Mathematical Physics, Analysis and Geometry 1: v, 1998. v Editorial This is the first issue of Mathematical Physics, Analysis and Geometry in its English-language, international form. A journal of the same name in the Russian language, having its roots in the long and splendid tradition of mathematical research in the former Soviet Union and, in particular, the Ukraine, was launched in 1994 by the Kharkov mathematical community and has published papers in the Ukrainian, Russian and English languages. The journal, in English only as of 1998, is intended to provide an international forum for important new results not only from the former Soviet Union but from all over the world. Mathematical Physics, Analysis and Geometry will publish research papers and review articles on new mathematical results with particular reference to complex function theory; operators in function space, especially operator algebras; ordinary and partial differential equations; differential and algebraic geometry; mathematical problems of statistical physics, fluids; etc. The Editors, supported and assisted by an international Editorial Board, will strive to maintain the highest quality. Papers which are too abstract will be discouraged. It is our purpose to make Mathematical Physics, Analysis and Geometry a leading journal of its kind attracting the best papers in the field. VLADIMIR A. MARCHENKO ANNE BOUTET de MONVEL HENRY McKEAN VTEXVR PIPS No: 168142 (mpagkap:mathfam) v.1.15 MPAGED1.tex; 14/05/1998; 16:06; p.1 Mathematical Physics, Analysis and Geometry 1: 1–22, 1998. 1 © 1998 Kluwer Academic Publishers. Printed in the Netherlands. Homogenization of Harmonic Vector Fields on Riemannian Manifolds with Complicated Microstructure L. BOUTET DE MONVEL Université de Paris 6, Institut de Mathématiques de Jussieu (UMR 9994 du CNRS), 4 place Jussieu, F-75252 Paris Cedex 05, France E. KHRUSLOV B. Verkin Institute for Low Temperatures, Mathematical Division, 47 Lenin Avenue, 310164 Kharkov, Ukraine (Received: 1 October 1997) Abstract. We study the asymptotic behaviour of harmonic vector fields with given fluxes or periods on special manifolds consisting of one or several copies of the Euclidean space, with a large number of small holes attached edge to edge by means of thin tubes (wormholes) when the number of holes tends to infinity. We obtain the homogenized equations describing the leading term of the asymptotics. Mathematics Subject Classifications (1991): 35B27, 35B40. Key words: electric field, homogenization, wormholes. According to the well-known ‘Wheeler picture’, electric fields can be represented as harmonic fields on special Riemannian manifolds Mϵ. Such a manifold can consist of one or several copies of the Euclidean space with small holes attached edge to edge by means of thin tubes. In geometrodynamics, these tubes are called ‘wormholes’ [1]. The flux of a vector field through a wormhole is interpreted as a charge of the electric field. Given fluxes through all wormholes of Mϵ determine a unique harmonic vector field on Mϵ vanishing at infinity. Such vector fields are also determined by their periods along cycles passing through wormholes. In this paper, we consider manifolds Mϵ depending on a small parameter ϵ > 0 such that the number of holes increases and their diameters vanish, as ϵ → 0. We study the asymptotic behaviour of harmonic vector fields on these manifolds with given fluxes or periods and obtain homogenized equations describing the leading term of the asymptotics. VTEX(EL) PIPS No.: 151041 (mpagkap:mathfam) v.1.15 MPAG009.tex; 14/05/1998; 16:03; p.1 2 L. BOUTET DE MONVEL AND E. KHRUSLOV 1. Description of the Problem n Let � be a fixed bounded domain in the space R (n ⩾ 3) and {Bϵi, i = 1 . . . N(ϵ)} be a family of closed pairwise disjoint balls in � depending on a small parameter ϵ > 0. We suppose that the total number N(ϵ) of balls tends to infinity and their diameters tend to zero, when ϵ → 0, and any open subdomain G ⊂ � contains some balls for sufficiently small ϵ. Let us consider the infinite domain N(ϵ) ⋃ n �ϵ = R \ Bϵi i=1 and the disjoint union of m copies (sheets) of �ϵ : �˜ϵ = �ϵ × {1 . . . m}. We construct a new manifold Mϵ by attaching to �˜ϵ some n-dimension tubes, gluing their boundaries to those of the holes of �˜ϵ. More precisely, Mϵ is defined by the following data: • for each ϵ > 0 the number mN(ϵ) is even and we are given a partition of the set of all pairs {(i, k) : i = 1 . . . N(ϵ), k = 1 . . . m} into subsets of two elements [(i, k), (j, l)]-linked pairs; kl • for each linked pair [(i, k), (j, l)], let T ϵij = Sn−1 ×[0, 1] be a spherical tube kl of dimension n (Sn−1 is unit (n − 1)-sphere). The boundary ∂T ϵij of this tube consists of two components: kl k l ∂T = Ŵ ∪ Ŵ ; ϵij ϵi ϵj • for each (i, k) we are given a diffeomorphism: k k k h : Ŵ ↔ ∂B , ϵi ϵi ϵi k k k where ∂B ϵi is a component of the boundary ∂�ϵ of kth sheet �ϵ = �ϵ × k. k kl The manifold Mϵ is the union of the sheets � ϵ (k = 1 . . . m) and tubes Tϵij identi- k fying boundary points pairwise according to the diffeomorphisms h . We suppose ϵi that Mϵ is an orientable manifold. Fragments of such a manifold are shown in Figures 1 and 2. We equip Mϵ with a differentiable structure inducing its canonical structure of k kl a differentiable manifold with a boundary on each sheet � and on each tube T . ϵ ϵij α We will usually denote x as the points of Mϵ and, when needed, x (α = 1 . . . n) as the coordinates in some local coordinate chart. Finally, we are given a Riemannian metric on Mϵ with a positively defined ϵ smooth metric tensor {g (x); α, β = 1 . . . n} inducing the standard flat metric αβ n of R outside some sufficiently small neighbourhoods of the tubes on Mϵ. For simplicity we will suppose that it induces such a flat metric on each whole sheet MPAG009.tex; 14/05/1998; 16:03; p.2 HARMONIC VECTOR FIELDS ON RIEMANNIAN MANIFOLDS 3 k Ωϵ i j kk T ϵij Figure 1. l Ωϵ j kl T ϵij i k Ωϵ Figure 2. k � . Its dependence on parameter ϵ on the tubes will be quantitatively characterized ϵ below (Section 2). We will study the harmonic vector fields on Mϵ and we may identify them with harmonic differential forms of degree 1 or n − 1. Let us recall some facts and notations from the theory of differentiable manifolds [2]. Differential forms of degree r (r-forms) are defined in local coordinates as follows: ∑ i1 ir ω(x) = ωi 1...ir (x) dx ∧ · · · ∧ dx , 16i1<··· 0 does not depend on ϵ. Here [(i, k), (j, l)] are linked pairs. kl Note, that solution v (x) of problem (2.1) minimizes the functional (2.2) over ϵij 1 kl k l class H (D ) of functions satisfying the boundary conditions on S and S . ϵij ϵi ϵj Taking this into account, it is easy to obtain the opposite inequality kl n−2 V ⩽ Caˇ , (2.3) ϵij ϵij where aˇϵij = min{aϵi, aϵj }. Therefore inequality (iv) means that radii of linked balls have the same order of magnitude when ϵ → 0. n n Now we introduce the distributions on R × R ∑ kl Vϵkl(x, y) = V ϵij δ(x − xϵi)δ(y − xϵj ); k, l = 1 . . . m, ij n kl where δ(x) is the delta-function on R , V = 0, if pairs [(i, k), (j, l)] are not ϵij linked. These distributions are non-negative and by virtue of (2.3) and (i)–(ii) ∫ ∫ Vϵkl(x, y) dx dy < C|�|; k, l = 1 . . . m, n where |�| is the volume of the domain � ⊂ R , C does not depend on ϵ. Thus, the sets of distributions {Vϵkl(x, y), ϵ > 0} (k, l = 1 . . . m) are compact in the sense of n n weak topology of distributions on R × R . We will assume that there exist weak limits (v) w − lim Vϵkl(x, y) = Vkl(x, y), k, l = 1 . . . m, ϵ→0 where Vkl(x, y) are non-negative distributions (densities of measures) with sup- ports in � × �. Symmetry Vkl(x, y) = Vlk(y, x) holds true and using (i), (ii), (v) and (2.3), one can show that integrals ∫ Vkl(x, y) dy = Ikl(x) are bounded functions of x: Ikl(x) ∈ L∞(�). MPAG009.tex; 14/05/1998; 16:03; p.7 8 L. BOUTET DE MONVEL AND E. KHRUSLOV n We further introduce the distributions on R : ∑ kl kl Pϵk(x) = V ϵijPϵij δ(x − xϵi) (k = 1 . . . m), ij and ∑ k �ϵk(x) = � ϵiδ(x − xϵi) (k = 1 . . . m), i kl k where P are periods of harmonic 1-form and � are periods of harmonic (n − ϵij ϵi 1)-form, which were described in Theorems 1 and 2, respectively. ′ n We will also assume that these distributions weakly converge (in D (R )) (j) w − lim Pϵk(x) = Pk(x) (k = 1 . . . m), ϵ→0 (jj) w − lim �ϵk(x) = �k(x) (k = 1 . . . m) ϵ→0 and, in addition, that the following inequalities hold (jjj) ∑ n−2 kl 2 a ϵi |Pϵij | < C1; i,j,k,l (jv) ∑ −n+2 k 2 a ϵi |�ϵi| < C2 i,k with constants C1, C2 independent of ϵ. It follows from (jjj), (jv) and (i), (ii) that limit distributions Pk(x), �k(x)(k = 1 . . . m) are functions of L2(�). (r) n (r) We denote L2(Mϵ) and L2(R ) the Hilbert spaces of r-forms (r = 1, n−1) n on Mϵ and R , respectively, with relation to the metrics scalar products [2]. (r) Let us introduce the operators Qϵk (k = 1 . . . m) mapping L2(Mϵ) into n (r) L2(R ) , defined by the formula: [ ] { k Qϵkvϵ (x) = vϵ(x), x × {k n} ∈ �ϵ, ⋃ 0, x ∈ R \ �ϵ = i Bϵi, (r) for any vϵ ∈ L2(Mϵ) . The main results of this paper are contained in the following theorems describing the asymptotic behaviour of harmonic 1-forms and (n − 1)-forms on Mϵ, as ϵ → 0. THEOREM 3. Let vϵ be the harmonic 1-form defined by Theorem 1 and let conditions (i)–(v) and (j), (jjj) be fulfilled, when ϵ → 0. Then, for any k n (1) Qϵkvϵ → vk weakly in L2(R ) , MPAG009.tex; 14/05/1998; 16:03; p.8 HARMONIC VECTOR FIELDS ON RIEMANNIAN MANIFOLDS 9 n where vk (k = 1 . . . m) are exact 1-forms on R such that vk = dUk and the col- lection of functions {Uk(x), k = 1 . . . m} is the generalized solution of the problem m ∫ ∑ n �Uk − Vkl(x, y)[Uk(x) − Ul(y)] dy = Pk(x), x ∈ R ; l=1 2−n Uk(x) = O(|x| ), x → ∞ (k = 1 . . . m). THEOREM 4. Let vϵ be the harmonic (n − 1)-form defined by Theorem 2 and let conditions (i)–(iv) and (jj), (jv) be fulfilled, when ϵ → 0. Then, for any k n (n−1) Qϵkvϵ → vk weakly in L2(R ) , n where vk (k = 1 . . . m) are (n−1)-forms on R such that vk = ∗ dUk and functions Uk(x) are the solutions of the problems n �Uk = �k(x), x ∈ R ; (2.4) 2−n Uk(x) = O(|x| ), x → ∞ (k = 1 . . . m). We will prove these theorems in Sections 3, 4 and 5. In Section 3 we will construct suitable representations for harmonic 1-forms and (n−1)-forms on Mϵ. Then using these representations, in Sections 4 and 5 we study the asymptotic behaviour of 1-form and (n − 1)-forms, respectively. 3. Representation of Harmonic Forms on Mϵ kl Let us make cuts on Mϵ along the middle section on each tube T ϵij , as indicated k l in Figure 3. We will denote by T and T the corresponding halves of the tube ϵi ϵj kl kl k l T (T = T ∪ T ). ϵij ϵij ϵi ϵj k Thus, we cut Mϵ in m components M ϵ (k = 1 . . . m) whose boundaries con- kl k n sist of N(ϵ) spheres S . Each M is homeomorphic to the space R with N(ϵ) ϵij ϵ k removed balls Bϵi. Therefore since n > 2, M ϵ has no 1-homology. k Hence, any closed 1-form vϵ on Mϵ is exact on each M ϵ , i.e., k vϵ = duϵk, x ∈ M ϵ (k = 1 . . . m). (3.1) k Here uϵk is a smooth function on M ϵ such that 2−n uϵk = O(|x| ), |x| → ∞ (3.2) and vϵ satisfying the vanishing condition (1.1). kl According to the definition of the periods P ϵij of vϵ and (3.1), (3.2), kl uϵk(x)| Skl − uϵl(x)|Slk = Pϵij . (3.3) ϵij ϵji MPAG009.tex; 14/05/1998; 16:03; p.9

Mathematical Physics, Analysis and Geometry - Volume 1

Similar Mathematical Physics, Analysis and Geometry - Volume 1