either u > 0 or u < 0. Aume, without lo of generality, that B δ (x) ext(d) contain point of N i only. Then u 0 on B δ (x) and by Harnack inequality, Theorem 2.1, u 0 on B δ/2 (x), which contradict the maximality of the et D, and hence alo the antithei (A). In thi cae our claim follow. 7

8 By the preceding reaoning it i ufficient to conider the following ituation. For any x D G and for any δ < δ 0, δ 0 > 0, the neighborhood B δ (x) G contain point of the nodal domain N i and N j for ome indice i and for ome indice j. We point out that there exit a fixed index pair (, t) {1,..., M } {1,..., M } and δ 0 > 0 uch that each B δ (x) contain point of N and N t, but there might be alo point of other nodal domain in B δ (x), for every δ < δ 0 ; thi i a conequence of the fact that the number of nodal domain i finite in our cae. We reaon a follow: We conider a point x D G and B δ (x), δ < δ 0. We then elect a decreaing equence {δ i } uch that δ i < δ 0 and lim i δ i = 0. For each δ i we may pick a pair of nodal domain, which we write a i := (N (δ i ), N t(δ i ) ), uch that B δi (x) contain point of both nodal domain. Since the number of all poible nodal domain pair i finite, there exit a pair which appear infinitely many time in the equence {a i }. We may hence chooe thi fixed pair (N, Nt ), where (δ ij ) = and t(δ ij ) = t, for ome ubequence {δ ij } uch that lim j δ ij = 0. It can be een from thi reaoning that the ame pair occur in any neighborhood B δ (x), δ < δ 0. We hall next bae our reaoning on ome topological argument. We write D A = {x D : x i acceible from D}, N i,a = {x N i : x i acceible from N i }, and correpondingly N j,a. By [16] acceible boundary point D A, N i,a, and N j,a are dene in D, N i, and N j, repectively. We will decribe a election proce which give a pair of point x 1 and x 2 uch that x 1, x 2 D A G and that the aociated pherical neighborhood B δ (x 1 ) and B δ (x 2 ), δ < δ 0, contain point of the ame nodal line N, {1,..., M } fixed. Moreover, it i aumed that B δ (x 1 ) B δ (x 2 ) =, and that B δ (x 1 ), B δ (x 2 ) G. Thi procedure i a follow: We elect a finite equence of point {x l }, each x l D A G. A pointed out earlier, for each x l there exit δ 0 > 0 uch that the pherical neighborhood B δ (x), δ < δ 0, contain point of N and Nt for ome {1,..., M } and t {1,..., M }. Since the number of all poible nodal domain pair a decribed above i finite, after finite number of tep the equence {x l } will contain a pair of point, denoted x 1 and x 2, which have the aforementioned propertie. We then elect x 3 B δ (x 1 ) N,A and x 4 B δ (x 2 ) N,A. We connect x 1 to x 2 by a cro-cut γ D in D, and x 3 to x 4 by a crocut γ N in N. We remark that x 3, and analogouly x 4, i acceible in N with a line egment (conult, e.g., Remark 3.3 in [10]). Alo x 1, and analogouly x 2, i acceible in D with a line egment. We fix uch 8

9 line egment to acce the point x 1, x 2, x 3, and x 4. In thi way the line egment contitute part of the cro-cut γ D and γ N, repectively. Since the boundary N i connected it i alo ε-connected for every ε > 0. Hence for each ε > 0 the point x 1 and x 3 can be joined by an ε-chain {a 1,..., a k } N G uch that x 1 = a 1, a 2,..., a k 1, a k = x 3. We conider a collection of open ball {B 3 ε(a i)} k i=1, a i N G, uch 2 that B 3 ε(a i) G, and a domain Uε 1 which i defined to be 2 U 1 ε = k i=1 B 3 2 ε(a i). Since U 1 ε i a domain there exit a Jordan arc, γ ε x 1 x 3, connecting x 1 to x 3 in U 1 ε. Correpondingly, the point x 2 and x 4 can be joined by an ε-chain in N and we obtain a domain U 2 ε and a Jordan arc γ ε x 2 x 4 connecting x 2 to x 4 in U 2 ε. It i worth noting that we have elected γ ε x 1 x 3 and γ ε x 2 x 4 uch that either of them doe not interect γ D or γ N, ave the point x 1 and x 2, and x 3 and x 4, repectively. Thi i poible becaue of the line egment contruction decribed above. From the preceding Jordan arc we obtain a Jordan curve Γ ε, and by light abue of notation we write it a a product Γ ε = γ ε x 1 x 3 γ N γ ε x 2 x 4 γ D. The Jordan curve Γ ε divide the plane into two dijoint domain, and Γ ε contitute the boundary of both domain. We conider the bounded domain, denoted by T ε, encloed by Γ ε. See Figure 1. We next deal with the Jordan domain T ε. There exit at leat one point y T ε uch that u(y) < 0, i.e. y N j 0 for ome fixed j 0 {1,..., M }. Aume that thi i not the cae: then u(x) 0 for every x T ε. A γ D i part of the boundary T ε contain alo point of D, and hence u vanihe at uch point. By Harnack inequality, Theorem 2.1, u 0 in T ε. Thi i, however, impoible ince γ N i part of the boundary of T ε, thu u > 0 on a ufficiently mall neighborhood of a point in γ N. In an analogou way, it i poible to how that there exit a point z N j 0 (G \ T ε ). We tre that it i crucial that the elected point z and y belong to the ame nodal domain N j 0. It i worth noting here that by the trong maximum principle Γ ε cannot encloe the nodal domain N j 0 containing the point y, and therefore G \ T ε mut alo contain point of N j 0. We then connect z and y in N j 0 by a Jordan arc γ zy. Oberve that u(x) < 0 for every x γ zy. 9

10 The Jordan arc γ zy a a connected et interect Γ ε at leat at one point. We then ditinguih the following four poible cae for the point of interection: The point of interection i contained in (1) γ D, (2) γ N, (3) γx ε 1 x 3, (4) γx ε 2 x 4. In the cae (1) and (2) we have reached a contradiction a u(x) = 0 for every x γ D and u(x) > 0 for every x γ N, repectively. Conider the cae (3) and cae (4). We denote the point of interection by x ε for every ε > 0. We can elect an appropriate ubequence {x εj } j=1, lim j ε j = 0, uch that for each j either x εj Uε 1 j or x εj Uε 2 j. We aume, without lo of generality, that x εj Uε 1 j. The equence {x εj } i clearly bounded, and hence there exit a ubequence, till denoted {x εj }, uch that lim j x εj = x 0. Oberve that each x εj B 3 2 ε (a j m) for ome a m N G in the ε j -chain. We note that u(a m ) = 0. Moreover, if there exited δ 0 and a ubequence, till denoted {x εj }, uch that u(x εj ) δ 0 > 0 for every x εj, thi would contradict with uniform continuity of u (note that u i uniformly continuou on compact ubet of G). We hence have that u(x 0 ) = lim j u(x εj ) = 0. In concluion, we have reached a contradiction ince u(x 0 ) = 0 but, on the other hand, x 0 γ zy and hence u(x 0 ) < 0. All four cae (1) (4) lead to a contradiction. Hence antithei (A) i fale, thu the claim follow. Figure 1. Jordan domain T ε and Jordan curve γ zy (dotted line) connecting z to y in N j 0. 10

11 To conclude, a non-trivial olution of (1.1) in G under (1.3) or (1.4) and ubject to the Dirichlet condition a decribed in Theorem 2.3 ha only finitely many nodal domain and all nodal line interect G. Thee two fact enure that uch olution doe not vanih in an open ubet of G. Reference [1] Aleandrini, G., Critical point of olution to the p-laplace equation in dimenion two, Boll. Un. Mat. Ital. A (7) 1 (1987), [2] Aleandrini, G., Critical point of olution of elliptic equation in two variable, Ann. Scuola Norm. Sup. Pia Cl. Sci. (4) 14 (1987), [3] Aleandrini, G., Strong unique continuation for general elliptic equation in 2D, J. Math. Anal. Appl. 386 (2012), [4] Aleandrini, G, Lupo, D., and Roet, E., Local behavior and geometric propertie of olution to degenerate quailinear elliptic equation in the plane, Appl. Anal. 50 (1993), [5] Aleandrini, G. and Sigalotti, M., Geometric propertie of olution to the aniotropic p-laplace equation in dimenion two, Ann. Acad. Sci. Fenn. Math. 26 (2001), [6] Armtrong, S. N. and Silvetre, L., Unique continuation for fully nonlinear elliptic equation, Math. Re. Lett. 18 (2011), [7] Bojarki, B. and Iwaniec, T., p-harmonic equation and quairegular mapping, Partial differential equation (Waraw, 1984), 25 38, Banach Center Publ., 19, PWN, Waraw, [8] Del Pezzo, L. M., Moquera, C.A. and Roi, J.D., The unique continuation property for a nonlinear equation on tree, Prerint [9] Garofalo, N. and Lin, F.-H., Unique continuation for elliptic operator: a geometric-variational approach, Comm. Pure Appl. Math. 40 (1987), [10] Granlund, S. and Marola, N., Unique continuation for the econd eigenfunction of the p-laplacian in the plane, Preprint, Submitted [11] Lindqvit, P., Note on the p-laplace Equation, Report 102, Univerity of Jyväkylä, Department of Mathematic and Statitic, [12] Manfredi, J. J., p-harmonic function in the plane, Proc. Amer. Math. Soc. 103 (1988), [13] Martio, O., Counterexample for unique continuation, Manucripta Math. 60 (1988), [14] Moer, J., On Harnack theorem for elliptic differential equation, Comm. Pure Appl. Math. 14 (1961), [15] Nehari, Z., Conformal Mapping, McGraw-Hill Book Co., Inc., New York, [16] Newman, M. H. A., Element of the Topology of Plane Set of Point, 2nd ed. Cambridge, [17] Pucci, P. and Serrin, J., The Maximum Principle, Progre in Nonlinear Differential Equation and their Application, 73. Birkhäuer Verlag, Bael, [18] Serrin, J., Local behavior of olution of quai-linear equation, Acta Math. 111 (1964), [19] Trudinger, N. S., On Harnack type inequalitie and their application to quailinear elliptic equation, Comm. Pure Appl. Math. 20 (1967),

12 (S.G.) Univerity of Helinki, Department of Mathematic and Statitic, P.O. Box 68, FI Univerity of Helinki, Finland addre: (N.M.) Univerity of Helinki, Department of Mathematic and Statitic, P.O. Box 68, FI Univerity of Helinki, Finland addre: 12