We study various lower and upper estimates for the first eigenvalue of Dirichlet Laplacians defined by positive Borel measures on bounded open subsets of Euclidean spaces. These Laplacians and the corresponding eigenvalue estimates differ from classical ones in that the defining measures can be singular. By using properties of self-similar measures, such as Strichartz\u27s second-order self-similar identities, we improve some of the eigenvalue estimates

Ngai, Sze-Man

English

Georgia Southern University: Digital Commons@Georgia Southern

Georgia Southern UniversityDigital Commons@Georgia SouthernMathematical Sciences Faculty Presentations Mathematical Sciences, Department of6-2014Eigenvalue Estimates of Laplacians Defined byFractal MeasuresSze-Man NgaiGeorgia Southern University, smngai@georgiasouthern.eduFollow this and additional works at: https://digitalcommons.georgiasouthern.edu/math-sci-facpresPart of the Mathematics CommonsThis presentation is brought to you for free and open access by the Mathematical Sciences, Department of at Digital Commons@Georgia Southern. Ithas been accepted for inclusion in Mathematical Sciences Faculty Presentations by an authorized administrator of Digital Commons@GeorgiaSouthern. For more information, please contact digitalcommons@georgiasouthern.edu.Recommended CitationNgai, Sze-Man. 2014. "Eigenvalue Estimates of Laplacians Defined by Fractal Measures." Mathematical Sciences Faculty Presentations.Presentation 5. source: http://www.math.cornell.edu/~fractals/5/ngai.pdfhttps://digitalcommons.georgiasouthern.edu/math-sci-facpres/5IntroductionStatement of main resultsProofs of main resultsEigenvalue Estimates of Laplacians defined byfractal measuresSze-Man NgaiGeorgia Southern University and Hunan Normal University5th Conference on Analysis, Probability and MathematicalPhysics on FractalsCornell University, Ithaca, NY, USAJune 11–15, 2014Joint with Da-Wen Deng (=Tai-Man Tang)IntroductionStatement of main resultsProofs of main resultsDefinition of Dirichlet Laplacian: weak formulationAssumptions throughout this talk:(1) Ω ⊆ Rn bounded, open, connected.(2) µ = regular Borel probability measure on Rn, supp(µ) ⊆ Ω,µ(Ω) > 0.Poincare´ type inequality:(PI) ∃ constant C > 0 s.t.∫Ω|u|2 dµ ≤ C∫Ω|∇u|2 dx ∀u ∈ C∞c (Ω). (1.1)IntroductionStatement of main resultsProofs of main resultsElements in the same equivalence class of H10 (Ω) can belong todifferent equivalence classes of L2(Ω, µ). (PI) ⇒ each equivalenceclass of u ∈ H10 (Ω) contains a unique u¯ ∈ L2(Ω, µ) satisfying (1.1).Define ι : H10 (Ω)→ L2(Ω, µ), ι(u) = u¯. Let N := ker ι. Thenι : N⊥ →֒ L2(Ω, µ).Define a nonnegative bilinear form E(·, ·) on L2(Ω, µ) byE(u, v) :=∫Ω∇u · ∇v dx ,with Dom(E) = N⊥ (more precisely ι(N⊥)).IntroductionStatement of main resultsProofs of main resultsFacts: Assume (PI).(a) Then E is closed. Denote the corresponding self-adjointoperator by: ∆µ, Dirichlet Laplacian with respect to µ.(b) Let u ∈ H10 (Ω) and f ∈ L2(Ω, µ). TFAE:(i) u ∈ Dom(∆µ) and ∆µu = f ;(ii) ∆u = f dµ as distributions.∆µ(u) = utt models a (nonhomogeneous) vibrating string ormembrane with mass distribution µ.IntroductionStatement of main resultsProofs of main resultsA sufficient condition for (PI)Definition: The lower L∞-dimension of µ:dim∞(µ) = limδ→0+ln(supx µ(Bδ(x)))ln δ.where the supremum is taken over all x ∈ supp(µ).IntroductionStatement of main resultsProofs of main resultsBased mainly on a result of Maz’ja, we obtained:Theorem(Hu-Lau-N., 06) Assume that dim∞(µ) > d − 2.(a) (PI) holds.(b) ∃ an orthonormal basis {un}∞n=1 of L2(Ω, µ) consisting ofeigenfunctions of −∆µ.(c) The eigenvalues of {λn}∞n=1 satisfy0 < λ1 ≤ λ2 ≤ · · · with limn→∞λn =∞.IntroductionStatement of main resultsProofs of main resultsMotivations:1) Estimate the Poincare´ constant = 1λµ1.2) Study existence of spectral gaps.3) Study of bounds for eigenfunctions.IntroductionStatement of main resultsProofs of main resultsClassical results:• Faber-Krahn, 23, 25: Let Ω ⊂ Rn and Vol(Ω) = Vol(Br (0)).λ1(Ω) ≥ λ1(Br (0)).Not true for manifolds!Figure: E. Calabi: dumbbell M homeomorphic to S2, λ1 → 0 asradius of connecting pipe (of fixed length) → 0.• Cheeger, 70: M be a compact Riemannian manifold.hD(M) := inf{Vol(∂U)Vol(U): U ⊂⊂ M}. Thenλ1 ≥ 14h2D(M).IntroductionStatement of main resultsProofs of main resultsTheorem 1. Let Ω = (0, 1). Then λµ1 ≥ π.We improve this theorem for the infinite Bernoulli convolutions:S1(x) = rx , S2(x) = rx + 1− r , 0 < r < 1,Let µ = µr ,p be the associated self-similar measure:µ = pµ ◦ S−11 + (1− p)µ ◦ S−12 , 0 < p < 1.suppµ ⊆ [0, 1].IntroductionStatement of main resultsProofs of main resultsProposition 2. Let µ be an infinite Bernoulli convolution on [0, 1]with p = 1/2. Then λµ1 > π. In fact,(a) for r ∈ (0, 1/2] (µ is a Cantor-type or Lebesgue), λµ1 ≥ 4/r ;(b) for r ∈ (1/2, 2/3], λµ1 ≥ 24/7 ≈ 3.428;(c) for r ∈ (2/3, 1), λµ1 ≥ 3.2.IntroductionStatement of main resultsProofs of main resultsDefinitionµ on Ω ⊆ Rn is upper s-regular if ∃c1 > 0 s.t.µ(F ) ≤ c1|F |s for all µ measurable subsets F ⊆ Ω.Theorem 3. Let Ω ⊂ R2. Suppose that µ is upper s-regular forsome s ≥ 1. Let β = c1|Ω|s−1/2 with c1 being the constant above.Thenλµ1 ≥√λ12β.where λ1 is the first eigenvalue of −∆.IntroductionStatement of main resultsProofs of main resultsΩ ⊂ Rn and µ absolutely continuous. By using Cheeger’sargument:Theorem 4. Let Ω ⊆ Rn, µ be absolutely continuous w.r.t.Lebesgue measure with a bounded density ρ, andβ := infU⊆Ω1µ(U)∫∂Uρ dHn−1.Then λµ1 ≥ β2/(4‖ρ‖L∞(dx)).IntroductionStatement of main resultsProofs of main resultsUpper estimatesIt is not clear how to obtain upper estimates in general.We can obtain upper estimates for infinite Bernoulli convolution1) Cantor type r < 1/2, and2) golden ratio (√5− 1)/2.• Erdo˝s, 39: µ is singular (also true for other Pisot numbers).• Lq-spectrum, multifractal formalism, and dimension (Lau-N.,98, 99, Feng 05).• Spectral dimension (N., 11).• Wave equation (Chan-Teplyaev-N., to appear).IntroductionStatement of main resultsProofs of main results0 0.2 0.4 0.6 0.8 10246810x 104Figure: First 100 Dirichlet eigenvalues for the Bernoulli convolutionassociated with the golden ratio. (Chen-N., 10)IntroductionStatement of main resultsProofs of main results0 1S1 AAUS2Second-order identities by Strichartz-Taylor-Zhang, 95:T1(x) = S1S1(x) = r2x ,T2(x) = S1S2S2(x) = S2S1S1(x) = r3x + r2,T3(x) = S2S2(x) = r2x + r .0 3T1 AAUT3?T21 20 3IntroductionStatement of main resultsProofs of main resultsFor any Borel subset A ⊆ [0, 1], µ(T1TjA)µ(T2TjA)µ(T3TjA) = Mi µ(T1A)µ(T2A)µ(T3A) , j = 1, 2, 3,where M1,M2,M3 are, respectively,p2 0 0(1− p)p2 (1− p)p 00 1− p 0,0 p2 00 (1− p)p 00 (1− p)2 0,0 p 00 (1− p)p (1− p)2p0 0 (1− p)2.IntroductionStatement of main resultsProofs of main resultsLet J = j1 · · · jm, ji ∈ {1, 2, 3}. Thenµ(TJA) = cJ µ(T1A)µ(T2A)µ(T3A) , where cJ = ej1Mj2 · · ·Mjm = (c1J , c2J , c3J ).Definew∗J := µ(TJ [0, 1]) =11− p + p2 cJ p2p(1− p)(1− p)2 .IntroductionStatement of main resultsProofs of main resultsTheorem 5. Let µ = µr ,p be an infinite Bernoulli convolution andfm be a piecewise linear approximate first eigenfunction obtainedby the finite element method.(a) If 0 < r ≤ 1/2, thenλµr,p1 ≤∑J∈Jm(fm(TJ(1))− fm(TJ(0)))2/(TJ(1)− TJ(0))∑J∈JmwJ min{fm(TJ(1))2, fm(TJ(0))2} ,for all m ≥ 1. For the standard Cantor measure, we have12 ≤ λµ1/3,1/21 ≤ 14.3865.(c.f.: numerical approximation by FEM with m = 7 is14.4353 . . . )IntroductionStatement of main resultsProofs of main results(b) If r = (√5− 1)/2, the reciprocal of the golden ratio, thenλµr,p1 ≤∑J∈Jm(fm(TJ(1)) − fm(TJ(0)))2/(TJ (1)− TJ(0))∑J∈Jmw∗J min{fm(TJ(1))2, fm(TJ(0))2}for all m ≥ 1. In particular, if p = 1/2, we have6.33437 ≤ λµr,1/21 ≤ 8.05171.(c.f.: numerical approximation by FEM with m = 7 is8.03475 . . . )IntroductionStatement of main resultsProofs of main resultsPreparations for proofs.DefinitionLet Ω ⊆ Rn be open (not necessarily bounded), ∅ 6= K ⊂ Ωcompact. Define the 1-capacity of K relative to Ω:Cap1(K ,Ω) := inf{∫Ω|∇ψ| dx : ψ ∈ C∞c (Ω), ψ ≥ 1 on K}.Examples:1) K ⊂ R compact ⇔ Cap1(K ,R) = 2.2) Cap1(K ,Rn) = 0 ⇔ Hn−1(K ) = 0.IntroductionStatement of main resultsProofs of main resultsProposition(Maz’ja, Sobolev spaces) Let Ω ⊆ Rn. If ∃ a constant β > 0 s.t. ∀compact F ⊂ Ω,µ(F ) ≤ β Cap1(F ,Ω),then ∫Ω|u| dµ ≤ β∫Ω|∇u| dx ,∀u ∈ C∞c (Ω),Bound of µ-measure by 1-capacity −→ 1-Poincare´ inequality.We also need Rayleigh’s formula:λµ1 = inff ∈DomE∫Ω(f′)2 dx∫Ω f2 dµ.IntroductionStatement of main resultsProofs of main resultsProof of Theorem 1:Step 1. By Rayleigh’s formula for λ1,λ1 ≤∫ 10 (f′)2 dx∫ 10 f2 dx, λµ1 =∫ 10 (f′)2 dx∫ 10 f2 dµ.Henceλµ1λ1≥∫ 10 f2 dx∫ 10 f2 dµ. (3.2)IntroductionStatement of main resultsProofs of main resultsStep 2. For F ⊂ Ω, µ(F ) ≤ 1 and Cap1(F ,R) = 2 impliesµ(F ) ≤ Cap1(F ,R)/2.Thus by the Proposition, For all u ∈ H10 (Ω),∫ 10|u| dµ ≤ 12∫ 10|u′| dx ∀u ∈ C∞c (Ω). (3.3)For u = [u¯] ∈ H10 (Ω), take {un} ∈ C∞c (Ω) that converges to u¯simultaneously in H10 (Ω) and L2(Ω, µ). Then µ(Ω) <∞ impliesthat un → u¯ in L1(Ω, µ) and ∇un → ∇u¯ in L1(Ω, dx). Takinglimit.IntroductionStatement of main resultsProofs of main resultsStep 3. (Cheeger’s argument: Convert L1-estimate toL2-estimate.) Taking u = f 2, we get2∫ 10f 2 dµ ≤∫ 102|f ||f ′| dx ≤ 2( ∫ 10f 2 dx)1/2( ∫ 10(f ′)2 dx)1/2.Henceλµ1∫ 10f 2 dµ =∫ 10(f ′)2 dx ≥ (∫ 10 f2 dµ)2∫ 10 f2 dx,implyingλµ1 ≥∫ 10 f2 dµ∫ 10 f2 dx. (3.4)Combining (3.2) and (3.4) gives (λµ1 )2 ≥ λ1 = π2.IntroductionStatement of main resultsProofs of main resultsProve of Theorem 3 is similar.Step 1: Use Rayleigh formula for λ1.Step 2: Prove that µ(F ) ≤ βCap1(F ; Ω) (needs upper s-regularity)and thus by Proposition∫Ω|u| dµ ≤ β∫Ω|∇u| dx .Step 3: Cheeger’s argument.IntroductionStatement of main resultsProofs of main resultsStep 2: Claim: µ upper s-regular s ≥ 1 (i.e., µ(F ) ≤ c1|F |s).Then for all compact F ⊂ Ω,µ(F ) ≤ βCap1(F ,Ω) ∀ compact F ⊂ Ω,where β = c1|Ω|s−1/2.Reason: Suppose F ⊂ Ω compact and connected. F a singleton:then µ(F ) = 0 and conclusion holds trivially for any β. F not asingleton: its connectedness implies that Cap1(F ; Ω) > 0. Let ballB|F | contain F . Then |F |s = π−s/2L2(B|F |)s/2. Notice:Cap1(F ; Ω) ≥ Cap1(F ;R2) = Cap1(co(F );R2) ≥ 2|F |.IntroductionStatement of main resultsProofs of main resultsHence|F |sCap1(F ; Ω)≤ π−s/2(L2(B|F |))s/22|F | =|F |s−12≤ |Ω|s−12.Together with upper s-regularity, we getµ(F )Cap1(F ; Ω)≤ C |Ω|s−12.For general compact sets, approximate from outside by the unionof countably many compact connected sets.IntroductionStatement of main resultsProofs of main resultsMain ideas in the proof of Proposition 2: (Use some results inBird-Teplyaev-N. 2003)µ =12µ ◦ S−11 +12µ ◦ S−12 .Let u be an λµ1 -eigenfunction corresponding to with u′(0) = 1.Known: u and u′ are continuous andu′(x) = u′(0) − λµ1∫ x0u(y) dµ(y).Also, u′(1/2) = 0 by symmetry.IntroductionStatement of main resultsProofs of main resultsHenceλµ1 =( ∫ 1/20u(y) dµ(y))−1=(∫ 1−r0u(y) dµ(y)+∫ 1/21−ru(y) dµ(y))−11) Use concavity of u and u′(0) = 1 to get u(y) ≤ y ∀y ∈ [0, 1].2) Fact:∫ 10 y dµ(y) = 1/2.3) Self-similar identity.IntroductionStatement of main resultsProofs of main resultsProof of Theorem 4. Step 1. Suppose ∃β > 0 s.t.β∫Ω|φ| dµ ≤∫Ω|∇φ| dµ ∀ φ ∈ C∞c (Ω) (hence H10 ). (3.5)Then λµ1 ≥β24‖ρ‖∞ .Take φ = f 2, f a λµ1 -eigenfunction. Assume φ ∈ H01(Ω);otherwise, take truncations of f and then take limit.β∫Ωf 2 dµ ≤ 2∫Ω|f ||∇f | dµ ≤ 2‖f ‖L2(µ)‖∇f ‖L2(µ),IntroductionStatement of main resultsProofs of main resultswhich implies thatβ24∫Ωf 2 dµ ≤∫Ω|∇f |2ρ dx ≤ ‖ρ‖∞∫Ω|∇f |2 dx .Thus,β24‖ρ‖∞∫Ωf 2 dµ ≤∫Ω|∇f |2 dx .Notice that λµ1∫Ω f2 dµ =∫Ω |∇f |2 dx .IntroductionStatement of main resultsProofs of main resultsStep 2. Proof of (3.5). Using the coarea formula,Let ψ be a nonnegative Borel measurable function on Ω and letu ∈ C 0,1(Ω). Then∫Ωψ(x)|∇u(x)| dx =∫ ∞−∞( ∫{|ψ|=t}ψ(x) dHn−1(x))dt.In its simplest form ψ ≡ 1, n = 2, u = “2-dim tent function”, theformula says the area of a disk can be evaluated by integrating thecircumferences of circles making up the disk.IntroductionStatement of main resultsProofs of main results∫Ω|∇φ| dµ =∫Ω|∇φ|ρ dx =∫ ∞0( ∫|φ|=tρ dHn−1)dt=∫ ∞0( 1µ{|φ| ≥ t}∫{|φ|=t}ρ dHn−1)µ{|φ| ≥ t} dt≥∫ ∞0( 1µ{|φ| ≥ t}∫∂{|φ|≥t}ρ dHn−1)µ{|φ| ≥ t} dt≥(infU⊆Ω1µ(U)∫∂Uρ dHn−1)∫ ∞0µ{|φ| ≥ t} dt= β∫Ω|φ| dµTheorem 4 proved.IntroductionStatement of main resultsProofs of main resultsProof of Theorem 5.• We choose any positive numerical first eigenfunction fm.• fm is piecewise linear and∫ 10 (f′m)2 dx can be evaluated exactly.• For ∫ 10 f 2m dµ, multiply the measure of an interval between twoconsecutive nodes by the minimum value of fm on thatinterval to get a lower bound s(fm).• Use Rayleigh’s formulaλµ1 = minu∈Dom(E)∫ 10 (u′)2 dx∫ 10 u2 dµ≤∫ 10 (f′m)2 dx∫ 10 f2m dµ≤∫ 10 (f′m)2 dxs(fm).IntroductionStatement of main resultsProofs of main resultsProblems for further study:1) Other eigenvalues λµn , n ≥ 2.2) Spectral gap conjecture.3) Relationship between eigenvalues and geometry of the set andmeasure-theoretic properties of µ.IntroductionStatement of main resultsProofs of main resultsThank you!

Eigenvalue Estimates of Laplacians Defined by Fractal Measures

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Eigenvalue Estimates of Laplacians Defined by Fractal Measures

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