PROCEEDINGSOFTHE AMERICANMATHEMATICALSOCIETY Volume00,Number0,Pages000–000 S0002-9939(XX)0000-0 SOBOLEV REGULARITY AND AN ENHANCED JENSEN INEQUALITY 7 0 0 MARKA.PELETIER,ROBERTPLANQUE´,ANDMATTHIASRO¨GER 2 n a Abstract. Wederiveanewcriterionforareal-valuedfunctionutobeinthe J Sobolev space W1,2(Rn). This criterion consists of comparing the value of a functionalRf(u)withthevaluesofthesamefunctionalappliedtoconvolutions 5 of u with a Dirac sequence. The difference of these values converges to zero 1 as the convolutions approach u, and we prove that the rate of convergence to zero is connected to regularity: u ∈ W1,2 if and only if the convergence ] A is sufficiently fast. We finally applyour criterium toa minimizationproblem with constraints, where regularity of minimizers cannot be deduced from the F Euler-Lagrangeequation. . h t a m [ 1. Introduction 1 Jensen’s inequality states that if f : R → R is convex and ϕ ∈ L1(Rn) with v ϕ≥0 and ϕ=1, then 2 1 R 4 f u(x) ϕ(x)dx≥f u(x)ϕ(x)dx , 1 ZRn (cid:18)ZRn (cid:19) 0 for any u : Rn → R for(cid:0)whic(cid:1)h the integrals make sense. A consequence of this 7 inequality is that 0 / h f(u) = f(u)∗ϕ ≥ f(u∗ϕ). (1.1) at ZRn ZRn ZRn m In this paper we investigate the inequality (1.1) more closely. In particular we : study the relationship between the regularity of a function u ∈ L2(Rn) and the v asymptotic behaviour as ε→0 of i X Tε(u) := f(u)−f(u∗ϕ ) . (1.2) r f ε a ZRn Here ϕ (y) = ε−nϕ(ε−1y) and f is a(cid:2)smooth function(cid:3), but now not necessarily ε convex. Since u∗ϕ → u almost everywhere, we find for ‘well-behaved’ f that ε Tε(u) → 0 as ε → 0. Our aim is to establish a connection between the rate of f convergence of Tε(u) to zero and the regularity of u. f ReceivedbytheeditorsFebruary2,2008. 2000 Mathematics Subject Classification. Primary46E35;Secondary 49J45,49J40. Key words and phrases. Jensen’sinequality,regularityofminimizers,parametricintegrals. MA and MR were supported by NWO grant 639.032.306. RP wishes to thank the Centrum voorWiskundeenInformaticafortheirfinancialsupport. (cid:13)c1997AmericanMathematicalSociety 1 2 PELETIER,PLANQUE´,ANDRO¨GER Such a connection between the decay rate of Tε(u) and the regularity of u is f suggested by the following informal arguments. Taking n = 1, and assuming ϕ to be even and u to be smooth, we develop u∗ϕ as ε ε2 u∗ϕ (x) = u(x−y)ϕ (y)dy ≈ u(x)+ u′′(x) y2ϕ(y)dy, (1.3) ε ε R 2 R Z Z so that Tε(u) ≈ −cε2 f′(u)u′′ = cε2 f′′(u)|u′|2, (1.4) f R R Z Z withc= 1 y2ϕ(y)dy. Thissuggeststhatifu∈W1,2(R)thenTε(u)isoforderε2. 2 R f ′′ Moreover,sincef ≥0inthecasethatf isconvex,(1.4)givesanenhancedversion R of (1.1). Conversely, for functions not in W1,2(Rn) we might not observe decay of Tε(u) f with the rate of ε2, as a simple example shows. Take f(u) = u2 and consider a functionwithajumpsingularitysuchasu(x)=H(x)−H(x−1)6∈W1,2(R),where H(x) is the Heaviside function. Now choose as regularizationkernels the functions ϕ (x) = 1 (H(x+ε)−H(x−ε)), after which an explicit calculation shows that ε 2ε Tε(u)=2ε/3. Here the decay is only of order ε. f The goalofthis paper is to provethe asymptotic development(1.4) in arbitrary dimension and to show that also the converse statement holds true: if Tε(u) is f of order ε2 then u ∈ W1,2(Rn). These results are stated below and proved in Sections 2 and 3. The fact that one can deduce regularity from the decay rate is the original motivation of this work. In Section 4 we illustrate the use of this result with a minimization problem in which the Euler-Lagrange equation provides no regular- ity for a minimizer u. Instead we estimate the decay of Tε(u) directly from the f minimization property and obtain that u is in W1,2. OurresultscanbecomparedtothecharacterisationofSobolevspacesintroduced by Bourgain, Brezis and Mironescu [2, 4]. In fact, the regularity conclusion in Theorem 1.2 could be derived from [2], with about the same amount of effort as the self-contained proof that we give here. 1.1. Notation and assumptions. Let ϕ∈L1(Rn) satisfy ϕ ≥ 0 in Rn, (1.5) ϕ = 1, (1.6) ZRn yϕ(y)dy = 0, (1.7) ZRn |y|2ϕ(y)dy < ∞. (1.8) ZRn For ε>0 we define the Dirac sequence 1 x ϕ (x) := ϕ . (1.9) ε εn ε (cid:16) (cid:17) SOBOLEV REGULARITY AND AN ENHANCED JENSEN INEQUALITY 3 Eventually we will restrict ourselves to the case that ϕ is rotationally symmetric, that is ϕ(x)=ϕ˜(|x|), where ϕ˜:R+ →R+. Then (1.6) is equivalent to 0 0 ∞ nω rn−1ϕ˜(r)dr = 1, (1.10) n ε Z0 where ω denotes the volume of the unit-ball, i.e. |B (0)|=ω . n 1 n For a function u ∈ L1(Rn) and 0 ≤ s ≤ 1 we define the convolution u and ε modified convolutions u by ε,s u (x) := u∗ϕ (x) = u(x−y)ϕ (y)dy, (1.11) ε ε ε ZRn (cid:0) (cid:1) u (x) := u∗ϕ (x) = u(x−y)ϕ (y)dy = u(x−sz)ϕ (z)dz. ε,s εs εs ε ZRn ZRn (cid:0) (cid:1) (1.12) Note that u = u ,u = u . ε ε,1 ε,0 We also use the notation a⊗b for the tensor product of a,b∈Rn. 1.2. Statement of main results. Our first result proves (1.4). Theorem 1.1. Let f ∈C2(R) have uniformly bounded second derivative and ′ f(0)=0, f (0) = 0. (1.13) Let (ϕ ) be a Dirac sequence as in Section 1.1. Then for any u∈W1,2(Rn), ε ε>0 1 1 ′′ lim f(u)−f(u ) dx= f (u(x))∇u(x)·A(ϕ)∇u(x)dx, (1.14) ε→0ε2 ZRn ε 2ZRn where (cid:2) (cid:3) A(ϕ) = y⊗y ϕ(y)dy. (1.15) ZRn If ϕ is rotationally symmetric, i.e. ϕ(x)(cid:0)=ϕ˜(|x(cid:1)|), then 1 1 lim f(u)−f(u ) dx= a(ϕ) f′′(u(x))|∇u(x)|2dx, (1.16) ε→0ε2 ZRn ε 2 ZRn with (cid:2) (cid:3) ∞ a(ϕ) = ω rn+1ϕ˜(r)dr. (1.17) n Z0 The second theorem shows that for uniformly convex f a decay of Tε(u) of f order ε2 implies that u∈W1,2(Rn). Theorem 1.2. Let f ∈ C2(R) have uniformly bounded second derivative, assume that (1.13) is satisfied, and that there is a positive number c >0 such that 1 f′′ ≥c on R. (1.18) 1 If for u∈L2(Rn), 1 liminf f(u(x))−f(u (x)) dx < ∞, (1.19) ε→0 ε2 ZRn ε then u∈W1,2(Rn). In particular(cid:2)(1.14) and (1.16) h(cid:3)old. Remark 1.3. Weprescribe(1.13)sinceingeneralthedifferencef(u)−f(u )need ε not have sufficient decay to be Lebesgue integrable. For functions u ∈ L1(Rn)∩ L2(Rn) Theorems 1.1 and 1.2 hold even without assuming (1.13). 4 PELETIER,PLANQUE´,ANDRO¨GER 2. Proof of Theorem 1.1 We first show that f(u)−f(u ) ∈ L1(Rn). Let η ∈ C0(Rn;[0,1]). Using the ε c Fundamental Theorem of Calculus we obtain that η(x) f(u(x))−f(u (x)) dx= ε ZRn (cid:12)(cid:12) 1 ∂(cid:12)(cid:12) = η(x) f u (x) ds dx ε,s ZRn (cid:12)Z0 ∂s (cid:12) 1 (cid:12) (cid:0) (cid:1) (cid:12) ≤ η(cid:12)(x) f′ u (x) ∇(cid:12)u(x−sy)·yϕ (y) dydxds ε,s ε Z0 ZRn ZRn(cid:12) (cid:12) ≤ 1 1 η(x) (cid:12)(cid:12) f(cid:0)′ u (x(cid:1)) 2|y|ϕ (y)dydxds (cid:12)(cid:12) ε,s ε 2Z0 ZRn ZRn 1 1 (cid:0) (cid:1) + η(x) |∇u(x−sy)|2|y|ϕ (y)dydxds, (2.1) ε 2Z0 ZRn ZRn where we have used Fubini’s Theorem and Young’s inequality. For the first term on the right-hand side we deduce by (1.8) and (1.13) that η(x)f′ uε,s(x) 2|y|ϕε(y)dydx ≤ Cε(ϕ)kf′′k2∞ uε,s(x)2dx ZRnZRn ZRn (cid:0) (cid:1) (1.1) ≤ C (ϕ)kf′′k2 u(x)2dx. (2.2) ε ∞ ZRn For the second term on the right-hand side of (2.1) we obtain similarly η(x)|∇u(x−sy)|2|y|ϕ (y)dydx ≤ C (ϕ) |∇u(x)|2dx. (2.3) ε ε ZRnZRn Z By (2.1)-(2.3) we therefore obtain η(x) f(u(x))−f(u (x)) dx ≤ C (ϕ) 1+kf′′k2 kuk2 < ∞. (2.4) ε ε ∞ W1,2(Rn) ZRn (cid:12) (cid:12) (cid:0) (cid:1) Letting η ր(cid:12) 1 we deduce tha(cid:12)t f(u)−f(u ) ∈ L1(Rn). Repeating some of the ε calculations above, we obtain that 1 ∂ Tε(u) = − f u (x) dsdx f ZRnZ0 ∂s ε,s 1 (cid:0) (cid:1) ′ = f u (x) ∇u(x−sy)·yϕ (y)dydxds (2.5) ε,s ε Z0 ZRnZRn (cid:0) (cid:1) and that s ∂ ′ ′ ′ f u (x) −f u (x−sy) = − f u (x−ry) dr ε,s ε,s ε,s ∂r Z0 (cid:0) (cid:1) (cid:0) (cid:1) s (cid:0) (cid:1) ′′ = f u (x−ry) ∇u (x−ry)·ydr. (2.6) ε,s ε,s Z0 (cid:0) (cid:1) SOBOLEV REGULARITY AND AN ENHANCED JENSEN INEQUALITY 5 Since for all 0≤s≤1 ′ f u (x−sy) ∇u(x−sy)·yϕ (y)dydx = ε,s ε ZRnZRn (cid:0) (cid:1) ′ = f u (z) ∇u(z)·yϕ (y)dzdy ε,s ε ZRnZRn (1.7) (cid:0) (cid:1) = 0, we deduce from (2.5), (2.6) that 1 f(u(x))−f(u (x)) dx= ε2 ZRn ε (cid:2) 1 1 (cid:3) s ′′ = f u (x−ry) ∇u (x−ry)·ydr ε2 Z0 ZRnZRnh(cid:16)Z0 (cid:0) ε,s (cid:1) ε,s (cid:17) ∇u(x−sy)·yϕ (y) dydxds ε 1 1 s i ′′ = f u (x) ∇u (x)·y ε2 Z0 Z0 ZRnZRnh (cid:0) ε,s (cid:1) ε,s ∇u(x−ry)·yϕ (y) dydxdrds ε 1 1 s i ′′ = f u (x) ∇u (x)· εn+2 Z0 Z0 ZRn ε,s ε,s (cid:0) (cid:1) y⊗y ·∇u(x−ry)ϕ(ε−1y)dydxdrds ZRn 1 1 s (cid:0) (cid:1) ′′ = f u (x) ∇u (x)· εn+2 Z0 Z0 ZRn ε,s ε,s (cid:0)1 (cid:1) z⊗z ·∇u(x−z)ϕ(ε−1r−1z)dzdxdrds rn+2 ZRn 1 s (cid:0) (cid:1) ′′ = f u (x) ∇u (x)· κ ∗∇u (x)dxdrds, (2.7) ε,s ε,s εr Z0 Z0 ZRn (cid:0) (cid:1) (cid:0) (cid:1) with the modified convolution kernel κ defined by εr z κ (z) := (εr)−nκ , (2.8) εr εr (cid:16) (cid:17) where κ is given by κ(y) := y⊗y ϕ(y). (2.9) As ε→0 we have that for all 0≤r, s≤(cid:0)1, (cid:1) u → u almost everywhere, ε,s ∇u → ∇u in L2(Rn), (2.10) ε,s κ ∗∇u → κ(y)dy ∇u in L2(Rn). (2.11) εr (cid:16)ZRn (cid:17) Moreover,theintegrandontheright-handsideof (2.7)isdominatedbythefunction 1 2kf′′kL∞(R) |∇uε,s|2+ κεr∗∇u 2 , (2.12) (cid:16) (cid:12) (cid:12) (cid:17) (cid:12) (cid:12) 6 PELETIER,PLANQUE´,ANDRO¨GER which converges strongly in L1(Rn) as ε → 0 by (2.10) and (2.11). We therefore deduce from (2.7) that 1 lim f(u(x))−f(u (x)) dx= ε→0ε2 ZRn ε (cid:2)1 s (cid:3) ′′ = f (u(x))∇u(x)· κ(y)dy ∇u(x)dxdrds Z0 Z0 ZRn (cid:16)ZRn (cid:17) 1 ′′ = f (u(x))∇u(x)· κ(y)dy ∇u(x)dx. (2.13) 2ZRn (cid:16)ZRn (cid:17) In the rotationally symmetric case we observe that ∞ y⊗y ϕ(y)dy = rn+1ϕ˜(r)dr θ⊗θdθ ZRn(cid:0) (cid:1) (cid:16)Z0 ∞ (cid:17)ZSn−1 = ω rn+1ϕ˜(r)dr Id. (2.14) n (cid:16)Z0 (cid:17) Here we used that for i,j =1,...,n, ω if i=j, n θ θ dθ = i j ZSn−1 (0 if i6=j. We therefore obtain from (2.7)-(2.14) that in the rotationally symmetric case ∞ 1 1 lim f(u)−f(u ) dx = ω rn+1ϕ˜(r)dr f′′(u)|∇u|2dx. ε→0ε2 ZRn(cid:2) ε (cid:3) n(cid:16)Z0 (cid:17)2ZRn (2.15) 3. Proof of Theorem 1.2 Let 1 Λ := liminf f(u(x))−f(u (x)) dx. (3.1) ε→0 ε2 ZRn ε (cid:2) (cid:3) We first remark that by assumption (1.18) the function r 7→f(r)− c1r2 is convex 2 and we deduce again by (1.1) that c c f(u)− 1u2 − f(u )− 1u2 ≥ 0. (3.2) ZRnh(cid:16) 2 (cid:17) (cid:16) ε 2 ε(cid:17)i This implies that 1 2 1 2 liminf u2−u2 ≤ liminf f(u)−f(u ) = Λ. (3.3) ε→0 ε2 ZRn(cid:16) ε(cid:17) c1 ε→0 ε2 ZRn(cid:2) ε (cid:3) c1 SOBOLEV REGULARITY AND AN ENHANCED JENSEN INEQUALITY 7 Next we consider δ > 0 and the regularizations u = u∗ϕ . Set γ := ϕ ∗ϕ . δ δ ε ε ε Then we obtain that u2(x)− u ∗ϕ 2(x) dx= δ δ ε ZRn (cid:2) (cid:0) (cid:1) (cid:3) = u2− u (x)u (y)γ (x−y)dydx δ δ δ ε ZRn ZRnZRn 1 2 = u (x)−u (y) γ (x−y)dydx δ δ ε 2ZRnZRn (1.1)1 (cid:0) (cid:1) 2 ≤ u(x−z)−u(y−z) ϕ (z)dzγ (x−y)dydx δ ε 2ZRnZRnZRn 1 (cid:0) 2 (cid:1) = u(ξ)−u(η) γ (ξ−η)dηdξϕ (z)dz ε δ 2ZRnZRnZRn 1 (cid:0) 2 (cid:1) = u(ξ)−u(η) γ (ξ−η)dηdξ ε 2ZRnZRn (cid:0) (cid:1) = u2(ξ)−u2(ξ) dξ. (3.4) ε ZRn (cid:2) (cid:3) We therefore deduce from Theorem 1.1 and (3.3), (3.4) that |∇u |2 = C(ϕ)liminf 1 u2(x)− u ∗ϕ 2(x) dx ZRn δ ε→0 ε2 ZRn δ δ ε 1 (cid:2) (cid:0) (cid:1) (cid:3)2 ≤ C(ϕ)liminf u2(x)−u2(x) dx ≤ ΛC(ϕ). (3.5) ε→0 ε2 ZRn ε c1 (cid:2) (cid:3) Since this estimate is uniform in δ > 0 it follows that u ∈ W1,2(Rn). By Theo- rem 1.1 we therefore deduce (1.14) and (1.16). 4. Application: regularity of minimizers in a lipid bilayer model We illustratethe utility ofTheorems1.1and1.2with anexample. The problem is to determine the regularity of solutions of the following minimization problem. Problem 4.1. Let α > 0,h > 0 and a kernel κ ∈ W1,1(R,R+) be given such that 0 κ′ ∈BV(R). Denote by τ the translation operator, h (τ u)(x):=u(x−h) for u:R→R. h Consider the set K ⊂R, K := u∈L1(R) u=1, u≥0, u+τ u≤1 a.e. (4.1) h R n (cid:12) Z o and a strictly convex, increasing(cid:12)(cid:12), and smooth function f with f(0) = f′(0) = 0. We then define a functional F :K →R by F(u):= f(u)− αuκ∗u (4.2) R R Z Z and consider the problem of finding a function u∈K that minimizes F in K, i.e. F(u)=min{F(v) | v ∈K}. (4.3) 8 PELETIER,PLANQUE´,ANDRO¨GER Remark 4.2. This problem arises in the modelling of lipid bilayers, biological membranes (see [1] for more details). There, a specific type of molecules is consid- ered, consisting of two beads connected by a rigid rod. The beads have a certain volume, but the rod occupies no space. The beads are assumed to be of sub-continuum size, but the rod length is non- negligeable at the continuum scale. In the one-dimensional case we also assume that the rods lie parallel to the single spatial axis. Combining these assumptions we model the distribution of such molecules over the real line by a variable u that representsthe volume fraction occupied by leftmost beads. The volume fraction of rightmost beads is then given by τ u, and in the condition u+τ u ≤ 1 we now h h recognize a volume constraint. The functional F represents a free energy. In the entropy term f(u) the func- tion f is strictly convex, increasing, and smooth and can be assumed to satisfy ′ R f(0)=f (0)=0. The destabilizing term − uκ∗u is a highly stylized representa- tion of the hydrophobic effect, which favours clustering of beads. R Without the constraints u ≥ 0 and u + τ u ≤ 1 present in (4.1), we would h immediately be able to infer that minimizers (even stationary points) are smooth usingasimplebootstrapargument: sinceu∈K wehaveu∈Lp(R)andκ∗u∈W1,p for all 1≤p≤∞, and therefore, using the Euler-Lagrangeequation, ′ f (u)−2ακ∗u=λ, for some λ∈R, we find u∈W1,p(R). Iterating this procedure we obtain that u∈ Wk,p(R) for all k ∈ N , 1≤ p≤ ∞. However, when including the two constraints, twoadditionalLagrangemultipliersµandν appearintheEuler-Lagrangeequation, ′ f (u)−2ακ∗u=λ+µ−ν−τ−hν. Hereµandν aremeasuresonR,andstandardtheoryprovidesnofurtherregularity thanthis. Inthiscasethelackofregularityintheright-handsideinterfereswiththe bootstrap process, and this equation therefore does not give rise to any additional regularity. The interest of Theorem 1.2 for this case lies in the fact that K is closed under convolution. In particular, if u is a minimizer of F in K, then u = ϕ ∗u is also ε ε admissible, and we can compare F(u ) with F(u). From this comparison and an ε application of Theorem 1.2 we deduce the regularity of u: Corollary 4.3. Let u be a minimizer of (4.3). Then u∈W1,2(R). Proof. Chooseafunctionϕ∈L1(R)asinSection1.1withDiracsequence(ϕ ) , ε ε>0 andsetu :=ϕ ∗u. Firstwe showthatthere existsa constantC ∈R suchthatfor ε ε all u∈L2(R) uκ∗u−u κ∗u ≤ Cε2 u2. (4.4) ε ε R R (cid:12)Z (cid:12) Z With this aim we obser(cid:12)ve th(cid:0)at (cid:1)(cid:12) (cid:12) (cid:12) uκ∗u−u κ∗u = u κ∗u−κ∗u∗γ (4.5) ε ε ε R R Z Z with (cid:0) (cid:1) (cid:0) (cid:1) 1 x γ := ϕ∗ϕ, γ (x) := ϕ ∗ϕ (x) = γ , ε ε ε ε ε (cid:16) (cid:17) and that γ,γ satisfy the assumptions in S(cid:0)ection(cid:1)1.1. ε SOBOLEV REGULARITY AND AN ENHANCED JENSEN INEQUALITY 9 Since κ′ ∈BV(R) we have (κ∗u)′′ =κ′′∗u∈L2(R) and ′′ ′′ kκ ∗ukL2(R) ≤ |κ | kukL2(R). R (cid:16)Z (cid:17) Repeating some arguments of Theorem 1.1 we calculate that 1 d (κ∗u)(x)−(κ∗u∗γ )(x)= − (κ∗u)(x−sy)γ (y)dyds ε ε Z0 dsZR 1 ′ = (κ∗u)(x−sy)yγ (y)dyds ε Z0 ZR 1 1 = (κ∗u)′′(x−rsy)sy2γ (y)dydsdr ε Z0 Z0 ZR 1 1 = ε2 (κ∗u)′′(x−εrsz)sz2γ(z)dzdrds, Z0 Z0 ZR and therefore that u κ∗u−κ∗u∗γ ≤ ε R (cid:12)Z (cid:12) ≤(cid:12)(cid:12)ε2 1(cid:0) 1 (κ∗(cid:1)u(cid:12)(cid:12))′′(x−εrsz)2dx 21 u(x)2dx 12sz2γ(z)dzdrds ≤ε2CZ0(γ)Zk0κ′Z′∗RuhkZLR2(R)kukL2(R) i hZR i ≤ε2C(ϕ,κ)kuk2 , L2(R) which by (4.5) implies (4.4). Since K is closed under convolution with ϕ , u is admissible, and therefore ε ε 0≤F(u )−F(u)≤ [f(u )−f(u)]+Cε2 u2. ε ε R R Z Z The last term is bounded by Cε2kuk kuk ≤Cε2, L1(R) L∞(R) which follows from combining both inequality constraints in (4.1). Therefore [f(u)−f(u )]=O(ε2), and from Theorem 1.2 we conclude that u∈W1,2(R). ε (cid:3) R As is described in more detail in the thesis [3], Corollory 4.3 paves the way towards rigorously deriving the Euler-Lagrange equations for Problem (4.3), and forms an important ingredient of the proof of existence of minimizers. References [1] J. G. Blomand M. A.Peletier, A continuum model of lipid bilayers, Euro. Jnl. Appl.Math. 15(2004), 487–508. [2] J. Bourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, Optimal control andpartialdifferentialequations(Jos´eLuis(ed.)etal.Menaldi,ed.),Amsterdam: IOSPress; Tokyo: Ohmsha,2001,pp.439–455. [3] R.Planqu´e,Constraintsinappliedmathematics: Rods,membranes,andcuckoos,Ph.D.thesis, Technische UniversiteitDelft,2005. [4] A.C.Ponce,Anew approach toSobolev spaces andconnections toΓ-convergence,Calc.Var. PartialDifferentialEquations 19(2004), no.3,229–255. 10 PELETIER,PLANQUE´,ANDRO¨GER TechnischeUniversiteitEindhoven,DenDolech2,P.O.Box513,5600MBEindhoven Departmentof Mathematics,Vrije Universiteit, De Boelelaan 1081a,1081HV Ams- terdam E-mail address: [email protected] MaxPlanckInstituteforMathematicsintheSciences,Inselstr. 22,D-04103Leipzig