We study the semi-classical limits of the first eigenfunction of a positive
second order operator on a compact Riemannian manifold when the diffusion
constant $\epsilon$ goes to zero. We assume that the first order term is given
by a vector field $b$, whose recurrent components are either hyperbolic points
or cycles or two dimensional torii. The limits of the normalized eigenfunctions
concentrate on the recurrent sets of maximal dimension where the topological
pressure \cite{Kifer90} is attained. On the cycles and torii, the limit
measures are absolutely continuous with respect to the invariant probability
measure on these sets. We have determined these limit measures, using a blow-up
analysis.Comment: Note to appear in C.R.A.

Holcman, D.

Kupka, I.

English

arXiv

Holcman, David

Kupka, Ivan

Numérisation de Documents Anciens Mathématiques

1/mannianenfunc-nvalues,e cycles. We haveer positif dust unmensions.la pressionla mesureC. R. Acad. Sci. Paris, Ser. I 341 (2005) 243–246http://france.elsevier.com/direct/CRASSDifferential GeometryConcentration of the first eigenfunctionfor a second order elliptic operatorDavid Holcmana, Ivan Kupkaba Department of Mathematics, Weizmann Institute of Science 76100 Rehovot, Israelb Département de mathématiques, 175, rue du Chevaleret, 75013 Paris, FranceReceived 9 June 2005; accepted 25 June 2005Available online 3 August 2005Presented by Thierry AubinAbstractWe study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemanifold when the diffusion constantε goes to zero. We assume that the first order term is given by a vector fieldb, whoserecurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigtions concentrate on the recurrent sets of maximal dimension where the topological pressure [Y. Kifer, Principal eigetopological pressure and stochastic stability of equilibrium states, Israel J. Math. 70 (1990) (1) 1–47] is attained. On thand torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these setsdetermined these limit measures, using a blow-up analysis.To cite this article: D. Holcman, I. Kupka, C. R. Acad. Sci. Paris,Ser. I 341 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméConcentration de la première fonction propre pour un opérateur elliptique du second ordre. Nous étudions sur unvariété Riemannienne compacte, les limites semiclassiques de la première fonction propre associée à un opérateusecond ordre positif divers quand la constante de diffusionε tend vers zéro. Nous supposons que le terme d’ordre un echamp de vecteurb, dont les ensembles récurrents sont des points hyperboliques ou des cycles ou des tores à deux diLes limites de la fonction propre normalisée sont concentrées sur les ensembles récurrents de dimension maximale oùtopologique est atteinte. Sur le cycles et les tores, les mesures limites sont absolument continues par rapport àde probabilitè invariante parb. Nous avons déterminé ces limites en utilisant une analyse de type blow-up.Po r citer cetarticle : D. Holcman, I. Kupka, C. R. Acad. Sci. Paris, Ser. I 341 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.E-mail addresses:david.holcman@weizmann.ac.il (D. Holcman), prukupka@aol.com (I. Kupka).1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.06.035244 D. Holcman, I. Kupka / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 243–246f aalalueeldof aeglobale limiterestingdy thisr ofere thebility1. Statement of the problemOn a compact Riemannian manifold(M,g), we study the semi-classical limits of the first eigenfunction opositive second order operatorLε = εg + θ(b) + c, (1)when the diffusion constantε goes to zero.g is the Laplace–Beltrami operator andθ(b) is the Lie derivative inthe directionb ∈ T M . The functionc is chosen so thatLε is positive. Note thatLε cannot be conjugated in generto a self-adjoint operator by scalar multiplication. The Krein–Rutman theorem implies that the first eigenvλεis real positive, simple and the associated eigenfunctionuε is of a fixed sign (see [6,7]). TheL2 normalized positiveeigenfunctionuε (∫Vnu2ε dVg = 1), is solution ofεguε + (b,∇uε) + cuε = λεuε.Let us introduce the following notation,Ω = b − ∇L, vε = e−L/(2ε)uε, cε = ε(c + gL2)+ ΨL, ΨL = 14(‖∇L‖2 + 2(∇L,Ω)),where the functionL :M → R belongs to a special class of Lyapunov functions, associated to the vector fib,as described in [5]. Eq. (2) is transformed intoLε(vε) = ε2gvε + ε(Ω,∇vε) + cεvε = ελεvε , and we impose thenormalization condition∫Vnv2ε dVg = 1.In this situation, a fundamental problem is to study (1) the limits ofλε (2) the limits of the measuresv2ε dVg , asε goes to zero.1.1. History of the problemThe study ofλε has its origin in the work of Kolmogorov about the mean first passage time (MFPT)random particle to the boundary of a bounded domain. For domains inRn, very interesting contributions to thfirst problem were made, in particular by Freidlin and Wentzell [4] and Devinatz and Friedman [1,3]. In thesituation (compact Riemannian manifold), Kifer in [6] made a fundamental contribution to the study of thof λε : When the recurrent set of the driftb is a finite union of hyperbolic components [8],λε has a unique limit,which can be explicitly determined in terms of the topological pressure of the components.1.2. Limits of the first eigenfunctionProblem (2) was much less studied than problem (1). However in the local situation, there are some intcontributions in the paper [2] by Devinatz and Friedman. Otherwise very little was known. Here, we stuproblem (2) in the spirit of Kifer. We consider driftb such that the recurrent set is the union of a finite numbehyperbolic components, which are either points, cycles or 2-dimensional torii.Let us mention that problem (2) is related to questions studied in the unique ergodic conjecture [9], whlimits of eigenfunctions are studied.2. Notation and assumptionsWe consider here vector fieldsb, related to the Morse–Smale fields, which satisfy some strong compaticonditions with respect to the metricg. More specifically we shall assume that for each fieldb:(I) The recurrent set is a finite union of stationary points, limit cycles and two-dimensional torii.D. Holcman, I. Kupka / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 243–246 245ceslessure is(II) The stationary points are hyperbolic and for each suchP the stable and unstable manifolds belonging toPare orthogonal atP with respect to the metricg.(III) Any limit cycle S has a tubular neighborhoodTS provided with a covering mapΦ :Rm−1 × R → TS havingthe following properties:(a) for all (x′, θ) ∈ Rm−1 × R,Φ−1 ◦ Φ(x′, θ) = {(x′, θ + nTS) | n ∈ Z}, TS minimal period ofS;(b) at any point(0, θ) ∈ Rm−1 × R, (Φ)∗g(0,θ) = ∑m−1n=1 dx2n + gmm(θ)dθ2 (θ = xm);(c) at any point(0, θ) ∈ Rm−1 ×R, (Φ)∗b = ∂∂θ +∑m−1i,j=1 Bij xj∂∂xi, up to term of order two inx′ = (x1, . . . ,xm−1), canonical coordinates onRm−1.(d) the(m−1)× (m−1) matrixB = {Bij | 1 i, j  m−1} is hyperbolic and its stable and instable spaare orthogonal with respect to the Euclidean metric∑m−1n=1 x2n on Rm−1.(IV) Any 2-dimensional torusR has a tubular neighborhoodTR provided with a diffeomorphismΦ :Rm−2 ×R2 → TR having the following properties:(a) at any point (0,θ) ∈ Rm−2 × R2, θ = (θ1, θ2), θ1, θ2 cyclic coordinates,(Φ−1)∗g(0,θ) =m−2∑n=1dx2n + a(θ)dθ21 + 2b(θ)dθ1 dθ2 + c(θ)dθ22;(b) at any point (0,θ) ∈ Rm−2 × R2, (Φ)∗(b)(0, θ) = k1 ∂∂θ1 + k2 ∂∂θ2 +∑m−2i,j=1 Bij xj∂∂xiwhere:(i) the (m − 2) × (m − 2) matrix B = {Bij | 1  i, j  m − 2} is hyperbolic and its stable and instabspaces are orthogonal with respect to the Euclidean metric∑m−2n=1 x2n on Rm−2,(ii) k1, k2 ∈ R and k1k2 ∈ R − Q. A torus with such a flow will be called an irrational torus.(iii) Small divisor condition: There exist constantsC > 0, α > 0 such that for allm1,m2 ∈ Z, |m1k1 +m2k2|  C(m21 + m22)α .3. Statement of the main resultThe spirit of our result can be summarized as follow:“On a Riemannian manifold, for any choice of a special Lyapunov functionL, vanishing at order2 on therecurrent sets of the field, the limits asε tends to0 of the normalized measures−L/(2ε)u2ε dVg are concentratedon the components of the recurrent sets which are of maximal dimension and where the topological preachieved.”The construction of the functionL is detailed in [5].Theorem 3.1. On a compact Riemannian manifold(M,g), let b be a Morse–Smale vector field andL bea special Lyapunov function forb. Consider the normalized positive eigenfunctionuε > 0 of the operatorLε = ε + θ(b) + c, associated to the first eigenvalueλε .(i) The recurrent setS of b is a union of a finite set of stationary pointsSs , a finite set of periodic orbitsSpand a finite set of two-dimensional irrational toriiSt . The limit set of a normalized measureu2εe−L/ε dVg∫Vnu2εe−L/ε dVg, iscontained in the set of probability measureµ of the formµ =∑P∈SstpcP δP +∑Γ ∈SptpaΓ δΓ +∑T∈SttpbTδTwhereSstp (resp.Sptp, Sttp) is the subset ofSs (resp.Sp, St ), where the topological pressure is attained.δP isthe Dirac measure atP .246 D. Holcman, I. Kupka / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 243–246eoperator,functioned,or not.a smalla Univ.e highestg, Newss and47.bridge,ment ofcaré Sect.For Γ ∈ Sp and h∈ C(V ), δΓ (h) =∫ TΓ0 fΓ (θ)h(Γ (θ))dθ , whereθ ∈ R →Γ (θ) ∈ V is a solution ofb repre-sentingΓ (see the notations for the precise definition ofθ ). The periodic functionfΓ is given byfΓ (θ) = exp{−θ∫0c(Γ (s))ds + θTΓTΓ∫0c(Γ (s))ds}andTΓ is the minimal period ofΓ .For T ∈ St andh ∈ C(V ), δT(h) =∫Th(θ1, θ2)fT(θ1, θ2)dST, wheredST is the unique probability measuron T invariant under the action of the field b andfT is the unique solution of maximum1, of the equationk1∂f∂θ1+ k2 ∂f∂θ2+ cf = µ2f whereµ2 =∫Tc dST.(ii) The coefficientscP , aΓ , bT obey the following rule: If at least one coefficientbT > 0, then for all cyclesaΓ = 0and all pointscP = 0. If all coefficientsbT = 0, and at least one coefficientaΓ > 0 then allcP = 0.4. Remark on the proofs and future perspectivesThe basic techniques are the Blow-up and stochastic analysis and the theory of Ornstein–Uhlenbeckbut they are too involved to be presented here. As a buy product, we estimate the decay of the first eigennear the recurrent sets.We conjecture that underLp normalization, wherep > 1, the limit measures are similar to the one we obtainexcept that the coefficients are different. To our knowledge, it is still unknown if the limit measure is uniqueBut this is a difficult problem as explained partially in [10] and in more details in [5].References[1] A. Devinatz, R. Ellis, A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators withparameter in the highest derivatives. II, Indiana Univ. Math. J. 23 (1973–1974) 991–1011.[2] A. Devinatz, A. Friedman, Asymptotic behavior of the principal eigenfunction for a singularly perturbed Dirichlet problem, IndianMath. J. 27 (1) (1978) 143–157.[3] A. Friedman, The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in thderivatives, Indiana Univ. Math. J. 22 (1972–1973) 1005–1015.[4] M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren Math. Wiss., vol. 260, Springer-VerlaYork, 1984.[5] D. Holcman, I. Kupka, Singular perturbation for the first eigenfunction and blow up analysis, Forum Math., May 2006, in prehttp://arxiv.org/pdf/math-ph/0412088.[6] Y. Kifer, Principal eigenvalues, topological pressure and stochastic stability of equilibrium states, Israel J. Math. 70 (1) (1990) 1–[7] R. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Stud. Adv. Math., vol. 45, Cambridge University Press, Cam1995.[8] C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, second ed., CRC Press, 1995.[9] P. Sarnak, Arithmetic quantum chaos, First annual R.A. Blygth Lectures, 15–19 March 1993, University of Toronto, DepartMathematics.[10] B. Simon, Semi-classical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions, Ann. Inst. H. PoinA (N.S.) 38 (3) (1983) 295–308.

Concentration of the first eigenfunction for a second order elliptic operator

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