In this section we give some estimates of the number n of sets minimizing the energyG∞. We will need some assumptions on the imagegin order to achieve the desired results.
Let A1, . . . , AN denote N sets of class C2(R2) such that for any i, j ∈ {1, . . . , N}, with i = j, the boundaries ∂Ai and ∂Aj intersect transversally at a finite number of points (see Figure 1). Moreover, let the sets be ordered accord- ing to the ordering relation (1.1):
A1< A2<ã ã ã< AN.
The functiong is assumed piecewise constant and it is defined by g=
N i=1
ciχA
i, (4.1)
where theci are positive constants, so thatgis constant on the visible partsAiof the regions.
The visible portion of the boundary of the regionAi is defined by (∂A1) =∂A1, (∂Ai) =∂Ai\
i)−1 j=1
Aj fori= 2, . . . , N.
A
A
A
1
2 3
Figure 1. The setsAi in the imageg.
By the assumptions on the setsAi, the visible portion of ∂Ai consists of a finite number ofC2arcs terminating in a finite setMi of endpoints. We set
Jg=∂A1 )N i=2
(∂Ai), M= )N i=2
Mi.
We now study the relation between the numbern minimizing the functional G∞
and the numberN of the regions which are actually present in the imageg.
Let {E1, . . . , En} be a minimizer of G∞; then, using (2.1) and (3.1), there existnsystems of curves Γi,i= 1, . . . , n, such that
Jg ⊆ )n i=1
(Γi), infG∞= n i=1
F(Γi). (4.2)
We then define
Γ ={Γ1, . . . ,Γn}, so thatJg⊆(Γ).
Let Σ denote the set of all finite familiesσ = {γ1, . . . , γm} of W2,p curves which connect pairwise all the endpoints inMin such a way that for any point p∈ Mi, the tangent vectors ofσand (∂Ai)inpare parallel, for anyi∈ {2, . . . , N}.
We have the following lemma.
Lemma 4.1. If Γ has a finite number of tangential self-intersections then there existsσ∈Σsuch that
F(Γ)≥
Jg
[1 +|κ|p]dH1+F(σ). (4.3) The proof of the lemma is based on the following argument. It can be proved [2] that each system of curves Γi has not transversal self-intersections. Using this
Properties of the Nitzberg-Mumford Variational Model 81 fact, the propertyJg ⊆(Γ), the hypothesis of a finite number of tangential self- intersections of Γ, and the assumptions on the imageg, a proof by induction (based on the finiteness hypothesis) shows that Γ not only coversJg, but also connects pairwise the endpoints inM by means of additional W2,p curves. Hence, there exists a familyσ∈Σ such that
Jg
)(σ)⊆(Γ),
and the estimate (4.3) is then obtained by a covering argument.
We are now in a position to state the following energy estimate.
Theorem 4.2. [Energy estimate] If Jg ⊆ +n
i=1∂Ei, with F(Ei) < +∞ for any i∈ {1, . . . , n}, then
infG∞≥
Jg
[1 +|κ|p]dH1+ inf
σ∈ΣF(σ). (4.4)
The proof of the estimate is based on the density result of Theorem 3.1, which permits us to remove from Lemma 4.1 the hypothesis of a finite number of tangential self-intersections of Γ. Then, using (4.2), Lemma 4.1 and taking the infimum over Σ, we have
G∞(n, E1, . . . , En) =F(Γ)≥
Jg
[1 +|κ|p]dH1+ inf
σ∈ΣF(σ), from which the estimate (4.4) follows.
Now we may constructN* setsE*1, . . . ,E*N*, withW2,pboundary, by connect- ing pairwise the endpoints inMby means of curves which minimize the energy
l(γ) 0
[1 +|κ|p]ds
with given tangent vectors inM(see Figure 2). In the casep= 2 the minimizing curves are called elastica because of their application to the theory of flexible inextensible rods [9].
Then we have Jg ⊆
N*
)
i=1
∂E*i, infG∞≤
N*
i=1
F(E*i).
In particular, there exists a finite family of curves*σ∈Σ such that
N*
)
i=1
∂E*i =Jg)
(σ),* and infG∞≤
Jg
[1 +|κ|p]dH1+F(*σ).
If*σminimizesF(σ) in Σ then, using Theorem 4.2, it follows infG∞=
Jg
[1 +|κ|p]dH1+F(*σ), so that the family of sets{E*1, . . . ,E*N*}minimizesG∞.
A
A
A
1
2 3
Figure 2. Endpoints inMconnected by elastica curves.
Hence, in this case the asymptotic functionalG∞is minimized by a family of sets with cardinalityn=N* and the number of regions that are actually present in the imagegis reconstructed. Note that the minimizer*σofF(σ) in Σ may be such thatN* =N. This may happen in the following case. Suppose that connecting the endpoints of the visible boundary (∂Ai) of the region Ai, a set E*i is obtained.
Analogously, a set E*j is obtained connecting the endpoints of (∂Aj) for some j =i. However, it may happen that the endpoints of (∂Ai) and (∂Aj) can be connected by forming a single setE*i,j such that
F(E*i,j)<F(E*i) +F(E*j).
This case corresponds to a family of curves*σ∈Σ such thatN < N* .
However, the assumption that there exists a family*σwith the above proper- ties and minimizingF(σ) in Σ, is not satisfied in general. We are able to improve such a result in the following way.
Let Σ0⊆Σ be the subset of the familiesσsuch that there exists a finite family of sets{E1, . . . , EM}, withF(Ei)<+∞for anyi∈ {1, . . . , M}andJg⊆ ∪Mi=1∂Ei, such that
M i=1
F(Ei) =
Jg
[1 +|κ|p]dH1+F(σ). (4.5) The case considered so far corresponds to*σ∈Σ0 andM =N.*
We have the following lemma.
Lemma 4.3. Ifσ∈Σ\Σ0 andE1, . . . , EM are such that Jg)
(σ)⊆ )M i=1
∂Ei,
Properties of the Nitzberg-Mumford Variational Model 83 then
M i=1
F(Ei)≥
Jg
[1 +|κ|p]dH1+F(σ) +c0π, (4.6) wherec0= [(p/p)1/p+ (p/p)1/p], with 1/p+ 1/p= 1.
The proof of the lemma is based on the following argument. IfF(Ei)<+∞ for anyi, by using (3.1) there exists a system of curves Γ such that
)M i=1
∂Ei⊆(Γ),
M i=1
F(Ei) =F(Γ).
Assume first that Γ has a finite number of tangential self-intersections. Then, if σ /∈ Σ0, we prove by means of an inductive method (based on the finiteness assumption) that the set (Γ)\(Jg∪(σ)) contains at least a closed curve. Hence, by the H¨older inequality the energy of a closed curve is greater than or equal to the constantc0π, and the inequality (4.6) then follows by a covering argument. Then the hypothesis of a finite number of tangential self-intersections of Γ is removed by means of the density result of Theorem 3.1.
Then, using Theorem 4.2, Lemma 4.3 and (4.5), we obtain the following result.
Proposition 4.4. If the inequality inf
σ∈Σ0F(σ)≤ inf
σ∈ΣF(σ) +c0π (4.7)
holds, then there exist a family of curves σ ∈ Σ0 and sets E1, . . . , EM, with F(Ei)<+∞andJg⊆ ∪Mi=1∂Ei, such that
M i=1
F(Ei) = infG∞, and
M i=1
F(Ei) =
Jg
[1 +|κ|p]dH1+F(σ).
If the assumption (4.7) of Proposition 4.4 is satisfied, it follows that the numberM of regions is reconstructed from the imageg. Then, the family of sets {E1, . . . , EM} can be endowed with the ordering relation
E1< E2<ã ã ã< EM , as follows
∂E1⊆Jg, ∂Ei\
i)−1 j=1
Ej⊆Jg fori= 2, . . . , M .
Remark 4.5. IfN = 2 in the definition (4.1) ofg andM consists of two points, then the assumption (4.7) is unnecessary,G∞is minimized by n= 2 and the two endpoints ofMare connected by an elastica.
References
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J. Math. Imaging Vision18(2003), 7–15.
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[9] D. Mumford,Elastica and computer vision. Algebraic Geometry and Applications, C. Bajaj ed., Springer-Verlag, Heidelberg, 1992.
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Giovanni Bellettini
Dipartimento di Matematica Universit`a di Roma “Tor Vergata”
Via della Ricerca Scientifica I-00133 Roma, Italy
e-mail:belletti@mat.uniroma2.it Riccardo March
Istituto per le Applicazioni del Calcolo, CNR Viale del Policlinico 137
I-00161 Roma, Italy e-mail:r.march@iac.cnr.it
International Series of Numerical Mathematics, Vol. 154, 85–94 c 2006 Birkh¨auser Verlag Basel/Switzerland
The ∞ -Laplacian First Eigenvalue Problem
Marino Belloni
Abstract. We review some results about the first eigenvalue of the infinity Laplacian operator and its first eigenfunctions in a general norm context.
Those results are obtained in collaboration with several authors: V. Ferone, P. Juutinen and B. Kawohl (see [BFK], [BK1], [BJK] and [BK2]). In Section 5 we make some remarks on the simplicity of the first eigenvalue of ∆∞: this will be the object of a joint work with A. Wagner (see [BW]).
Mathematics Subject Classification (2000).Primary 35P30; Secondary 35J70, 49L25, 49R50.
Keywords.Nonlinear eigenvalue problems, Degenerate elliptic equation, Vari- ational methods, Viscosity solutions.
1. Introduction
Imagine a nonlinear elastic membrane, fixed on a boundary∂Ω of a plane domain Ω. Ifu(x) denotes its vertical displacement, and if its deformation energy is given by
Ω|∇u|pdx, then a minimizer of the Rayleigh quotient
Ω|∇u|p dx,
Ω|u|p dx onW01,p(Ω) satisfies the Euler-Lagrange equation
−∆pu=λp |u|p−2u in Ω, (1.1) where ∆pu= div(|∇u|p−2∇u) is the well-knownp-Laplace operator. This eigen- value problem has been extensively studied in the literature, see [L3]. A somewhat strange recent result is that (asp→ ∞) the limit equation reads
min{ |∇u| −Λ∞u, −∆∞u}= 0. (1.2) Here ∆∞u=
i,juxiuxjuxixj is the∞-Laplacian, Λ∞= limp→∞Λp where Λp= λ1/pp (see [JLM1, FIN]).
Now suppose that the membrane is not isotropic. It is for instance woven out of elastic strings like a piece of material. Then the deformation energy can
This work was completed with the support of the research project Calcolo delle Variazioni e Teoria Geometrica della Misura.
be anisotropic, see [BK2, BJK]. We are mainly interested in generalizing the result on eigenfunctions for the p-Laplacian to the situation, where Ω ⊂ Rn is no longer equipped with the Euclidean norm, but instead with a general norm
| ã |, for instance with |x| = (n
i=1|xi|q)1/q and q ∈ (1,∞). In that case a Lip- schitz continuous function u : Ω → R (in a convex domain Ω) has Lipschitz constantL= supz∈Ω|∇u(z)|∗, where| ã |∗ denotes the dual norm to| ã |, because
|u(x)−u(y)| ≤L|x−y|with thisL. Then we study the asymptotic behavior of the first eigenvalue (eigenfunctions) whenp→ ∞. The case whenp→1 and the corre- sponding limiting problem for the first eigenvalue is not considered, see [KF, KLR].
It is well known, that the infinite-Laplacian operator ∆∞ is closely related to finding a minimal Lipschitz extension of a given functionφ∈C0,1(∂Ω) into Ω:
see Section 2. In [BFK] the eigenvalue problem was carried over to a general norm and studied for finite p, while in [BK2] the eigenvalue problem was investigated first for finitepand the special non-Euclidean norm|x|= (n
i=1|xi|p)1/p withp conjugate top, and then for the limitp→ ∞. In [BJK] the eigenvalue problem was investigated for general strictly convex norm|x|, and then for the limitp→ ∞.
This paper is organized as follows.
In Section 2 we introduce the∞-Laplacian operator and we survey some old and recent results.
In Section 3 we introduce the first eigenvalue of the operator ∆∞, and we survey some results obtained in [BJK, BFK, BK1] and [BK2].
In Section 4 we expose some examples related to the results quoted in Section 3 (see [BJK, BK2]).
In Section 5 we expose some unpublished material on the simplicity of the first eigenvalue obtained in [BW], a joint work in preparation with A. Wagner.