Minimal convex combinations of sequential Laplace-Dirichlet eigenvalues

1/ 21 Minimal convex combinations of sequential Laplace-Dirichlet eigenvalues Braxton Osting 1 and Chiu-Yen Kao 2 1. University of California, Los Angeles 2. Claremont McKenna College October 27, 2012 AMS Fall Western Sectional Meeting Special Session on The Ubiquitous Laplacian: Theory, Applications, and Computations

2/ 21 Extremum problems for Laplace-Dirichlet eigenvalues Consider the general shape optimization problem inf { f (Λ(Ω)) : Ω open set in R d, Ω = 1 } where Λ(Ω) = {λ k (Ω)} k=1 are the Laplace-Dirichlet eigenvalues of Ω, satisfying ψ j (x) = λ j ψ j (x) ψ j (x) = 0 x Ω x Ω, f is a real-valued function on the eigenvalue sequences, and denotes the Lebesgue measure. For an introduction and survey of work on this topic, consult Antoine Henrot s (2006) book.

/ 21 Extremum problems for Laplace-Dirichlet eigenvalues inf { f (Λ(Ω)) : Ω open set in R d, Ω = 1 } Mostly, I ll focus on dimension d = 2 and f = C j α,β is the convex combination of three eigenvalues, C j α,β (Ω) = αλ j(ω) + βλ j+1 (Ω) + (1 α β)λ j+2 (Ω), for j N and (α, β) T := {(α, β) R 2 : α 0, β 0, α + β 1}. Denoting C j, α,β = inf{cj α,β (Ω): Ω A, Ω = 1} where A is an admissible class of domains, we ll view this as an (α, β)-parameterized shape optimization problem and study properties of the minimal values, C j, α,β, and minimizers, ˆΩ j α,β, as a function of (α, β).

/ 21 Outline 1. Spoiler: computational results 2. Previous results 3. Two analytical results 4. The restricted admissible set consisting of disjoint unions of balls 5. Computational methods 6. Revisit computational results Many open problems in this area...

Spoiler: computational results C j α,β = αλ j + βλ j+1 + (1 α β)λ j+2 1 0.875 0.75 j=1 β 0.625 0.5 0.375 0.25 0.125 0 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 α 1 0.875 0.75 j=2 β 0.625 0.5 0.375 0.25 0.125 0 5/ 21 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 α properties: connectedness, symmetry, balls, smooth varying w.r.t (α, β),...

6/ 21 Previous results for single eigenvalues: inf { λ k (Ω): Ω open set in R d, Ω = 1 } If the area of a membrane be given, there must evidently be some form of boundary for which the pitch (of the principal tone) is the gravest possible, and this form can be no other than the circle. Lord Rayleigh, The Theory of Sound (1877) 1923 Faber and Krahn prove that a ball uniquely minimizes λ 1 (Ω) among domains of unit measure, i.e., λ 1 πj 2 0,1. 1926 Krahn proves that two disjoint balls of equal measure uniquely minimizes λ 2 (Ω) among domains of unit measure, i.e., λ 2 2πj 2 0,1. 1994 Wolf+Keller show that any minimizer for λ 3 (Ω) is connected. Show that a ball is a local minimizer among star-shaped, bounded domains with smooth boundary. 2000 Bucur + Henrot prove that the minimizer of λ 3 exists 2011 Bucur and Mazzoleni + Pratelli prove the infimum for λ k, k > 3 exists over quasi-open sets and every minimizer has finite perimeter? Conjectured that the minimizer of λ 4 is the disjoint union of two balls with radii which have the ratio j 0,1 /j 1,1.

Computational results for single eigenvalues TITLE WILL BE SET BY THE PUBLISHER Oudet (2004) Antunes + Freitas (2012) No Optimal union of discs Computed shapes i Ω multiplicity λ i Oudet s result 3 4 46.125 46.125 64.293 64.293 5 2 78.20 78.47 6 3 88.52 88.96 5 82.462 78.47 7 3 106.14 107.47 6 92.250 88.96 8 3 118.90 119.9 7 110.42 107.47 9 3 132.68 133.52 8 127.88 119.9 10 4 142.72 143.45 9 138.37 133.52 11 4 159.39-10 154.62 143.45 12 4 172.85-13 4 186.97 - Fig. 5. Best-known shapes The level set method is used to represent the domains Relaxed formulation used to compute eigenvalues Fig. 6. λ1 (left) and λ2 (right) The k-th eigenvalue of the minimizer is multiple [6] M. G. Crandall and P. L. Lions, Viscosity Solutions of Hamilton-Jacobi Equations, Tran. AMS 277 (1983), 1-43. [7] G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitz. Ber. Bayer. Akad. Wiss. (1923), 169-172. 14 4 198.96-15 5 209.63 - Eigenvalues computed via meshless method Table 2: Dirichlet minimizers with the optimal values for λ i and the corresponding multiplicity. Domains parameterized using Fourier coefficients do this, we find that the results obtained do not differ in a significant way and, in particular, the numerical for λ13 remains without any symmetries. k = 13 minimizer is not symmetric 6 Symmetries, multiplicities and TRIANGULAR domains [8] S. Finzi Vita, Constrained shape optimization for Dirichlets problems : Discretization via relaxation, Adv. in Math. Sci. and Appl. 9 (1999), 581-596. An analysis of the optimizers obtained suggests several remarks and directions for future study, both numer [9] A. Henrot, Minimization problems of eigenvalues of the laplacian, to appear in Journal of Evol. Eq. 7/ 21

Previous results for the range of the first two eigenvalues, (λ 1 (Ω), λ 2 (Ω)) Wolf + Keller (1994), Antunes + Henrot (2010) RANGE OF DIRICHLET AND NEUMANN EIGENVALUES 3 Boundary consists of two rays: 60 Λ2 j1,12 j0,1 2 Λ1 {(λ 1, λ 2 ): λ 2 = λ 1 and λ 1 πj 2 0,1} 50 ε D {(λ 1, λ 2 ): λ 2 = j2 1,1 j 2 λ 1 and λ 2 2πj 2 0,1} 0,1 40 Λ2 Λ1 30 15 20 25 30 35 40 45 50 (given by Ashbaugh-Benguria, Faber-Krahn, and Krahn inequalities) and a curve connecting their endpoints Figure 1. The region E D. Curve determined by minimizers of convex comb. Cα,1 α 1 n eigenvalues of convex domains. We prove, in particular, that the conain which maximizes the second (non trivial) Neumann eigenvalue is not a. 2. The Dirichlet case with convex domains is section we are interested in the subset of E D obtained for convex sets, = { (x, y) R 2 : (x, y) = (λ 1(Ω), λ 2(Ω)), Ω R 2, Ω = 1, Ω convex }. tart by some numerical results obtained for particular classes of convex s. e/ 21 case of convex polygons. In this section we present some numerical, α [0, 1] Observe that for α > 0, the minimizer is connected, while it is known (Krahn) that for α = 0, the minimizer is the disjoint union of two equal-measure balls. = topological change as α 0 The minimizer for λ 3 (Ω) is connected and the ball is a local minimum The minimum of λ k (Ω) for k = 1 : 17 is found when Ω is restricted to be a union of disjoint circular discs or a union of disjoint rectangles

9/ 21 Some results for C j α,β, j = 1 Theorem. Iversen + Pratelli (2012) The following are properties of the minimizer of Cα,β 1 (Ω) (a) Any minimizer is connected for each of the following cases: (i) α + β = 1, α > 0, (ii) 0.35 α 1 (ii) 0 β 0.725(1 α) b 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 a (b) Any disconnected minimizer, Ω, satisfies λ 1 (Ω) = λ 2 (Ω) and has exactly two components. Furthermore, the disjoint union of two balls can be optimal only if β = 1. (c) If any minimizer is connected for α = 0 and each β [0, 1), then any minimizer is connected unless β = 1. Iversen + Pratelli conjecture that the minimizer is connected unless β = 1.

10/ 21 Review: parameterized objective functions, correspondences, and hemicontinuity { } C j, α,β := inf Ω A Cj α,β (Ω) and ˆΩj α,β := Ω A: C j α,β (Ω) = Cj, α,β We view C α,β as an (α, β)-parameterized optimization problem and consider (α, β) C j, α,β (α, β) ˆΩ j α,β Note that ˆΩ j α,β is a set-valued function (a.k.a. correspondence). Recall: A set-valued function Γ: A B is upper hemicontinuous at a point a A if for all sequences {a n } n such that a n a and all sequences {b n } n such that b n Γ(a n ), there exists a b Γ(a) such that b n b. Note: In a sequentially compact space, uhc closed graph

1/ 21 Minimal convex combinations of sequential eigenvalues The minimum values and minimizers inherit some continuity properties from the objective function. Theorem. O. + Kao (2012) Consider the (α, β)-parameterized shape optimization problem { } C j, α,β := inf Ω A Cj α,β (Ω) and ˆΩj α,β := Ω A: C j α,β (Ω) = Cj, α,β C j α,β (Ω) := αλ j(ω) + βλ j+1 (Ω) + (1 α β)λ j+2 (Ω) A := {Ω: Ω quasi-open and Ω 1}. For each k N the following statements hold: 1. For each (α, β) T, C j, α,β exists and ˆΩ j α,β is a non-empty and closed set. Furthermore, every Ω ˆΩ j α,β has finite perimeter. 2. The optimal value, C j, α,β, is a non-increasing, Lipschitz continuous, and concave function in both α and β. 3. As a set-valued function of (α, β), ˆΩ j α,β is upper hemicontinuous.

2/ 21 Sketch of proof. 1. Since C j α,β (Ω) is a non-decreasing and Lipschitz continuous function of the Laplace-Dirichlet eigenvalues, the recent results of Bucur (2011) and Mazzoleni + Pratelli (2011) show that the infimum exists and that every minimizer has finite perimeter. 2. I ll show that C j, α,β is a non-increasing function in α. The other proofs are similar. Fix β [0, 1] and let 0 α 1 < α 2 1 such that β + α 2 1. Let Ω α1,β ˆΩ j α 1,β and assume Cj, α < 1,β Cj, α. Then 2,β C j, α 2,β > Cj α 1,β (Ω α 1,β) C j α 2,β (Ω α 1,β), since C j α,β is non-increasing in α. But this contradicts the optimality of C j, α 2,β. Thus, Cj, α,β is non-increasing in α.

13/ 21 Sketch of proof. 3. Let M > 0 be such that diam(ω) M for all Ω ˆΩ j α,β, (α, β) T. Let (α n, β n ) (α, β) and Ω n ˆΩ j α n,β n be sequences. Since the diameter of Ω n is uniformly bounded, there exists a domain Ω A and a subsequence (which we denote by Ω n ) such that Ω n Ω in the sense of weak γ-convergence [see Bucur + Buttazzo (2005)]. Suppose Ω / ˆΩ j α,β, i.e., there exists Ω such that C j α,β (Ω ) < C j α,β (Ω). Using the continuity of C j α,β (Ω) in (α, β) and Ω, we compute lim n Cj α n,β n (Ω ) = C j α,β (Ω ) < C j α,β (Ω) = lim n Cj α n,β n (Ω n ). For sufficiently large n, this implies C j α n,β n (Ω ) < C j α n,β n (Ω n ), contradicting the optimality of Ω n. Thus, ˆΩ j α,β is upper hemicontinuous in (α, β).

14/ 21 For j = 1 and some (α, β), the ball is a local minimizer C 1 α,β = αλ 1 + βλ 2 + (1 α β)λ 3 Theorem. O. + Kao (2012) For for the (α, β)-set {(α, β) T : α + 2β 1}, the ball is a local minimizer of Cα,β 1 over the admissible class of equal measure, star-shaped, bounded domains with smooth boundary. Proof. Generalization of the Wolf+Keller (1994) proof that the ball is a local minimum of λ 3 (Ω). Based on asymptotic formulas for Ω ɛ λ k (Ω ɛ ) where Ω ɛ is a nearly-circular domain.

15/ 21 Minimum over two balls: inf{c j α,β (Ω): Ω B B}, j = 1, 2, (α, β) T Let Ω = B(r) B( π 1 r 2 ) where r 2 [ 0, (2π) 1] j=1 j=2

16/ 21 Computational methods Domain parameterization. Define a domain Ω to be F N representable if Ω = {(r, θ) : r R N (θ), θ [0, 2π]}, where N R N (θ) = a k e ıkθ and a k = a k. k= N Define the set of all F N representable domains by F N = {Ω A: Ω is F N representable}. We use N = 10. Note: F is the class of star-shaped, bounded domains with smooth boundary. Eigenvalue computation. Eigenvalues are computed using the Matlab toolbox mpspack 1. The weighted-neumann-to-dirichlet scaling method is chosen with the argument ntd and M = 100 quadrature points are used. 1 http://code.google.com/p/mpspack/

17/ 21 Computational methods Optimization. Use line-search-based BFGS algorithm implemented in HANSO 2. This quasi-newton method has proven to be effective for non-smooth optimization problems such as this. If λ j is simple, the derivatives of λ j (Ω) with respect to the coefficients a k describing Ω are λ j a k = 2π 0 R N (θ)e ıkθ ψ j (R N (θ), θ) 2 dθ. In our computations, the Neumann data u j is evaluated at the quadrature points and the above integral is evaluated via quadrature. Note that while the derivative of an eigenvalue with higher multiplicity can be computed, in numerical computations, roundoff error causes all eigenvalues to be simple. 2 http://www.cs.nyu.edu/faculty/overton/software/hanso/index.html

Computational Results: Minimal convex combinations of two sequential eigenvalues MINIMAL CONVEX COMBINATIONS OF SEQUENTIAL LAPLACE-DIRICHLET EIGENVALUES 17 40 50 C j, 1 β,β := min (1 β)λ j(ω)+βλ j+1 (Ω) Ω =1 35 45 * C 1,γ * C 3,γ 30 25 20 15 65 60 55 50 45 0 0.2 0.4 0.6 0.8 1 γ 0 0.2 0.4 0.6 0.8 1 γ 95 * C 2,γ * C 4,γ 40 35 30 85 80 75 70 65 60 0 0.2 0.4 0.6 0.8 1 γ 0 0.2 0.4 0.6 0.8 1 γ For some values of j, the number of connected components of the minimizer varies with the convex combination parameter, β. For j = 2 : 5, the optimizer for β = 1, is also the optimizer on an interval β [1 δ, 1] for some δ > 0 and C j, 1 β,β is constant. = λ j (Ω j, 0,1 ) = λ j+1(ω j, 0,1 ). * C 5,γ 90 85 80 75 70 0 0.2 0.4 0.6 0.8 1 γ is non-decreasing, Lipschitz continuous, and concave. Minimizer smoothly varies except at isolated points. C j, 1 β,β 18/ 21 Figure 5.4. Plots of Ck,γ (blue asterisks) and the minimizers Ω k,γ and γ =0:.1 :1.See 5andTable3. for k =1:5

19/ 21 Computational results: min{c j α,β (Ω): Ω F 10 F 10 }, j = 1, 2, (α, β) T 1 0.875 0.75 j=1 β 0.625 0.5 0.375 0.25 0.125 0 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 α 1 0.875 0.75 j=2 β 0.625 0.5 0.375 0.25 0.125 0 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 α

20/ 21 For j = 2, zoom in on region near (α, β) = (0, 0) where the minimizer is disconnected 1 0.875 0.75 0.625 β 0.5 0.375 0.25 0.125 0 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 α 0.5 0.4 0.3 β 0.2 0.1 0 0 0.02 0.04 0.06 0.08 0.1 α

Discussion Shape optimization problem: Minimize convex combinations of sequential Laplace-Dirichlet eigenvalues as a function of the domain. Showed that as a function of (α, β), the minimum value and minimizer inherit some continuity properties from the objective function C j α,β. Showed that for j = 1 and some values (α, β), the ball is a local min. Studied properties of the minimizers computationally, including uniqueness, connectivity, symmetry, and eigenvalue multiplicity. Many open questions remain... References: B. Osting and C.-Y. Kao. Minimial Convex Combinations of Sequential Laplace-Dirichlet Eigenvalues, submitted (2012). B. Osting and C.-Y. Kao. Minimial Convex Combinations of Three Sequential Laplace-Dirichlet Eigenvalues, in preparation (2012). Thanks! Questions? 1/ 21 Email: braxton@math.ucla.edu