FRACTAL LAPLACIANS ON THE UNIT INTERVAL

Transcription

1 Ann. Sci. Math. Québec 7 3, no.,. FRACTAL LAPLACIANS ON THE UNIT INTERVAL ERIK J. BIRD, SZE-MAN NGAI AND ALEXANDER TEPLYAEV RÉSUMÉ. Nous étudions les valeurs/fonctions propres du laplacien sur [, ] définies par des mesures positives µ bornées, continues, supportées par [, ] et par la forme de Dirichlet classique sur [, ]. Nous donnons des preuves simples d existence, d unicité, de concavité ainsi que des propriétés des zéros de ces fonctions propres. Par une récritures des équations définissant le laplacien comme une équation intégrale de Volterra-Stieltjes, nous étudions les comportements asymptotique des premières valeurs/fonctions propres de Neumann et de Dirichlet, lorsque la mesure µ varie. Nous étudions les bornes du domaine des valeurs propres, dès que µ possède une structure autosimilaire finie post-critique. Lorsque µ appartient à la classe des mesures autosimilaires sur [, ], nous décrivons à la fois, la méthode des éléments finis et la méthode des approximations par différences, afin d obtenir des approximations numériques des valeurs/fonctions propres. Les fonctions propres en question, peuvent être considérées comme des analogues fractals du sinus et du cosinus de l analyse de Fourier. Nous notons l existence d une sous-suite de fonctions propres à décroissance rapide indexées par les nombres de Fibonacci. ABSTRACT. We study the eigenvalues and eigenfunctions of the Laplacians on [, ] which are defined by bounded continuous positive measures µ supported on [, ] and the usual Dirichlet form on [, ]. We provide simple proofs of the existence, uniqueness, concavity, and properties of zeros of the eigenfunctions. By rewriting the equation defining the Laplacian as a Volterra-Stieltjes integral equation, we study asymptotic behaviors of the first Neumann and Dirichlet eigenvalues and eigenfunctions as the measure µ varies. For µ defined by a class of post critically finite self-similar structures, we also study asymptotic bounds of the eigenvalues. By restricting µ to a class of singular self-similar measures on [, ], we describe both the finite element and the difference approximation methods to approximate numerically the eigenvalues and eigenfunctions. These eigenfunctions can be considered fractal analogs of the classical Fourier sine and cosine functions. We note the existence of a subsequence of rapidly decaying eigenfunctions that are numbered by the Fibonacci numbers.. Introduction. Let µ be a continuous positive finite measure with support suppµ = [, ]. In particular we are interested in the case when µ is a self-similar fractal measure. In this paper we study the eigenvalues λ and eigenfunctions u of the following equation:. u xv x dx = λ uxvx dµx, Reçu le 9 septembre 3 et, sous forme définitive, le 3 octobre 3. c Association mathématique du Québec

2 Fractal Laplacians on the Unit Interval where the equality holds for all v C,, the space of all infinitely differentiable functions with support contained in,. We impose either the Neumann boundary conditions. u = u = or the Dirichlet boundary conditions.3 u = u =. The left side of. is the standard Dirichlet form Eu, v := u xv x dx. In the Neumann case, the domain of E, DomE, is the Sobolev space W,, of functions u whose distributional derivative u belongs to L,, dx. Such functions must be continuous and representable as ux = c + x gy dy, x [, ], where g L,, dx and u = g. In the Dirichlet case, DomE = W,, := { u W,, : u = u = }. See, e.g., [D]. Throughout this paper we let and denote the L,, µ and the µ-essential supremum norms, respectively. Since C,, W W, L,, µ, DomE is dense in L,, µ. Moreover, the embeddings W, L,, µ and W, L,, µ are compact. Therefore, the quadratic form E is closed. Hence, equation. defines a Laplacian µ u as a distribution such that u v dx = µ uv dµ for all v C,. We can rewrite. as µ u = λu. In general, the Laplacian µ with domain Dom µ is defined as follows: for a continuous f and u Dom µ we have.4 µ u = f if and only if u = fdµ in the distributional sense Theorem.. The domains of the Dirichlet Laplacian D µ and Neumann Laplacian N µ will be characterized in Section 3. We remark that Freiberg

3 E. J. Bird, S.-M. Ngai and A. Teplyaev 3 [F, F, F3] has recently developed a theory for a general class of Laplace operators that includes the Laplacian µ. One of the main motivations for studying equation. comes from the study of similar problems on the Sierpinski gasket. In [DSV], eigenfunctions are computed explicitly. However, because of the high multiplicities, no effective algorithm has been found for the expansion of an arbitrary function in terms of the eigenfunctions. The equation we consider here provides a simpler model on a self-similar set. In Section we give a concise summary of the fundamental properties of the eigenvalues and eigenfunctions. We show that u and u are continuous and they have only isolated zeros unless u is constant. Moreover, the Dirichlet and Neumann eigenvalues are simple. We also study the Sturm-Liouville theory and show that an nth Neumann eigenfunction has n zeros and its derivative has n + zeros, while the nth Dirichlet eigenfunction has n + zeros and its derivative has n zeros. Furthermore, for each eigenfunction u the zeros of u and u alternate, and the zeros of the nth Dirichlet and Neumann eigenfunctions alternate. By converting equation. into an integral equation see Section 3, results in this section can be derived from classical known results see Atkinson [A]. For completeness, we include short proofs of the fundamental results. Equation. can be written as a Volterra-Stieltjes integral equation see Section 3:.5 ux = u + u x λ x x yuy dµy for all x. We interpret u and u as the left-hand and right-hand derivatives, respectively. It is known see [A] that a solution u of.5 is differentiable and the derivative satisfies.6 u x = u λ x uy dµy for all x. Conversely, any solution of.6 is also a solution of.5 see Theorem 3.. In view of equations.5 and.6, equation. is a generalization of the classical Sturm-Liouville equation.7 u x = λgxux for all x, where u is assumed to exist at every x [, ] and g L [, ]. If we define, for all Borel subsets E [, ], µe = gx dx, then, since u is differentiable at every x [, ] and gu L [, ], we have u x u = x u y dy = λ E x gyuy dy = λ x uy dµy. Hence, u satisfies.6. Conversely, if µ is absolutely continuous with Radon-Nikodym derivative g L [, ], then a solution u of.6 satisfies u x = u λ x gyuy dy

4 4 Fractal Laplacians on the Unit Interval with gu L [, ]. Hence u x = λgxux for a.e. x [, ]. Thus, u satisfies.7. In the special case gx = on [, ], solutions of.7 are the classical Fourier sine and cosine functions. Thus, for µ singular, solutions of. are the eigenfunctions of a vibrating string with end-points at, and with a mass distribution given by the singular measure µ. For the sine and cosine functions, the regions bounded by these functions and the interval on the x-axis between two successive zeros is convex. By using equations.5 and.6 we can easily obtain similar concavity results for the eigenfunctions of. see Propositions 3.3 and 3.4. We are interested in how eigenvalues λ and eigenfunctions u in equation. are affected by varying the measure µ. Assume {µ p : < p < } is a family of measures such that for each fixed c,, µ p [, c] as p. A typical example for such a class is provided by the iterated function system consisting of two similitudes S x = r x, S x = r x + r, < r, r <, and with µ = µ p defined by µ = pµ S + pµ S, < p <. Let z = z p be the zero of the first Neumann eigenfunction u p, and let λ p be the first eigenvalue. In Section 4 we prove that z p as p and that λ p as p. If we assume in addition that the first Neumann eigenfunctions u p are normalized such that u p = for all p,, then we show that u p as p. For the Dirichlet case, we show that the results lim p λ p = and lim p u p = also hold for the first eigenvalue and normalized eigenfunction. However, if z = z p, is such that u p z is the maximum of the first Dirichlet eigenfunction, we show in Example 4.7 that the analog lim p u p z = need not hold. Although many results concerning equations. and.3 have been obtained for general measures µ see [A], the restriction of µ to self-similar measures allows us to obtain good numerical approximations to the solutions and enables us to observe some interesting phenomena. To study the asymptotics of the eigenvalues, we restrict µ to be a self-similar measure defined by a post-critically finite see [K, K] self-similar structure. Let.8 S x = rx, S x = rx + r, < r <. Let µ be the self-similar measure satisfying the identity.9 µ = pµ S + pµ S, < p <. The spectral dimension see [K], d s, for this µ is given by pr d s/ + p r d s / =. Let R, Q, Z, N denote respectively the sets of reals, rationals, integers, and natural numbers. The eigenvalue counting function ρ : R N is defined as ρx = #{λ : λ is an eigenvalue and λ x}.

5 E. J. Bird, S.-M. Ngai and A. Teplyaev 5 Let λ n be the nth Dirichlet or Neumann eigenvalue. In Section 5 we study some asymptotic properties of ρλ n and λ n. In Section 6 we discuss numerical solutions to equation.. To approximate the eigenvalues and eigenfunctions numerically, we further restrict µ to be a self-similar measure defined by the iterated function system. S x = x, S x = x +. By making use of the identity fx dµ = p fs x dµ + p fs x dµ, we can solve equation. for u, v S m, the space of bounded continuous piecewise linear functions with knots at k/ m, k =,,..., m, where m is any positive integer. This allows us to set up a generalized eigenvalue system M m u = λn m u, with the solutions u approximating the Neumann eigenfunctions u. In addition to the finite element method above, we also describe the difference approximation method see Section 6. We show graphs of some approximate eigenfunctions. We observe that there exists a subsequence of rapidly decaying eigenfunctions that are numbered by the Fibonacci numbers.. Fundamental properties of solutions. In this section we summarize the basic existence and uniqueness results and study the oscillation properties of the eigenfunctions. Theorem.. Let µ be a positive bounded continuous measure on [, ] such that suppµ = [, ]. Then the following hold. a The initial value problem µ u = λu uy = α u y = β has a unique solution. In particular, if uy = u y = then ux. b µ u = λu if and only if u = λµu in the sense of generalized functions. c u is continuously differentiable; u is differentiable a.e. on [, ]. d u and u have only isolated zeros unless u constant. Proof. a In Section 3 we will prove that this initial value problem is equivalent to an integral equation, for which the existence and uniqueness of solutions is known see, for example, [A]. There is a different approach that is more relevant to analysis on fractals see [K] for more details. We define Green s function for the interval [, ] by { x y if x y gx, y = y x if x y

6 6 Fractal Laplacians on the Unit Interval and Green s operator by G µ fx = gx, yfydµy. For any continuous function f, we have that G µ f is Lipschitz continuous, has zero boundary values and satisfies µ G µ f = f. For any subinterval [t, t + δ] we can define Green s operator G µ,[t,t+δ] by G µ,[t,t+δ] fx = δ t+δ t g x t δ, y t δ fydµy. If δ is small enough depending on λ, then u has a unique representation on [t, t + δ] as. u = n λgµ,[t,t+δ] h n= where hx is a linear function such that ht = ut, ht + δ = ut + δ. Let u be the solution with boundary values {, } on [t, t + δ]. If δ is small enough depending on λ, then u t. This implies that ut = u t = if and only if u on [t, t + δ]. Hence we have uniqueness of the initial value problem on [t, t + δ]. It is easy to extend it to [, ] by dividing into small subintervals. b We have that µ u = λu if and only if u v dx = λ uvdµ for any test function, which means u = λµu as generalized functions. c The equation u = zµu implies that u is of bounded variation and hence u is differentiable a.e. on [, ]. d If δ is small enough, then on an interval [s, s + δ] the Dirichlet and the non-zero Neumann eigenvalues are larger than λ to see this, estimate G µ,[t,t+δ]. Hence u and u can have at most one zero each in [s, s + δ]. Proposition.. Let u = λµu. Then. u u = + λµ u u.3 u u = λµ u u as generalized functions. Proof. Direct computation. Proposition.3. Comparison of solutions Let u i suppose that a a, λ λ, = λ i u i, u i, i =, and b i = min{, {s > a i : u i s = }}

7 O / E. J. Bird, S.-M. Ngai and A. Teplyaev 7 Then c i = min{, {s > a i : u is = }}. a If u a = u a =, then c c. b If u a = u a =, then b b. c If λ = λ >, a = a =, u =, u, u =, u, then b b, c c. Proof. a According to., u i /u i is an increasing function on [a i, c i and so u u u u on [a, c [a, c, as on the illustration below. u u u u a a c c Statements b and c are proved similarly by. and.3. Proposition.4. Fix the initial conditions u, u and let u = λµu. Then a Zeros of u and u move to the left as λ increases. b fλ = u u is an increasing function on each interval of λ where u. c gλ = u is a decreasing function on each interval of λ where u. u Proof. By Proposition.3 consider the intervals on which neither u nor u have zeros. These intervals move to the left as λ increases. Theorem.5. Assuming the same hypotheses as in Theorem., the following hold: a The nth Neumann eigenfunction has n zeros and its derivative has n + zeros. b The nth Dirichlet eigenfunction has n + zeros and its derivative has n zeros. c For each eigenfunction u, zeros of u and u alternate. d Zeros of nth Dirichlet and Neumann eigenfunctions alternate. e There exists a complete orthonormal basis consisting of the Dirichlet Neumann eigenfunctions. The Dirichlet Neumann eigenvalues λ n are simple. Moreover, lim n λ n =. Proof. Parts a to d follows from Propositions.3 and.4 consider u/u and u /u as λ increases.

8 8 Fractal Laplacians on the Unit Interval e The eigenfunctions form a complete orthonormal set because the Laplacian has a compact resolvent, which is Green s operator defined in the proof of Theorem.. The eigenvalues tend to infinity by the same reason. Green s operator is compact because its kernel, Green s function gx, y defined in the proof of Theorem., is continuous see also [K, Theorem B..3]. Another approach can be found in [A]. The simplicity of the eigenvalues follows from the uniqueness of the initial value problem in Theorem. a. Remark. If the support of µ is smaller than [, ], then the results of this section are still true with the following modification. A problem is that u is linear on the intervals that does not intersect the support of µ. The uniqueness implies that u cannot have an interval of zero values. However u can have an interval of zero values, and in the results above such an interval should be taken as a single zero. 3. Volterra-Stieltjes integral equation and concavity of solutions. Assuming that µ is a bounded continuous positive measure with suppµ [, ], we will show that. is equivalent to the following Volterra-Stieltjes integral equation see [A, Chapter ]: 3. ux = u + u x λ x x yuy dµy, x. It is known that a solution u of 3. is differentiable and the derivative satisfies 3. u x = u λ x uy dµy, x. See [A, Theorem..]. The converse is also true. There is also a Green s function representation of u, essentially given in [S], as follows 3.3 ux = u + u u x + λ gx, yuy dµy, x, where gx, y is as defined in Section. We summarize these in the following theorem and give a brief proof for completeness; details in a more general setting can be found in [F] and [F]. Theorem 3.. Let µ be a bounded continuous positive Borel measure with suppµ [, ]. Then equations., 3., 3. and 3.3 are equivalent. Proof. Since the implication from 3. to 3. is known, we will show 3. implies 3.. Let u be a solution of 3.. Since u is continuous and µ is bounded, it follows from Fubini s Theorem that u x L [, ]. By using Fubini s Theorem again, ux u = x = u x λ Hence u satisfies 3.. u s ds = x x y x u λ s uy dµy ds uy dsdµy = u x λ x x yuy dµy.

9 E. J. Bird, S.-M. Ngai and A. Teplyaev 9 To see that. and 3. are equivalent, let u be a solution of 3.. Then for any v C,, we have u v dx = = u = λ = λ u λ x v x dx λ y uy dµy v x dx x uyv x dx dµy uyvy dµy. uyv x dµy dx Fubini s Theorem Hence, ux satisfies.. The above also proves the converse. Lastly, by using the definition of gx, y, we can rewrite equation 3.3 as ux = u + u u x + x yuy dµy λ x x yuy dµy. It is easy to see that this is equivalent to 3.. The following corollary follows from the proof of Theorem 3.. Corollary 3.. Assume the same hypotheses on µ as in Theorem 3., and let u L [, ], µ. Let f be a continuous function on [, ]. Then the following conditions are equivalent. a For all v C,, u v dx = vf dµ. b ux = u + u x + c u x = u + x x x yfy dµy, x. fy dµy, x. d ux = u + u u x gx, yfy dµy, x. Let D µ and N µ denote the Laplacian µ under the Dirichlet and Neumann boundary conditions respectively. Then their domains can be characterized, using Corollary 3., as follows. u Dom D µ resp. u Dom N µ if and only if u = u = resp. u = u = and there exists a continuous function f on [, ] f = D µ u or f = N µ u respectively satisfying any of the conditions in Corollary 3.. We now turn to concavity of the eigenfunctions. Although these results can be proved using the approach of Section, here we present another short proof, which is based in the use of integral equations.

10 Fractal Laplacians on the Unit Interval Proposition 3.3. Let µ be a bounded continuous positive Borel measure with suppµ [, ], and let u be an nth Neumann eigenfunction of 3. satisfying u >. Let z < z < < z n be the zeros of u and write z = and z n+ =. Then u is decreasing on [z i, z i+ ] for even i i < n + and u is increasing on [z i, z i+ ] for odd i i < n +. Consequently, a If i i < n + is even, then u has a local minimum at z i and u is concave downward on [z i, z i+ ]. b If i < i < n + is odd, then u has a local maximum at z i and u is concave upward on [z i, z i+ ]. Proof. Let i i < n + be even and let z i < x < x < z i+. Since ux > on z i, z i+, by using 3. we have x u x = λ λ zi zi uy dµy = λ uy dµy λ x z i x uy dµy λ uy dµy = λ z i x uy dµy uy dµy = u x. Hence u is decreasing on [z i, z i+ ]. Now let i < i < n + be odd and let z i < x < x < z i+. Since uy < on z i, z i+, a similar argument shows that u x u x. Therefore u is increasing on [z i, z i+ ]. Consequences a and b follow directly from these results. Analogous results hold for Dirichlet eigenfunctions. Proposition 3.4. Let µ be a bounded continuous positive measure with suppµ [, ] and let u be an nth Dirichlet eigenfunction of 3. satisfying u + >. Let = z < z < z < < z n = be the zeros of u. Then u is increasing on [z i, z i+ ] for even i i < n and u is decreasing on [z i, z i+ ] for odd i i < n. Consequently, a If i i < n is even, then u has a local maximum at z i and u is concave downward on [z i, z i+ ]. b If i < i < n is odd, then u has a local minimum at z i and u is concave upward on [z i, z i+ ]. Proof. Use u x = u x uy dy instead of u x = x uy dy and use the same argument as in the proof of Proposition The first eigenvalues and eigenfunctions. In this section we study behaviors of the eigenfunctions and eigenvalues as the measure varies. We will focus on properties of the first Dirichlet and non-zero Neumann eigenvalues and the associated eigenfunctions. We will restrict our attention to families {µ p : < p < } of bounded continuous measures satisfying the condition that for any fix c,, µ p [, c] as p.

11 E. J. Bird, S.-M. Ngai and A. Teplyaev An example of such a family is provided by self-similar measures defined by an iterated function system of two similitudes of the form S x = r x, S x = r x + r, < r, r <. Let µ be the self-similar measure satisfying the identity 4. µ = pµ S + pµ S, < p <. Note that µ depends on p. Proposition 4.. Let µ = µ p be defined as in 4.. Then for any fixed c,, µ[, c] as p. Proof. Applying 4., we get µ[, c] = pµ[, r c] + pµ[ r /r, c/r r /r ] p + pµ[, c/r r /r ]. Define c = c, c = c /r r /r, and in general define c i+ = c i /r r /r. Then by applying the above calculations repeatedly, we have µ[, c] p + p p + p p + + p p i + p i+ µ[, c i+ ]. It is easy to see that c i c i+ = r /r i+ c. In fact, c i c i+ = c i c i = = c c = r r r r i+ c. Consequently there must exist some i o such that c io + < and therefore p i o+ µ[, c] p = p io+, p which tends to as p. Let λ denote the first Dirichlet or nonzero Neumann eigenvalue and let u denote a corresponding eigenfunction. According to Theorem. e, λ is a simple eigenvalue and therefore all associated eigenfunctions are scalar multiples of each other. Moreover, according to Theorem.5 a, the first Neumann eigenfunctions have a unique common zero in,, which we will denote by z. Figure shows the behavior of the first Neumann eigenfunctions as the measure varies. Note that equation 3. and the boundary condition u = together imply that u dµ = and consequently 4. z u dµ = z u dµ.

12 Fractal Laplacians on the Unit Interval a p = /, λ = π 9.87 b p = /4, λ c p = /8, λ 6.5 d p = /6, λ 7. Figure. Approximate first L µ-normalized Neumann eigenfunctions u and eigenvalues λ as p varies, where µ is defined by 4. with r = / Theorem 4.. Let {µ p : < p < } be a family of continuous probability measures with suppµ p [, ] such that for any fixed c,, µ p [, c] as p. Let z = z p be the common zero of the Neumann eigenfunctions associated to the first eigenvalue. Then z as p. Proof. For each p, we let u = u p be the first eigenfunction satisfying u = and hence 4.3 ux = λ x x yuy dµy. Suppose z p does not tend to as p. Then there would exist < b < and a sequence {p k } such that p k and z = z p k b for all k. We notice by assumptions that z Hence by 4., u pk dµ pk b dµ pk = µ pk [, b] as k. 4.4 z u pk dµ pk = z u pk dµ pk as k.

13 E. J. Bird, S.-M. Ngai and A. Teplyaev 3 Since µ pk [z, ] µ pk [b, ] as k, 4.4 implies that for all x [z, ], 4.5 u pk x u pk as k, for otherwise, the concavity of u pk on [z, ] would force the integral on the left-hand side of 4.4 to be greater than some positive constant independent of k. Let c = b + /. Then the concavity of u pk and 4.5 imply that 4.6 u p k c u p k c c z as k. We will show that 4.6 is impossible. By Proposition 3.3, u p k is increasing on [, z ]. By applying the Mean-Value Theorem to this interval, we get 4.7 Also, by using 4., z u p k z = λ pk u p k z z b >. = λ pk c u pk y dµ pk y = λ pk z u pk y dµ pk y + λ pk with the second term dominating. In fact, c z u pk y dµ pk y c u p k y dµ pk y u c µ [z pk pk, c] u pk c µ pk [c, ] Combining these observations with 4.7 leads to lim u p k c = lim λ pk k k c z u pk y dµ pk y upk y dµ pk y = lim contradicting 4.6. The first equality is because for any p c u pk y dµ pk y, as k. k u p k z c z c u pc = λ p u p y dµ p y = λ p u p y dµ p y λ p = λ p This proves the result. z u p y dµ p y λ p c z u p y dµ p y = λ p b, z u p y dµ p y c u p y dµ p y. Theorem 4.3. Assume the same hypotheses as in Theorem 4. and let λ = λ p be the first Neumann eigenvalue of the corresponding equation 3.. Then λ as p. Proof. We assume, as in the proof of Theorem 4., that for all p, u p are chosen so that u p = and 4.3 holds. We will prove the assertion by contradiction. Suppose there

14 4 Fractal Laplacians on the Unit Interval exists a sequence {p k } with p k such that λ pk is bounded by some M >. Then, by 3., for all x [, z ], 4.8 u p k x M µ pk [, x] M. The first inequality is because ux u = on [, z ]. Now, fix any number N sufficiently large so that N > M and /N >. Then for any x [, /N], u pk x [ Mµpk [, x] Mµ pk, ] as k. N Hence, u pk uniformly on [, /N] as k. In particular, there exists some k o N such that for all k k o, u pk x > [ on, ]. N Since uz = and z <, the Mean-Value Theorem implies that there exists some ξ pk [ /N, z ] such that u ξ pk / /N = N > M, contradicting 4.8. Thus λ p as p. Theorem 4.4. Assume the same hypotheses as in Theorem 4. and assume in addition that u p =. Then u p = u p z = λ z u p y dµy = λ up y dµp y as p. z Proof. We prove by contradiction. Suppose there exists a subsequence {u pk }, with p k as k, and a positive constant C such that u p k z C for all k. Obviously, this can happen only if u pk as k. Hence, for all k sufficiently large we have 4.9 By assumption, 4. z u pk y dµ pk y z Therefore by 4., 4.9, and 4. we have z upk y dµpk y as k. u pk y dµ pk y + u pk y dµ pk y =. z z z u pk y dµ pk y as k u pk y dµ pk y as k.

15 E. J. Bird, S.-M. Ngai and A. Teplyaev 5 Let x = x p k [, z ] be the unique number such that u pk x =. Such an x exists; otherwise we would have u pk and therefore z u pk y dµ pk y z u pk y dµ pk y as k, contradicting 4.. Consider the following two cases. Case. x z as k. In this case we have lim k x p k = lim k z = and therefore lim k u p k z =, a contradiction. Case. x z as k. By taking a subsequence if necessary we may assume that x p k c < for some c and for all p k. In this case, z x This forces u pk y dµ pk y x z x u pk y dµ pk y as k by 4.. u pk y dµ pk y as k by 4.. But µ pk [, x ] µ pk [, c] as k, and therefore u pk, forcing u p k ξ pk for some ξ pk [, x ]. Consequently, lim k u p k z = because u p k x is decreasing on [, z ]. This again contradicts the assumption and the proof is complete. We now turn to properties of the first Dirichlet eigenvalues and eigenfunctions. Figure shows rather striking behavior of the first Dirichlet eigenfunctions as the measure varies. Theorem 4.5. Let {µ p : < p < } be a family of continuous probability measures with suppµ [, ] such that for any fixed c,, µ p [, c] as p. Let λ p be the first Dirichlet eigenvalue. Then lim p λ p =. Proof. For < p <, let u p be the first Dirichlet eigenfunction satisfying u p =. Then the concavity of u p forces 4.3 u p x for all x. Also, by putting x = into 3., we have 4.4 λ p yu p y dµ p y =. We prove the assertion by contradiction. Suppose there exists a constant C > and a sequence {p k } such that lim k p k = but λ pk C for all k. By replacing with a subsequence if necessary, we can assume that there exists a sequence {c pk } with < c pk < such that 4.5 lim k c p k =

16 6 Fractal Laplacians on the Unit Interval a p = /, λ = π 9.87 b p = /4, λ and c p = /8, λ 6.5 d p = /6, λ 7. Figure. Approximate first L µ-normalized Dirichlet eigenfunctions u and eigenvalues λ as p varies, where µ is defined by 4. with r = / 4.6 µ pk [, c pk ] C. Then by 4.3, 4.4 and 4.6, cpk = λ pk yu pk y dµ pk y + λ pk λ pk µ pk [, c pk ] + λ pk c pk µ pk [c pk, ] + C c p k, c pk yu pk y dµ pk y which is impossible by 4.5. This contradiction completes the proof. Theorem 4.6. Assume the same hypotheses as in Theorem 4.5 and assume in addition that each eigenfunction u p satisfies u p > and is normalized, i.e., u p =. Then 4.7 lim p u p =. Proof. Let ξ p be the unique zero of u p in,. The normalization condition u p = implies that 4.8 u p = u p ξ p.

17 E. J. Bird, S.-M. Ngai and A. Teplyaev 7 We suffices to prove that 4.7 holds for each sequence p k. We prove this by contradiction. Suppose there exist a sequence {p k } and a positive constant C such that p k but 4.9 u p k C for all k. If there exists some subsequence {p kj } such that ξ pkj or u pkj, then obviously we would have u p kj cf. 4.8, contradicting 4.9. Hence we will assume there exist positive constants C and C 3 such that ξ pk C < and u pk C 3 for all k. Now, for any fixed c C,, we have = u pk = = C 3 c c u pk y dµ pk y + upk y dµpk y + C3 C 3 C 3µ pk [, c] + C 3 c Consequently, for all k sufficiently large, we have c c u pk y dµ pk y upk y dµpk y C 3 u pk C 3 dµ pk y C 3µ pk [, c] + C 3 u pk c. u pk c C 3. This obviously implies that u p k, contradicting 4.8 again. This completes the proof. Example 4.7. Assume the same hypotheses of Theorem 4.6 and let z = z p, be such that u p z is the maximum of the first Dirichlet eigenfunction. In this example we describe two situations such that: A lim p z p = and B lim p z p =. Therefore the analog of Theorem 4. does not hold for the first Dirichlet eigenfunction. In fact, one can construct an example such that lim p z p =. A To simplify computations, instead of the interval [, ] we consider the interval [, + p ], and the zero boundary conditions are at and at + p. Clearly, this will not change the limiting behavior of z p as p. Let measure µ p have a density with respect to Lebesgue measure that is equal to p on [, and equal to /p on [, + p ]. This µ p is not a probability measure, but again it will not change the limiting behavior of z p as p. Equation.4 or Theorem.b implies that u p x = C sin x λp on the interval [, ], and u p x = C sin λ + p x/p on the interval [, + p ]. The function u p x, as well as its derivative, must be continuous at x =, which means C sin λp = C sin p λ and C p p cos λp = C cos p λ.

18 8 Fractal Laplacians on the Unit Interval Therefore λ is the lowest positive solution of the equation tan λp = p p tan p λ. If pλ = y, then y is the lowest positive solution of the equation tany = p p tan y p. It is easy to see that lim p pλ = limp y = π. This implies lim p z p =. In this example we have lim p u p x = uniformly on any compact subset of,. However, lim p up x/ u p = sinπx uniformly on [, ]. B Here, in order to simplify computations, we consider the interval [, + p] instead of the interval [, ], and measure µ p that has density p on [, and density /p on [, + p]. The zero boundary conditions are at and at + p. Then we have u p x = C sin x λp on the interval [, ], and u p x = C sin λ+p x/p on the interval [, + p]. Moreover, C sin λp = C sin pλ Therefore pλ = π and lim p z p =. and C p cos λp = C cos pλ. In fact, a simple computation shows that lim p u p x = sin π x uniformly on any compact subset of [,. In particular, lim p u p = lim p u p z =. B Here we modify example B so that lim p z p = and lim p u p x = x uniformly on any compact subset of [,. We consider the interval [, + p] instead of the interval [, ], and measure µ p with density p on [, and density p on [, + p]. The zero boundary conditions are at and at + p. Then λ is the lowest positive solution of the equation tan λp = p p tan λ p. It is easy to see that lim p λ p = π. This implies lim p z p =, and also lim p u p x = x for x <. Example 4.7 shows that under the same hypotheses as in Theorem 4.6, we can have lim sup p u p = as well as lim sup p u p <. The same is true for the first Neumann eigenfunction. Conjectures 4.8. We conjecture that lim p u p = if µ p and u p are as in Figures and. However, the numerical approximations show that the growth rate is very low. We also conjecture that in the same situation there is a limit lim p u p x/ u p uniform on any compact subset of [,. Moreover, this limit is a convex nonnegative piecewise linear function different in Dirichlet and Neumann cases. In addition, we conjecture that lim p z p = in the Dirichlet case, and lim p u p = in the Neumann case. 5. Eigenvalues and their asymptotics. We begin by recovering two well-known properties of the eigenvalues for the Laplacians corresponding to general measures on [, ].

19 E. J. Bird, S.-M. Ngai and A. Teplyaev 9 Proposition 5.. Let µ be a bounded positive continuous measure with suppµ = [, ]. Let λ D n and λ N n denote the nth Dirichlet and Neumann eigenvalues, respectively. Then a λ N n λ D n ; b The Dirichlet and Neumann eigenvalues are discrete. Proof. Part a follows from the variational formula see e.g., [D, K] because W, W,. Part b follows from Theorem.5 e. The discreteness of the Dirichlet eigenvalues follows from the inequality λ n= n gx, y dµx dµy <, a consequence of Bessel s inequality and the Green s function representation of the eigenfunctions Theorem 3.. The discreteness of the Neumann eigenvalues follows by combining this with part a. For the rest of this section we restrict our attention to a p.c.f. self-similar structure. Consider the iterated function system 5. S x = rx, S x = rx + r, < r <, and let µ be the self-similar measure defined by 5. µ = pµ S + pµ S, < p <. Recall that the spectral dimension see [K], d s, for this µ is given by 5.3 pr d s/ + p r d s / =. Let ρ : R N be the eigenvalue counting function defined as ρx = #{λ : λ is an eigenvalue and λ x}. Then according to a theorem of Kigami and Lapidus see [Kl, K, SV], ρx is related to d s as follows: i Non-arithmetic case If logpr/ log p r R\Q, then lim x ρx/xd s/ exists and is positive and finite. ii Arithmetic case On the other hand, suppose logpr/ log p r Q and suppose T > is the generator of the group logpr/z + log p r/z i.e. T Z equals the group. Then log x ρx = G + o x ds/, where G is a non-zero bounded periodic function of period T. Using the above result we can prove

20 Fractal Laplacians on the Unit Interval Proposition 5.. Fix < p < and let d s be the spectral dimension of the measure µ in 5.. Let ρ be the eigenvalue counting function for the Laplacian defined in. with the Dirichlet or Neumann boundary condition. Then a b lim n lim n ρprλ n = pr ds/. ρλ n ρ p rλ n ρλ n = p r d s /. Proof. We only prove a; the proof of b is similar. We first consider the non-arithmetic case. By i above and the fact that lim n λ n = Theorem.5 e, we have ρprλ n ρλ n = ρprλ nprλ n d s/ pr d s/ ρλ n λ d s/ n pr ds/ as n. For the arithmetic case, by using ii above, we have ρprλ n G ρλ n = logprλ n +o prλ n ds/ G = logλ n +o prλ n ds/. G logλ n +o λ ds/ n G logλ n +o λ ds/ n The last equality follows from the periodicity of G, because logpr/ = kt for some k Z. Again ρprλ n /ρλ n pr ds/ as n. Remark. lim n ρprλ n /ρλ n resp. lim n ρ p rλ n /ρλ n is the asymptotic ratio of the number of zeros of an eigenfunction in [, r] resp. [r, ] to the number of zeros in [, ]. In the case r = /, it is observed that if this ratio is equal to α, where α = 5 / be the reciprocal of the golden ratio, then there exists a subsequence of rapidly decaying Neumann eigenfunctions. To find the corresponding value of p, we set the result in Proposition 5. equal to α to get p ds / 5.4 = α. 5.3 then implies that p ds/ 5.5 = α = α. Taking logarithms on both sides of 5.4 and 5.5 and dividing, we get logp/ log p/ = p = 3. The graphs in Section 6 are plotted with this value of p. See Figures 3 and 4. Notation. Given two real functions f and g, we denote by fx gx as x if there exist two constants C, C > such that C gx fx C gx for all sufficiently large x. Similarly, if {a n }, {b n } are two sequences, then a n b n as n means that there exist two positive constants C, C such that C b n a n C b n for all sufficiently large n.

21 E. J. Bird, S.-M. Ngai and A. Teplyaev Proposition 5.3. Fix < p < and let µ = µ p be a self-similar measure defined by 5., let d s denote the spectral dimension of µ, and let λ n denote the nth Dirichlet or Neumann eigenvalue. Then λ n n /d s. Proof. First, consider the non-arithmetic case. From the result in [KL] again, lim x ρx = L for some < L <. xds/ In particular, for all x sufficiently large, we have L ɛ ρx x d s/ L + ɛ, where ɛ > is small enough so that L ɛ >. This implies that for all n sufficiently large, 5.6 L ɛ ρλ n λ d s/ n L + ɛ. We consider the Neumann case ρλ n = n + ; the Dirichlet case ρλ n = n is similar. For all n sufficiently large, L ɛ n + λ ds/ L + ɛ n + /ds n L + ɛ n /d s λ n /ds L ɛ /ds L + ɛ n/d s λ n /ds + n /ds n /d s L ɛ /ds /d s n /d s. This proves the assertion for the non-arithmetic case. For the arithmetic case, ρx = G log x + o x ds/. Since G is a bounded non-zero periodic function and o as x, there exist constants C, C > such that for all sufficiently large x, we have C ρx/x ds/ C. In particular, for all n sufficiently large, C ρλ n /λ ds/ n C. We are back to 5.6 and the assertion follows from the same argument. 6. Numerical Approximations. Let {S, S } be defined as in. and let µ be the corresponding self-similar measure as defined in.9. Then for any f L [, ], µ, 6. fx dµ = p fs x dµ + p fs x dµ.

22 Fractal Laplacians on the Unit Interval All integrals in this section are over the interval [, ]. The following identities can be derived by using 6. see [S]: a x dµ = p b x dµ = p3 p 3 6. c x dµ = p + p 3 d x x dµ = p p. 3 For a multi-index J = j,..., j m, j i = or, we let J denote the length of J and define S J = S j S jm and m J = m J m := #{i : i m, j i = }. i.e., m J is the number of S s in the composition S j S jm. By iterating 6., we get the following useful identity 6.3 f dµ = p m J p m m J fs J x dµ, m, J =m where the summation runs over all indices J = j,..., j m with j i = or. Identities 6. and 6.3 are useful in the finite element method, one of the methods we use to obtain approximate eigenvalues and eigenfunctions. 6.. The finite element method. To numerically approximate the solutions of., we first solve it for u, v S m, the space of bounded continuous piecewise linear functions with knots at points k/ m, k =,,..., m. Suppose u, v S m satisfy k k u m = a k, v m = b k, k =,,..., m. That is, for all x [k / m, k/ m ], k =,,..., m, ux = m a k a k x k m + a k vx = m b k b k x k m + b k. Then m u v dx = m a k a k b k b k 6.4 k= m m = m a k a k b k m a k a k b k k= k= m = m a a b + m -a k + a k a k+ b k + m -a m + a mb m. k=

23 E. J. Bird, S.-M. Ngai and A. Teplyaev 3 On the other hand, by using identity 6.3 we can express the right-hand side of. as λ uv dµ = λ p m J p m m J us J xvs J x dµ 6.5 = λ m k= J =m p m k p m m k ak x + a k x b k x + b k x dµ, where m k = m k m := m J, and J is the unique index such that S J [, ] = [k / m, k/ m ]. The second equality in 6.5 follows from the definitions of u and v and the formula S J x = x/ m + k / m. Substituting the identities in 6. into 6.5 and regrouping terms, we get 3 uv dµ = m k= p m k p m m k p + pa k b k + p pa k b k + a k b k + p3 pa k b k = + pp m+ a + pp m+ m { a b + p +m k p m+ m k a k + + pp +m k+ p m m k+ + 3 pp m k p m+ m k a k + p +m k+ p m+ m k+ a k+ }b k + p p m+ a m + 3 p p m+ a m b m. Comparing coefficients of the b k in this equation and in 6.4 leads to the following system of linear equations + pp m+ a + pp m+ a k= 6.6 m a a = λ 3 m a k + a k a k+ = λ p +m k p m+ m k a k pp +m k+ p m m k+ + 3 pp m k p m+ m k a k + p +m k+ p m+ m k a k+, k =,..., m m a m + a m = λ p p m+ a 3 m + 3 p p m+ a m. By writing u = [u, u,..., u m] T, we can express these equations in matrix form as 6.7 M m u = λn m u. For example, M =, M =,

24 4 Fractal Laplacians on the Unit Interval and N, N are respectively p + p p p p p 4p p p p, 3 p p 3 p p p 3 + p p 3 p p 3 p 4p p p p p p p p3 4p +4p p p 3 p p 4p p p p 3. p p 3 3 p p 3 We remark that both M m and N m are symmetric, tridiagonal, and of order m +. Moreover, they satisfy the recursive relations in the following proposition. Proposition 6.. [ ] Mm M m+ = [ Nm N m+ = p [ + M m ] + p ] [ N m ], where,, and are zero matrices of orders m m, m m+, and m+ m respectively. Proof. The first identity is quite obvious. The second one follows by using induction and applying the following simple relation to 6.6: { mk m + for k =,..., m m k m + = m k m for k = m +,..., m Difference approximation method. There is an alternative way to approximate the solutions of. by making use of the discrete Laplacian on the set of knots. Define V m = { k m : k =,,..., m}, m =,,.... For any function u : V m R, the discrete Laplacian of u, denoted by H m u, is defined as 6.8 H m u = u + u m k k k H m u m = u m u m H m u = u m u. k + + u m, k m

25 E. J. Bird, S.-M. Ngai and A. Teplyaev 5 Let Ψ m k/, k =,,..., m, be a sequence of triangular piecewise linear functions m defined as { m Ψ m x +, x / m x =, otherwise, m x k, k / m x k/ m Ψ m k/ mx = m x + k +, k/ m x k + / m k =,..., m, otherwise, Ψ m mx = { m x m +, / m x, otherwise. Then, solutions of. can also be approximated by using the formula see [K]: k k 6.9 H m u m = λ Ψ m k/ mx dµ u m, k =,,..., m. The left-hand side of 6.9 is determined by 6.8; the right-hand side is determined by the following proposition. Proposition 6.. Let m and for k =,..., m, let J, J be the unique indices such that [ k S J [, ] = m, k ] [ k m and S J [, ] = m, k + ] m. Then a b c Ψ m dµ = p m+. Ψ m k/ m dµ = pm J p m+ m J + p m J + p m m J, k =,..., m. Ψ m dµ = p m+. Proof. We will prove b; the proofs of a and c are similar and easier. Using 6.3 and the fact that Ψ m k/ m is supported on [k / m, k + / m ], we have Ψ m k/ m dµ = J =m p m J p m m J Ψ m k/ ms Jx dµ = p m J p m m J Ψ m k/ ms J x dµ + p m J p m m J Ψ m k/ ms J x dµ = p m J p m m J x Ψ m k/ m m + k m dµ

26 6 Fractal Laplacians on the Unit Interval + p m J p m m J x Ψ m k/ m m + k m dµ = p m J p m m J x dµ + p m J p m m J x + dµ by definition of Ψ m k/ m = p m J p m+ m J + p m J + p m m J by 6.. By combining 6.8 and Proposition 6. we can express 6.9 in a matrix form as 6. M m ũ = λñ m ũ, where M m = M m and, for example, p Ñ = p p p p 3 p p Ñ = p p + p p p p. p 3 We remark that Ñ m is a diagonal matrix of order m +. As in the finite element method, there is a recursive relation governing Ñ m. We justify this in the following Proposition 6.3. For m =,,..., we have Ψ m+ k/ dµ = m+ p Ψ m dµ + p Ψ m dµ, p Ψ m k/ m dµ, if k =,..., m if k = m p Ψ m k m / m dµ, if k = m +,..., m+. Proof. If k =, then it follows easily from Proposition 6. a that Ψ m+ dµ = p m+ = p Ψ m dµ. For k =,..., m, Proposition 6. b gives 6. Ψ m+ k/ dµ = p m m+ J p m+ m J + p m J + p m+ m J, where S J [, ] = [ k m, k m ] and S J [, ] = [ k m, k + m ].

27 E. J. Bird, S.-M. Ngai and A. Teplyaev 7 Since S J [, ] and S J [, ] are on the left-hand side of the unit interval [, ], J, J can be expressed as J =, J and J =, J, where S J [, ] = [ k m, k m ] and S J [, ] = [ k m, k + ] m. Hence 6. becomes Ψ m+ k/ dµ = p m m+ J + m+ m p J + m + p J + m+ m p J + = p p m J p m+ m J + p m J + m m p J = p Ψ m k/ m dµ. With a few modifications, equalities for the other cases can also be established similarly. By using Proposition 6.3, we obtain the following recursive relation for Ñ m : 6. Ñ m+ = p where,, are defined as in Proposition 6.. [ ] [ ] Ñm + p, Ñ m 6.3. Normalization. There are several different ways that we have used to normalize the eigenvectors numerically. We summarize these methods below. N Suppose u n is an eigenvector not yet normalized. Since u n dx = λ n u n dµ, we have cu n λ /. dµ = c = u n dx cu n now is normalized by taking the above value of c. This method has the advantage that integration is with respect to Lebesgue measure. N Note that /. = cu n dµ c = u n dµ Another method we use is to create a list of µ measures on the dyadic intervals and find the approximate normalization factor directly by u n dµ m k= un k + u n k + [ k µ m, k + ] m. N3 Here we proceed as in N but integrate u n dµ exactly using the fact that u n is a piecewise quadratic function. This can be done since u n is piecewise linear and we know the integral of all quadratic functions from the identities in 6..

28 8 Fractal Laplacians on the Unit Interval 6.4. Rapidly decaying eigenfunctions. Figure 3 shows the numerical solutions for the first 8 approximate normalized Neumann eigenfunctions for µ defined by equations.9 and. and p = 3 see the remark following Proposition 5.. It is observed that for this particular value of p, there exists a distinct subsequence of rapidly decaying eigenfunctions u nk where n k satisfies the recursive Fibonacci type relation n =, n =,..., n k+ = n k + n k for k. Alternatively, one can say that these eigenfunctions are strongly localized. The same phenomenon occurs for Dirichlet eigenfunctions we do not include the pictures for this case because of space limitations. Figure 4 shows the first ten Neumann eigenfunctions in this subsequence. Conjectures 6.4. We conjecture that, as the graphs suggest, u nk as k for the normalized eigenfunctions. We also conjecture that max x [ε,] u nk x as k for any ε >, which means a strong localization Error estimates. We finish this section by discussing the error estimates of the finite element method. For any u DomE, and x, y [, ], 6.3 uy ux y = u y / t dt y x / u dt and so for all x, y [, ] we have 6.4 uy ux y x Eu, u x In particular, for each u DomE that has a zero in [, ] including all Dirichlet and Neumann eigenfunctions, we have u Eu, u / u Eu, u /. Moreover, by using the proof of [SU, Lemma 4.6], we have 6.7 Eu, u / µ u. Using equations and the proof of [SU, Theorem 4.8 and Corollary 4.9] with j = see also [GRS], we have Theorem 6.5. Let u be an L µ-normalized Dirichlet or Neumann eigenfunction of. with eigenvalue λ. Then there exists ũ S m such that a Eu ũ, u ũ / λρ m/, b u ũ λ u ρ m, c u ũ λρ m/, where ρ = max{p/, p/}. Moreover, ũ may be taken to be the spline that interpolates u on V m := {k/ m : k =,,..., m }. Proof. Part a follows from the proof of [SU, Theorem 4.8] by using 6.7 and observing that for each multi-index J with J = m, u ũ S J vanishes on the boundary {, }. Part b follows from a and the same proof as [SU, Corollary 4.9]. Part c follows by combining 6.6 and a. x

29 E. J. Bird, S.-M. Ngai and A. Teplyaev Figure 3. Approximate nth Neumann eigenfunctions of equation. for µ defined by equations.9 and. with p = 3, plotted with m =.

30 3 Fractal Laplacians on the Unit Interval Figure 4. The sequence of rapidly decaying Neumann eigenfunctions.

31 E. J. Bird, S.-M. Ngai and A. Teplyaev 3 Acknowledgments. Part of this research was carried out while the second authors was visiting the Department of Mathematics of Cornell University and the Southeast Applied Analysis Center at the School of Mathematics, Georgia Institute of Technology. He would like to thank both institutions for their hospitality and support. The authors are especially grateful to Robert Strichartz for suggesting this project to them and showing them how to use the finite element method to obtain numerical results. The authors also thank Jun Kigami and Jiaxin Hu for some valuable comments and discussions. The research of the first author is supported by the National Science Foundation through the Research Experiences for Undergraduates REU Program. The research of the second author is supported in part by the National Science Foundation, grant DMS The research of the last author is supported in part by the National Science Foundation through a Mathematical Sciences Postdoctoral Fellowship. Résumé substantiel en français. Soit µ une mesure finie continue et positive à support suppµ = [, ]. Typiquement, µ est une mesure auto-similaire. Nous étudions les valeurs propres λ et les fonctions propres u de l équation., c est-à-dire u xv x dx = λ uxvx dµx, pour v C,, l espace des fonctions infiniement différentiables à support dans,. Nous imposons soit la condition de Neumann u = u = à la frontière, soit celle de Dirichlet u = u =. L équation. définie comme distribution un laplacien µ u pour lequel u v dx = µ u v dµ lorsque v C,. Les fonctions propres sont les analogues fractals des fonctions sinus et cosinus classiques de Fourier. Pour une étude complète, nous démontrons aisément, en plus des résultats suivants essentiellement connus voir, par exemple, [A], l existence d une solution unique et lisse de l équation précédente. La n-ième fonction propre de Neumann admet n zéros et sa dérivée n + zéros. La n-ième fonction propre de Dirichlet admet n + zéros et sa dérivée n zéros. Pour une fonction propre quelconque u, les zéros de u et de u alternent entre eux et pareillement pour les zéros de la n-ième fonction propre de Dirichlet et de Neumann. Il existe une base orthonormale complète composée des fonctions propres de Dirichlet Neumann. Les valeurs propres λ n de Dirichlet Neumann sont simples. De plus, on a lim n λ n =. L équation. est équivalente à l équation intégrale de Volterra-Steiltjes voir [A, Chapitre ]: ux = u + u x λ x x yuy dµy, x. Il est connu qu une solution quelconque u est différentiable et que la dérivée satisfait u x = u λ x uy dµy, x.

32 3 Fractal Laplacians on the Unit Interval Voir [A, Théorème..]. L inverse est aussi vrai. De plus, il existe une fonction de Green pour représenter u, qui est essentiellement celle que l on trouve dans [S, K] et donnée par ux = u + u u x + λ gx, yuy dµy, x. où gx, y est telle que définie dans Section. On utilise ces formes équivalentes de l équation pour obtenir des résultats de concavité pour les fonctions propres Propositions 3.3 et 3.4 et d autres résultats sur les propriétés des valeurs propres et des fonctions propres. On étudie la première valeur propre et la première fonction propre lorsque la mesure varie. Soit {µ p : < p < } une famille de mesures de probabilités continues avec supp µ p [, ] et telles que, pour c, fixé quelconque, µp [, c] lorsque p. Dans le cas Neumann, soit z = z p le zéro commun à toutes fonctions propres de Neumann associées à la première valeur propre. Alors z lorsque p Théorème 4.. Soit λ = λ p la première valeure propre de Neumann. Alors λ lorsque p Théorème 4.3. On montre Théorème 4.4 que u p = u p z z = λ u p y dµy = λ z u p y dµ p y lorsque p. Si les fonctions propres sont normalisées, alors u p =. Soit λ p la première valeur propre de Dirichlet. Nous montrons que lim p λ p = Théorème 4.5. Pour la fonction propre de Dirichlet normalisée u p, nous montrons que lim p u p = Théorème 4.6. Nous donnons aussi un exemple qui montre que l analogue du Théorème 4. est faux pour la première fonction propre de Dirichlet Exemple 4.7. Nous étudions aussi le comportement asymptotique des valeurs propres. Nous nous restreignons à une structure p.c.f. auto-similaire définie par le système itératif S x = rx, S x = rx + r, < r <, et on prend pour µ la mesure auto-similaire définie par µ = pµ S + pµ S, < p <. Utilisant un résultat de Kigami et Lapidus [KL], on obtient des bornes asymptotiques pour les valeurs propres définies par cette classe de mesures auto-similaires. Nous faisons la description des solutions numériques de l équation.. Pour estimer les valeurs propres et fonctions propres numériquement, nous restreignons µ davantage au cas d une mesure auto-similaire définie par le système itératif Moyennant l identité fx dµ = p S x = x, S x = x +. f S x dµ + p f S x dµ,