Approximation of the second eigenvalue of the p -Laplace operator in symmetric domains
AAPPROXIMATION OF THE SECOND EIGENVALUEOF THE p -LAPLACE OPERATOR IN SYMMETRICDOMAINS FARID BOZORGNIA
Abstract.
A new idea to approximate the second eigenfunctionand the second eigenvalue of p -Laplace operator is given. In thecase of Dirichlet boundary condition, the scheme has the restrictionthat the positive and the negative part of the second eigenfunctionhave equal L p -norm, however in the case of Neumann boundarycondition, our algorithm has not such restriction. Our algorithmgenerates a descending sequence of positive numbers that convergesto the second eigenvalue. We give various examples and computa-tional tests. Keywords:
Nonlinear Eigenvalue, p-Laplace, Numerical methods.
Introduction
The p -Laplace operator is a homogeneous nonlinear operator whicharises frequently in various applications in physics, mechanics, and im-age processing. Derivation of the p -Laplace operator from a nonlinearDarcy law and the continuity equation has been described in [5]. Theeigenvalues and the corresponding eigenfunctions of the p -Laplace op-erator have been much discussed in the literature, due to mathemat-ical challenges, open questions cf. [3, 14, 24, 25], and regarding relatedapplications in image processing we refer to [15, 17]. In particular,the second eigenvalue and eigenfunction have been studied extensively,see [1, 12, 27].For dimension higher than one, except for the case p = 2, the struc-ture of higher eigenfunctions are not well understood. For the one-dimensional case see [13, 16, 26]. In this work , we are interested innumerical approximation of the second eigenvalue λ and the corre-sponding eigenfunction of the p -Laplace operator. We treat the secondeigenvalue as an optimal bi-partition of the first eigenvalue. There arevarious problems in mathematical physics and probability theory re-lated to optimal partitioning of the first eigenvalue, see [4, 6, 7]. In [18] Date : February 24, 2020. a r X i v : . [ m a t h . SP ] F e b FARID BOZORGNIA investigated Constrained Descent Method and the Constrained Moun-tain Pass Algorithm to approximate the two smallest eigenvalue for1 . ≤ p ≤ . The structure of this paper is as follows. In Section 2, we reviewmathematical background and characterization of the second eigen-value. Next in Section 3, we present our numerical approximationalong the proof of convergence. In Section 4, as an application of ouralgorithm, we study the spectral clustering. Last section deals with thenumerical implementation.2.
Mathematical Background
In this section, we briefly review some known results about the firstand second eigenfunction of p -Laplace operator with zero Dirichletboundary condition in bounded domain.For 1 ≤ p < ∞ the first eigenvalue of the p -Laplace operator in W ,p (Ω) , denoted by λ ,p (Ω) is given by(2.1) λ ,p (Ω) := min u ∈ W ,p (Ω) u (cid:54) =0 (cid:82) Ω |∇ u ( x ) | p dx (cid:82) Ω | u ( x ) | p dx . For every 1 < p < ∞ , the first eigenvalue is simple and isolated andthe first eigenfunction doesn’t change the sign. The correspondingminimizer satisfies the Euler-Lagrange equation(2.2) (cid:26) − ∆ p u = λ | u | p − u in Ω ,u = 0 on ∂ Ω . Here ∆ p u = div( |∇ u | p − ∇ u ) which for p = 2 , we have Laplace operator. Definition 2.1.
A non zero function u ∈ W ,p (Ω) ∩ C (Ω) , is called a p -eigenfunction in the weak sense if there exist a λ ∈ R such that(2.3) (cid:90) Ω |∇ u | p − ∇ u · ∇ φ dx = λ (cid:90) Ω | u | p − u φ dx, ∀ φ ∈ W ,p (Ω) . The associated number λ is called a p -eigenvalue. It is not, however,known whether every such quantity is a ”variational eigenvalue” likefor the case p = 2 . In [1] Anane and Tsouli gave a characterization ofthe variational eigenvalues of Problem (2.2) by the following minimaxprinciple. To do this, first define Krasnoselskii genus of a set A ⊆ W ,p (Ω) by γ ( A ) = min { k ∈ N : there exist f : A → R k \ , f continuous and odd } . For k ∈ N defineΓ k := { A ⊆ W ,p (Ω) , symmetric, compact and γ ( A ) ≥ k } . HE SECOND EIGENVALUE OF p -LAPLACE OPERATOR 3 Then the eigenvalues of the p -Laplace are(2.4) λ k,p (Ω) = min A ∈ Γ k sup u ∈ A (cid:82) Ω |∇ u ( x ) | p dx (cid:82) Ω | u ( x ) | p dx , which satisfying 0 < λ < λ ≤ · · · ≤ λ k → ∞ , as k tends to infinity, see [1, 22, 24, 25].It is shown by Anane and Tsouli that λ defined by (2.4) is essen-tially the second eigenvalue of the Dirichlet p -Laplace, means that theeigenvalue problem (2.2) has no other eigenvalue between λ and λ .In the one dimensional case, Ω = ( a, b ) ⊂ R , it is known that alleigenvalues are simple and the eigenfunction corresponding to λ n hasexactly n + 1 zeros, counting the ending boundary points a, b . Theeigenvalues can be computed explicitly by variational formula and thecorresponding eigenfunctions are obtained in terms of the Gaussianhypergeometric function, see [16, 26]).Here, we consider another characterization of the second eigenvaluewhich we use for our numerical simulation. Definition 2.2.
Given a bounded open set Ω ⊂ R d , a class bi-partitionof Ω (or decomposition) is a family of pairwise disjoint, open and con-nected subsets { Ω , Ω } such thatΩ , Ω ⊆ Ω , Ω ∩ Ω = ∅ , Ω = Ω ∪ Ω . By D we mean the set of all bi-partition of Ω.For any arbitrary partition D = (Ω , Ω ) ∈ D , we defineΛ ( D ) = max ( λ (Ω ) , λ (Ω )) . Also let L (Ω) denote the infimum of Λ ( D ) , over all the bi-partitioni.e.,(2.5) L (Ω) = inf D ∈ D Λ ( D ) . An optimal bi-partition is a partition which realizes the infimum in(2.5). For p = 2 the optimal partition of the first eigenvalue has beenstudied extensively, see [6, 11].We know that the second eigenfunction changes its sign on the do-main, i.e., the second eigenfunction can be written as u = u + − u − , where u + = max( u, , u − = max( − u, . Obviously u + , u − ≥ , u + · u − = 0 in Ω . Nodal domains of u denoted by Ω + and Ω − are defined as the supportof positive and negative part of u Ω + = { x ∈ Ω : u ( x ) > } , Ω − = { x ∈ Ω : u ( x ) < } . FARID BOZORGNIA
The following Lemma in [12] shows existence for minimal two parti-tions and implies that λ (Ω) = L (Ω) . Lemma 2.1.
There exists u ∈ W ,p (Ω) such that ( { u + > } , { u − > } ) achieves infimum in (2.5). Furthermore, λ ( { u + > } ) = λ ( { u − > } ) . It is also known that any eigenfunction associated to an eigenvaluedifferent from λ changes sign. The following properties for secondeigenvalue hold: • If Ω ⊆ Ω ⊆ Ω , then λ ( p, Ω ) ≤ λ ( p, Ω ) . • Let Ω be a bounded domain in R d , then the eigenfunction as-sociated to λ ( p, Ω) admits exactly two nodal domains. • The Courant theorem implies that in the linear case p = 2 , thenumber of nodal domains of an eigenfunction associated to λ is exactly 2 . • In [2, 3] is shown that the second eigenfunctions are not radialin ball.The limiting cases p → p → ∞ are more complicated andrequires tools from non smooth critical point theory and the conceptviscosity solutions. Note for the limiting case p tends to one, thereare several ways to define the second eigenfunction of the 1-Laplaceoperator which it does not satisfy many of the properties of the secondeigenfunction of the p-Laplace operator in general [27]. As p tends toone, in some cases the second eigenfunction takes the form u = c χ C − c χ C . Here χ A is the characteristic function of given set A , the pair ( C , C )is so called Cheeger-2-cluster of Ω, while in other cases functions ofthat type can’t be eigenfunctions at all.In the limiting case as p tends to infinity, the second eigenvalue hasa geometric characterization. Following Section 4 of [23] define(2.6) Λ = 1 r , where r = sup { r ∈ R + : there are disjoint balls B , B ⊂ Ω with radius r } . Then the following lemma from [23] states that the second eigenvalueof infinity Laplace is Λ . HE SECOND EIGENVALUE OF p -LAPLACE OPERATOR 5 Lemma 2.2.
Let λ ( p ) be the second p -eigenvalue in Ω . Then it holdsthat lim p →∞ λ ( p ) p → Λ . Λ ∈ R is the second eigenvalue of the infinity Laplace. Furthermore, it is shown in [23] that the second eigenfunction of theinfinity Laplace operator can be obtained as a viscosity solution of thefollowing equation(2.7) F Λ ( x, u, ∇ u, D u ) = 0 for x ∈ Ω , for which F Λ is given by(2.8) F Λ ( x, u, ∇ u, D u ) = min {|∇ u | − Λ u, − ∆ ∞ u } u ( x ) > , − ∆ ∞ u u ( x ) = 0 , max {−|∇ u | − Λ u, − ∆ ∞ u } u ( x ) < . Here Λ = Λ ∈ R denotes the second eigenvalue of the infinity Laplacegiven by (2.6). 3. An iterative scheme
In this section we discuss our algorithm to approximate the sec-ond eigenvalue λ and corresponding eigenfunction denoted by u . ForDirichlet boundary condition our main assumption is that domain Ωhas the following symmetric property (cid:107) u + (cid:107) L p (Ω) = (cid:107) u − (cid:107) L p (Ω) . In [10] the authors studied gradient flows of p -homogeneous func-tionals on a Hilbert space and proved that after suitable rescaling theflow always converges to a nonlinear eigenfunction of the associatedsubdifferential operator. They also gave conditions for convergence tothe first eigenfunction.The inverse power method is known to be an efficient method to ap-proximate the first eigenvalue of given operator see [8, 9, 19]. Our aimhere is to extend this method combining with Lemma 2.1 to approxi-mate the second eigenvalue.Following Lemma 2.1 and notation introduced before this Lema, weknow that restriction of the second eigenfunction on each nodal domainis the first eigenfunction, i.e., λ (Ω) = λ (Ω + ) = λ (Ω − ) . So the second eigenvalue problem can be written as(3.1) (cid:26) − ∆ p ( u + − u − ) = λ (Ω + ) u p − − λ (Ω − ) u p − − in Ω ,u + = u − = 0 on ∂ Ω . The main steps are as follows:
FARID BOZORGNIA (1) Choose arbitrary initial bi-partition Ω and Ω − . Since Ω andΩ − are disjoint and connected components of Ω this prevent ofobtaining higher eigenfunctions than second one.(2) Given Ω k + and Ω k − , k = 0 , , · · · , obtain the first eigenvaluesdenoted by λ k (Ω + ) , λ k (Ω − ) and first eigenfunctions ( u k + , u k − )normalized in L p related to (Ω k + , Ω k − ) . (3) Solve the following boundary value problem (cid:26) − ∆ p u = λ k (Ω + )( u k + ) p − − λ (Ω − )( u k − ) p − in Ω ,u = 0 on ∂ Ω . (4) Update (Ω + , Ω − ) as supports of positive part and negative partof the solution u .(5) Go to step (2).Note that in step 2, given Ω k + and Ω k − , one needs to calculate λ (Ω k + )and λ (Ω k − ) and corresponding first eigenvalues for each sub-domain.This step can be modified by implementing the inverse power methodalong our Algorithm. Assume u k + and u k − are given normalized in L p with disjoint supports, then define values λ k + and λ k − by(3.2) λ k + = λ k + (Ω k + ) = (cid:82) Ω k + |∇ u k + ( x ) | p dx,λ k − = λ k − (Ω k − ) = (cid:82) Ω k − |∇ u k − ( x ) | p dx. The algorithm to approximate the second eigenvalue and the secondeigenfunction are as follows:
Algorithm 1:
Second eigenvalue algorithm inputs : u = u − u − , (cid:15) . output : Approximation of second eigenvalue and second eigenfunction.(1) Set k = 0, choose initial arbitrary guesses u > , u − > L p (Ω) and vanishing on the boundary with Ω + and Ω − as the supports of functions u and u − respectively.(2) Given u k = u k + − u k − where u k + and u k − are normalized in L p , with disjointsupports, then obtain λ k + and λ k − by (3.2). ;(3) Solve(3.3) − ∆ p u = | u k | p − (cid:18) λ k + u k + − λ k − u k − (cid:19) in Ω ,u = 0 on ∂ Ω . (4) Set u k +1+ and u k +1 − as positive and negative part of the solution of (3.3).Normalized u k +1+ and u k +1 − in L p . Calculate λ k +11 (Ω ) , λ k +11 (Ω ) . (5) If | λ k +11 (Ω + ) − λ k (Ω + ) | ≥ (cid:15), & | λ k +11 (Ω − ) − λ k (Ω − ) | ≥ (cid:15) then(6) Set k = k + 1 and go step (2) Remark 1.
Note that Ω + and Ω − change in iterations but for simplic-ity we write Ω + and Ω − instead of Ω k + and Ω k − . HE SECOND EIGENVALUE OF p -LAPLACE OPERATOR 7 Remark 2.
Consider the case p = 2 . Let denote the first eigenfunctionby w . The second eigenvalue is given by (3.4) λ (Ω) = inf u ⊥ w (cid:82) Ω |∇ u ( x ) | dx (cid:82) Ω | u ( x ) | dx . Assume the initial guess u be chosen such that ( λ u +0 − λ − u − ) ⊥ w then the Algorithm generates sequence { u n } which are orthogonal to w .Multiply the equation − ∆ u = λ (Ω + ) u − λ − (Ω − ) u − by w and integrating by parts two times on left side, implies (cid:90) Ω u (∆ w ) dx = (cid:90) Ω ( λ u − λ − u − ) w dx = 0 . This shows (cid:90) Ω u w dx = 0 . Neumann case.
In this part, we consider the p -Laplace eigen-value problem with Neumann boundary.(3.5) (cid:40) − ∆ p u = λ | u | p − u in Ω , |∇ u | p − ∇ u · ν = 0 on ∂ Ω , where 1 < p < ∞ and ν is the unit normal vector to ∂ Ω.A non zero function u ∈ W ,p (Ω) is called a p -eigenfunction of (3.5)in the weak sense, if there exist a λ ≥ (cid:90) Ω |∇ u | p − ∇ u · ∇ φ dx = λ (cid:90) Ω | u | p − u φ dx, φ ∈ W ,p (Ω) . The number λ is called a p -eigenvalue. The first eigenfunction is aconstant function and λ = 0. Note that by testing (3.6) with φ = 1we obtain that any eigenfunction with λ > (cid:90) Ω | u | p − u dx = 0 . Equivalently, (cid:107) u + (cid:107) L p − (Ω) = (cid:107) u − (cid:107) L p − (Ω) . This motivates the following definition:The variational second p -eigenvalue is defined as(3.8) λ ,p (Ω) := inf (cid:26) (cid:82) Ω |∇ u ( x ) | p dx (cid:82) Ω | u ( x ) | p dx : (cid:90) Ω | u | p − u = 0 (cid:27) . Any u ∈ W ,p (Ω) \{ } realizing the infimum is called variational second p -eigenfunction. FARID BOZORGNIA
Definition 3.1.
The p -mean of a function u ∈ L p (Ω) is defined as(3.9) mean p ( u ) := 1 | Ω | (cid:90) Ω | u | p − u d x. Remark . The following p -Laplace problem(3.10) (cid:40) − ∆ p u = f on Ω , |∇ u | p − ∇ u · ν = 0 on ∂ Ω , has a one parameter family of solutions; adding any constant to asolution will be a solution. Furthermore, the datum f has to admit the compatibility condition mean ( f ) = 0.The algorithm for Neumann case is as following. Algorithm 2:
Second eigenvalue for Neumann boundary inputs : u = u − u − , (cid:15) . output : Approximation of second eigenvalue and second eigenfunction. while | λ k +11 (Ω + ) − λ k (Ω + ) | ≥ (cid:15), & | λ k +11 (Ω − ) − λ k (Ω − ) | ≥ (cid:15) do (1) Set k = 0. Initialize with arbitrary guess u (0) ∈ W ,p (Ω) with mean p ( u (0) ) = 0.(2) For k ≥
0, we set u k ± := max( ± u k ) ,
0) and define λ k ± := (cid:82) Ω |∇ u k ± | p d x (cid:82) Ω | u k ± | p d x ,f k := λ k + ( u k + ) p − − λ k − ( u k − ) p − . (3)(4) Next define u ( k +1) as the unique solution to the problem(3.11) − ∆ p u = f k in Ω |∇ u | p − ∇ u · ν = 0 on ∂ Ωmean p ( u ) = 0 . (5) Set k = k + 1 and go back to (2). end Remark . To enforce the condition mean p ( u ) = 0 numerically in(3.11), we fix the value of the solution at a single node of the gridto an arbitrary value, which yields a unique solution ˜ u . Then we set u := ˜ u − c where c is such that mean p ( u ) = 0 holds.3.2. Convergence of the Algorithm.
In the sequel, for any v ∈ W ,p (Ω) by ˜ v we mean the normalized in L p (Ω)˜ v = v (cid:107) v (cid:107) L p (Ω) . Furthermore, we define u k + , u k − ∈ W ,p (Ω) as the positive and negativeparts of the solution of the following Dirichlet problem inductively,(3.12) (cid:26) − ∆ p u k = λ k − (˜ u k − ) p − − λ k − − (˜ u k − − ) p − in Ω ,u k = 0 on ∂ Ω . Here ˜ u k − = ˜ u k − − ˜ u k − − . HE SECOND EIGENVALUE OF p -LAPLACE OPERATOR 9 Proposition 3.1.
With the introduced notations, the following factshold: | ˜ u k − | = | ˜ u k − − ˜ u k − − | = ˜ u k − + ˜ u k − − , | ˜ u k − | p − = (˜ u k − ) p − + (˜ u k − − ) p − , (cid:107) λ k − ˜ u k − − λ k − − ˜ u k − − (cid:107) pL p = ( λ k − ) p + ( λ k − − ) p , (cid:107) λ k − ∇ ˜ u k − − λ k − − ∇ ˜ u k − − (cid:107) pL p = ( λ k − ) p +1 + ( λ k − − ) p +1 . The next Lemma shows the monotonicity of sequence of approxima-tions of second eigenvalues as p tends to one. Lemma 3.2.
Let λ k + (Ω + ) and λ k − (Ω − ) be obtained by Algorithm 1.Then as p → the following holds max (cid:0) λ k + (Ω + ) , λ k − (Ω − ) (cid:1) ≤ max (cid:0) λ k − (Ω + ) , λ k − − (Ω − ) (cid:1) , for every k ≥ .Proof. To start, multiply the first equation (3.12) by u k + and integrateover Ω to deduce (cid:90) Ω |∇ u k + | p dx = (cid:90) Ω u k + | ˜ u k − | p − (cid:20) λ k − ˜ u k − − λ k − − ˜ u k − − (cid:21) dx = (cid:90) Ω u k + (cid:20) λ k − (˜ u k − ) p − − λ k − − (˜ u k − − ) p − (cid:21) dx. (3.13)H¨older inequality on right hand side gives (cid:90) Ω |∇ u k + | p dx ≤ (cid:107) u k + (cid:107) L p (cid:107) λ k − (˜ u k − ) p − − λ k − − (˜ u k − − ) p − (cid:107) L pp − . By Proposition (3.1) we obtain(3.14) (cid:107)∇ u k + (cid:107) pL p ≤ (cid:107) u k + (cid:107) L p (cid:20) ( λ k − ) pp − + ( λ k − − ) pp − (cid:21) p − p . The same argument as above indicates(3.15) (cid:107)∇ u k − (cid:107) pL p ≤ (cid:107) u k − (cid:107) L p (cid:20) ( λ k − ) pp − + ( λ k − − ) pp − (cid:21) p − p . The inequalities (3.14) and (3.15) implymax (cid:18) (cid:107)∇ u k + (cid:107) pL p (cid:107) u k + (cid:107) L p , (cid:107)∇ u k − (cid:107) pL p (cid:107) u k − (cid:107) L p (cid:19) ≤ (cid:20) ( λ k − ) pp − + ( λ k − − ) pp − (cid:21) p − p . Let p → + and using the factlim α →∞ ( a α + b α ) α = max( a, b ) , complete the proof. (cid:3) Lemma 3.3.
With same assumptions as before the sequence u k isbounded from below for every p ≥ . Proof.
Multiply equation (3.12) by u k , integrate over Ω and H¨olderinequality give(3.16) (cid:90) Ω |∇ u k | p dx ≤ (cid:107) u k (cid:107) L p (cid:107) λ k − (˜ u k − ) p − − λ k − − (˜ u k − − ) p − (cid:107) L pp − . (cid:107)∇ u k (cid:107) pL p ≤ (cid:107) u k (cid:107) L p (cid:20) ( λ k − ) pp − + ( λ k − − ) pp − (cid:21) p − p . Note that u k satisfies λ (cid:107) u k (cid:107) pL p ≤ (cid:107)∇ u k (cid:107) pL p . This shows λ (cid:107) u k (cid:107) pL p ≤ (cid:107) u k (cid:107) L p (cid:20) ( λ k − ) pp − + ( λ k − − ) pp − (cid:21) p − p , which implies(3.17) (cid:107) u k (cid:107) L p ≤ λ (cid:20) ( λ k − ) pp − + ( λ k − − ) pp − (cid:21) p . Multiply (3.12) by ˜ u k − and integrate over Ω to deduce (cid:90) Ω |∇ u k | p − ∇ u k · ∇ ˜ u k − dx = λ k − . Here we used the facts that˜ u k − · ˜ u k − − = 0 , (cid:90) Ω (˜ u k − ) p dx = 1 . Form here we get λ k − ≤ (cid:107)∇ u k (cid:107) p − L p (cid:107)∇ ˜ u k − (cid:107) L p , the same argument shows λ k − − ≤ (cid:107)∇ u k (cid:107) p − L p (cid:107)∇ ˜ u k − − (cid:107) L p , Considering (cid:107)∇ ˜ u k − (cid:107) L p = ( λ k − ) p and (cid:107)∇ ˜ u k − − (cid:107) L p = ( λ k − − ) p implies(3.18) ( λ k − ) p − p ≤ (cid:107)∇ u k (cid:107) p − L p , ( λ k − − ) p − p ≤ (cid:107)∇ u k (cid:107) p − L p . Inserting the inequalities (3.18) in (3.16) yields(3.19) 1 ≤ p − p (cid:107) u k (cid:107) L p . Also from inequalities (3.18) and (3.14) it follows
HE SECOND EIGENVALUE OF p -LAPLACE OPERATOR 11 (3.20) (cid:107)∇ u k + (cid:107) pL p ≤ p − p (cid:107) u k + (cid:107) L p (cid:107)∇ u k (cid:107) pL p , (cid:107)∇ u k − (cid:107) pL p ≤ p − p (cid:107) u k − (cid:107) L p (cid:107)∇ u k (cid:107) pL p . (cid:3) Lemma 3.4.
For every p > the following holds λ k ≤ max (cid:0) λ k − (Ω + ) , λ k − − (Ω − ) (cid:1) , where λ k = (cid:82) Ω |∇ u k | p dx (cid:82) Ω | u k | p dx . Proof.
We rewrite the right hand side of (3.16) as(3.21) (cid:107)∇ u k (cid:107) pL p ≤ (cid:107) u k (cid:107) L p (cid:13)(cid:13)(cid:13)(cid:13) λ k − (˜ u k − ) p − − λ k − − (˜ u k − − ) p − (cid:13)(cid:13)(cid:13)(cid:13) pp − L pp − (cid:13)(cid:13)(cid:13)(cid:13) λ k − (˜ u k − ) p − − λ k − − (˜ u k − − ) p − (cid:13)(cid:13)(cid:13)(cid:13) p − L pp − Next by Proposition 3.1 we have (cid:13)(cid:13)(cid:13)(cid:13) λ k − (˜ u k − ) p − − λ k − − (˜ u k − − ) p − (cid:13)(cid:13)(cid:13)(cid:13) p − L pp − = (cid:18) ( λ k − ) pp − + ( λ k − ) pp − (cid:19) p . Also (cid:107) λ k − (˜ u k − ) p − − λ k − − (˜ u k − − ) p − (cid:107) pp − L pp − = (cid:90) Ω ( λ k − ) pp − (˜ u k − ) p + ( λ k − − ) pp − (˜ u k − − ) p dx = (cid:90) Ω (cid:20) λ k − (˜ u k − ) p − − λ k − − (˜ u k − − ) p − (cid:21) (cid:20) ( λ k − ) p − ˜ u k − − ( λ k − − ) p − ˜ u k − − (cid:21) dx = (cid:90) Ω ( − ∆ p u k ) [( λ k − ) p − ˜ u k − − ( λ k − − ) p − ˜ u k − − ] dx = (cid:90) Ω |∇ u k | p − ∇ u k · [( λ k − ) p − ∇ ˜ u k − − ( λ k − − ) p − λ k − ∇ ˜ u k − − ] dx ≤ (cid:107)∇ u k (cid:107) p − L p (cid:107) ( λ k − ) p − ∇ ˜ u k − − ( λ k − − ) p − ∇ ˜ u k − − (cid:107) L p . Inserting the last inequality in (3.21) and dividing by (cid:107) u k (cid:107) pL p we obtain (cid:82) Ω |∇ u k | p dx (cid:82) Ω | u k | p dx ≤ ( (cid:82) Ω |∇ u k | p dx (cid:82) Ω | u k | p dx ) p − p (cid:107) ( λ k − ) p − ∇ ˜ u k − − ( λ k − − ) p − ∇ ˜ u k − − (cid:107) L p (( λ k − ) pp − + ( λ k − ) pp − ) p . The inequality above yields (cid:82) Ω |∇ u k | p dx (cid:82) Ω | u k | p dx ≤ (cid:107) ( λ k − ) p − ∇ ˜ u k − − ( λ k − − ) p − ∇ ˜ u k − − (cid:107) pL p ( λ k − ) pp − + ( λ k − ) pp − . Thus(3.22) (cid:82) Ω |∇ u k | p dx (cid:82) Ω | u k | p dx ≤ ( λ k − ) pp − +1 + ( λ k − − ) pp − +1 ( λ k − ) pp − + ( λ k − ) pp − . Form (3.22) we infer(3.23) λ k ≤ max (cid:0) λ k − , λ k − − (cid:1) . (cid:3) Graph p -Laplacian The aim of this part is to perform data clustering by using our algo-rithm. Given some data and a notion of similarity, we aim to partitionthe input data into maximally homogeneous groups (i.e. clusters). Themain idea is to find a low-dimensional embedding and then to projectdata points to new space. Recursive bi-partitioning method is widelyused in many clustering algorithm for the multi-class problem. As abasic idea, for given graph G a traditional spectral clustering algorithmuses the first k eigenvectors for ∆ G as a low-dimensional embedding ofthe graph. Different graph Laplacian and their basic properties, spec-tral clustering algorithms are described in [20, 21].Let G = ( V, E ) be an undirected graph with vertex set V = { v , · · · , v n } or simply V = { , · · · , n } , and E is the set of edges. The weighted ad-jacency matrix W encode the similarity of pairwise data points, orweight w ij ≥ v i and v j ; W = ( w ij ) i, j = 1 , · · · , n. Note that G being undirected means w ij = w ji . The degree of a vertex i ∈ V denoted by d i is d i = (cid:88) j ∈ V w ij . The degree matrix D is defined as the diagonal matrix with the degrees d , ..., d n on the diagonal.For given graph ( V, E ) and a subset of vertex C ⊂ V the Cut( C, C c )(or the perimeter | ∂C | ) is defined byCut( C, C c ) := (cid:88) i ∈ C,j ∈ C c w ij , where C c is the complement of C . The ratio Cheeger cut RCC( C, C c )and normalized Cheeger cut NCC( C, C c ) are defined respectively, byRCC( C, C c ) = cut( C, C c )min( | C | , | C c | ) , NCC(
C, C c ) = cut( C, C c )min(vol | C | , vol | C c | ) . The minimum is achieved if | C | = | C c | . HE SECOND EIGENVALUE OF p -LAPLACE OPERATOR 13 For function f : V → R , the unnormalized p -Laplace operator ∆ up and normalized p -Laplace operator ∆ np are defined as follow(dependson the choice of inner product) :(∆ up f ) i = (cid:88) j φ p ( f i − f j ) , (∆ np f ) i = 1 d i (cid:88) j φ p ( f i − f j ) , where φ p ( · ) = | · | p − sign( · ) , for p = 2 we get φ p ( x ) = x. Also for functions f, g : V → R the innerproduct is defined by < f, g > = (cid:88) i ∈ V d i f i g i . Next consider the following problem. Find f : V → R and λ ∈ R suchthat (∆ up f ) i = λφ p ( f i ) , ∀ i ∈ V. We call λ an eigenvalue of ∆ up associated with eigenvector f. In thecase p = 2 , the operator ∆ up is the regular graph Laplacian given by∆ u = L = D − W. The Rayleigh quotient is defined by R p ( f ) = Q p ( f ) (cid:107) f (cid:107) pp = (cid:80) ij ( f i − f j ) p (cid:107) f (cid:107) p , with Q p ( f ) := < f, ∆ up f > = 12 (cid:88) i,j w ij | f i − f j | p , (cid:107) f (cid:107) pp = (cid:88) i | f i | p . Similar to continuous case, f is a p -eigenfunction of ∆ up if and only ifthe functional R p has a critical point at f ∈ R | V | . The correspondingeigenvalue λ p is given as λ p = R p ( f ) . The first eigenvalue is zero and the first eigenvector is a constant vector.Then to find the second eigenvalue we minimize the Rayleigh quotient R p ( f ) over all f such that (cid:88) i ∈ V f i = 0 , which is the requirement of Algorithm 2 presented in previous section.We point out that in [11] is shown that the cut obtained by threshold-ing the second eigenvector of p -Laplace converges to optimal Cheegercut as p tends to 1 and λ (∆ ) = RCC ∗ . However, finding optimal ratio Cheeger cut
RCC ∗ = min C ⊂ V RCC is NP-hard problem. To see more about minimization of R p ( f ) with con-straint (cid:80) i ∈ V f i = 0 , see [11].5. Numerical implementation
In this section, we briefly explain about numerical approximation ofthe following problem(5.1) (cid:26) − ∆ p u = f ( x ) in ∂ Ω ,u = 0 on ∂ Ω . For p > E ( u ) = min u ∈ W ,p (Ω) u (cid:54) =0 (cid:90) Ω p |∇ u ( x ) | p − f ( x ) u ( x ) dx. Equation in (5.1) is understood in weak sense:(5.2) (cid:90) Ω |∇ u | p − ∇ u · ∇ φ dx = (cid:90) Ω f ( x ) φ dx, ∀ φ ∈ W ,p (Ω) . The discretization of problem is as follows. Let T h be a regular trian-gulation of Ω h which is composed of disjoint open regular triangles T i , that is, Ω h = n (cid:91) i =1 T i . Considering the regularity for the solution of the p -Laplace equation,we deal with continuous piecewise linear element. Consider a finite di-mensional subspace V h of C (Ω h ) , such that the restriction on elementsof T h , where P is the linear function space: V h = { v ∈ H : v (cid:98) T ∈ P , ∀ T ∈ T h } . Assume u n ∈ V h be the current approximation the, we associate theresidual denoted by R n R n = f ( x ) − ∇ ( |∇ u n | p − ∇ u n ) . Equivalently, (cid:90) Ω R n φ dx = (cid:90) Ω ( f ( x ) φ − |∇ u n | p − ∇ u n · ∇ φ ) dx. Then to update u n and obtaining next approximation denoted by u n +1 u n +1 = u n + α n w n , where w n is determined solution of linearized p -Laplace and the sourceterm residual, i.e., HE SECOND EIGENVALUE OF p -LAPLACE OPERATOR 15 (5.3) (cid:26) − ∆ w n = R n in Ω ,w n = 0 on ∂ Ω , Or (cid:90) Ω ( ε + |∇ u n | ) p − ∇ w n · ∇ φ dx = (cid:90) Ω R n φ dx. The step length α n in search direction w n can be obtained as E ( u n + α n w n ) = min α E ( u n + αw n ) . Numerical Examples.
This section provides some examples ofnumerical approximations to the p -Laplace eigenvalue problem for dif-ferent values of p . For initial guess we use the second eigenfunction ofLaplace operator. Example 5.1.
Here, we verify our algorithm by invoking it in dimen-sion one. Let λ p denotes the second eigenvalue in the interval ( a b ) thenby result in [13, 26] we have(5.4) p (cid:112) λ p = p √ p − π ( b − a ) p sin( πp ) . Let Ω = ( − ,
2) then for p = 100 the above formula gives: (cid:112) λ = 2 . . For p = 50 , p = 100 our approximation of second eigenvalue are (cid:112) λ = 2 . (cid:112) λ = 2 . , with asymptotic given by (2.6) or (5.4)lim p →∞ p (cid:112) λ p = 2 . Figure 1.
The second eigenfunction for p = 100. Example 5.2.
Let p = 2 and the domain be Ω = [ − × [ − λ = π . We set (cid:52) x = (cid:52) y = . . Theinitial value is given in Figure 2. and our approximate value after 50iterations is: 3.0843295. with | λ − λ (50)2 | ≤ . . Figure 2.
Initial guessIn Figure 3, and 4 the second eigenfunctions for Dirichlet and Neu-mann cases are shown.
Figure 3.
Surface of u . Example 5.3.
Let domain be the square [0 2] × [0 2]. Picture 5shows the second eigenfunction for p = 10 . HE SECOND EIGENVALUE OF p -LAPLACE OPERATOR 17 Figure 4.
Surface of u for Neumann case. Figure 5.
Surface of u for p = 10. Example 5.4.
In this experiment, we use ε -graph. The points aregenerated by standard uniform distribution. We choose n = 5000, and ε = . Acknowledgements
F. Bozorgnia was supported by the FCT post-doctoral fellowship SFRH/BPD/33962/2009and by Marie Sk(cid:32)lodowska-Curie grant agreement No. 777826 (NoMADS). The author isthankful to Leon Bungert for insightful discussion contributing to deepen on subsection3.1.
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CAMGSD, Department of Mathematics, Instituto Superior T´ecnico,Lisbon
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