# The Second Eigenvalue of thep-Laplacian as p Goes to 1

Authors: Enea Parini
Journal: International Journal of Differential Equations
 (2010)

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## Abstract

The asymptotic behaviour of the second eigenvalue of thep-Laplacian operator as p goes to 1 is investigated. The limit setting depends only on the geometry of the domain. In the particular case of a planar disc, it is possible to show that the second eigenfunctions are nonradial if p is close enough to 1.


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## Body

## 1. Introduction

LetΩ⊂ℝn be a bounded open domain with Lipschitz boundary. We consider the following nonlinear eigenvalue problem:(1.1)-Δpu=λ|u|p-2uinΩ,u=0on∂Ω,
where λ∈ℝ, p∈(1,+∞), and Δpu=div(|∇u|p-2∇u) is the p-Laplacian operator. Notice that for p=2 we recover the usual Laplacian. A real number λ is said to be an eigenvalue if there exists a function u∈W01,p(Ω)∖{0} (called eigenfunction) satisfying (1.1) in the weak sense, which means (1.2)∫Ω|∇u|p-2∇u∇v=λ∫Ω|u|p-2uv∀v∈W01,p(Ω).A sequence of eigenvalues{λk(p;Ω)}k=1∞ can be obtained by means of a minimax principle, as shown, for instance, in [1] and explained later in Section 5. The sequence is such that(1.3)λ1(p;Ω)<λ2(p;Ω)≤λ3(p;Ω)≤⋯
and λk(p;Ω)→+∞ as k→+∞.In this paper we will mainly focus on the asymptotic behaviour of the second eigenvalue of thep-Laplacian as p goes to 1. Our aim is to extend the results found in [2], where it was shown that the first eigenvalue converges to the so-called Cheeger constant defined as (1.4)h1(Ω)=infE⊂ΩPer(E;ℝn)V(E).
Here Per(E;ℝn) is the perimeter of E measured with respect to ℝn and defined in the distributional sense (see [3]), while V(E) stands for the n-dimensional Lebesgue measure of E. The task of finding a set for which the infimum is attained is called Cheeger problem. We will show that a similar result holds also for the second eigenvalue; moreover, we are able to state that the second eigenfunctions of the p-Laplacian in a planar disc cannot be radial, if p is sufficiently close to 1.The paper is structured as follows. After recalling some known results about the Cheeger problem (Section2), in Sections 3 and 4 we deal with a geometrical problem, which will turn out to be crucial in order to describe the asymptotic behaviour of λ2(p;Ω) as p→1 (Section 5). Finally, in Section 6 we apply the results to the particular case where Ω is a planar disc.


## 2. Known Facts on the Cheeger Problem

In this section we will recall some known results about the Cheeger problem. We say that a functionu∈L1(Ω) has bounded variation if the quantity (2.1)∥Du∥(Ω)=sup{∫Ωudivφdx∣φ∈𝒞c1(Ω;ℝn),|φ|≤1},
called total variation, is finite; in this case we will write u∈BV(Ω). Since we assume Ω to have a Lipschitz boundary, it can be proved that u∈BV(Ω) implies u∈BV(ℝn) (we consider u equal to zero outside Ω). The space BV(Ω) is compactly embedded in L1(Ω). Moreover, the total variation is a lower semicontinuous functional with respect to the L1-convergence, that is,(2.2)uk→uinL1(Ω)⇒∥Du∥(Ω)≤lim infk→∞∥Duk∥(Ω).
A set E⊂ℝn has finite perimeter (measured with respect to ℝn) if (2.3)Per(E;ℝn)=∥DχE∥(ℝn)<∞,
where χE is the characteristic function of E. For the sake of simplicity, in the following we will set Per(E)=Per(E;ℝn).A setC⊂Ω such that (2.4)Per(C)V(C)=h1(Ω)=infE⊂ΩPer(E)V(E)
is called a Cheeger set for Ω. The existence of a Cheeger set for every domain Ω follows easily from the coarea formula and from the fact that actually (2.5)h1(Ω)=infu∈BV(Ω)∖{0}∥Du∥(ℝn)∥u∥1.The following proposition is a useful approximation result, whose proof can be found in [4].Proposition 2.1.
LetE⊂ℝn be a set of finite perimeter. Then there exists a sequence of sets of finite perimeter {Ek}k=1+∞ such that:  (1)
∂Ek is smooth
for every k;(2)
Ek⊂⊂E for every k;(3)
χEk→χE in Lloc1(ℝn) as k→+∞;(4)
Per(Ek)→Per(E) as k→+∞.Proposition 2.2.
The following equalities hold:(2.6)infE⊂ΩPer(E)V(E)=infE⊂⊂ΩPer(E)V(E)=infE⊂⊂Ω∂EsmoothPer(E)V(E).Proof.
It is clear that(2.7)infE⊂ΩPer(E)V(E)≤infE⊂⊂ΩPer(E)V(E)≤infE⊂⊂Ω∂EsmoothPer(E)V(E).
Let F be a Cheeger set for Ω; by Proposition 2.1 we can approximate F with a sequence of smooth sets Fk⊂⊂F such that Per(Fk)→Per(F) and V(Fk)→V(F). This yields
(2.8)infE⊂⊂Ω∂EsmoothPer(E)V(E)≤Per(F)V(F)=infE⊂ΩPer(E)V(E),
so that the claim is proved.In the following we will mention some geometric properties of Cheeger sets.Proposition 2.3.
LetE be a Cheeger set for Ω; then ∂E∩∂Ω≠∅.Proof.
Let us suppose that this is not the case. ThenE is compactly contained in Ω, which means that there exists a number λ>1 such that the set λE={λx∣x∈E} is contained in Ω. But then
(2.9)Per(λE)V(λE)=1λPer(E)V(E)<Per(E)V(E)
which contradicts the fact that E is a Cheeger set.Proposition 2.4.
LetE be a Cheeger set for Ω; then  (1)
∂E∩Ω is analytical, up to a singular set of Hausdorff dimension n-8;(2)
the mean curvature in every regular point of∂E∩Ω is equal to h1(Ω);(3)
letx∈∂E∩∂Ω be a regular point for ∂Ω; then x is a regular point for ∂E.Proof.
The proof can be found in [5]. As a consequence, the boundary of E must touch the boundary of Ω tangentially.Proposition 2.5.
LetΩ⊂ℝn be a convex domain. Then there exists an unique Cheeger set E for Ω. Moreover, E is convex.Proof.
A proof of the existence of a convex Cheeger set can be found in [2, Remark 10]. Uniqueness has been proved in [6] for the case n=2, and in [7] for general n.Remark 2.6.
Ifn=2 and Ω is convex, then according to [6] the Cheeger set is the union of balls of suitable radius contained in Ω (where “suitable” means in this case equal to h1(Ω)-1). It seems that the hypothesis of convexity cannot be dropped; there are examples of star-shaped domains which admit infinitely many Cheeger sets (see [8]). However, it was proved that “almost all” bounded domains admit a unique Cheeger set (see [9]).We will often make use of the following property.Proposition 2.7.
LetΩ⊂ℝn, and let B⊂ℝn be a ball such that |B|=|Ω|. Then
(2.10)h1(B)≤h1(Ω).Proof.
The proof is a consequence of the well-known isoperimetric property of the ball (see e.g., [10]).Remark 2.8.
There are some domains whose Cheeger set coincides with the wholeΩ; this is of course the case of balls, but also of annuli and other domains satisfying a condition on the curvature of the boundary (see [6]).


## 3. Higher Cheeger Constants

LetΩ⊂ℝn be an open bounded domain with Lipschitz boundary. We define, for k∈ℕ, (3.1)hk(Ω)=inf{λ∈ℝ+∣∃E1,…,Ek⊂Ω,Ei∩Ej=∅fori≠j,maxi=1,…,kPer(Ei)V(Ei)≤λ},
with the convention that (3.2)Per(E)V(E)=+∞
whenever V(E)=0. We will call hk(Ω) the kthCheeger constant for Ω. Notice that, for k=1, we recover the definition of the Cheeger constant h1(Ω). By Proposition 2.1 it is possible to take the infimum on sets compactly contained in Ω, or even on sets compactly contained in Ω with smooth boundary.Theorem 3.1.
For everyk, there exist k pairwise disjoint subsets E1,…,Ek contained in Ω such that
(3.3)maxi=1,…,kPer(Ei)V(Ei)≤hk(Ω).Proof.
Let us consider minimizing sequences of pairwise disjoint setsE1,n,…,Ek,n for n=1,2,…, corresponding to the value μn, where
(3.4)μn=maxi=1,…,kPer(Ei,n)V(Ei,n).
Set χi,n=χEi,n for i=1,…,k. Fix R as the radius of k equal disjoint balls of fixed arbitrary volume V0>0 contained in Ω. We are going to show that we can consider V(Ei,n)≥V0 for every i,n. Indeed, if we had V(Eî,n̂)<V0 for some values of î and n̂, then by Proposition 2.7 we would surely have
(3.5)Per(Eî,n̂)V(Eî,n̂)≥h1(Br),
where Br is a ball with the same volume as V(Eî,n̂) and so of radius r<R. As a consequence, μn̂>h1(BR), which means that we can actually discard the k-tuple of sets E1,n̂,…,Ek,n̂. Because of the compact embedding of BV(Ω) in L1(Ω), there exist E1,…,Ek such that, up to a subsequence, χi,n→χEi almost everywhere on Ω. Moreover, V(Ei)≥V0>0. Denote with N the negligible set of nonconvergence. From the lower semicontinuity of the total variation, it follows that
(3.6)Per(Ei)V(Ei)≤hk(Ω)
for every i=1,…,k. We are going to show that the Ei are pairwise disjoint: suppose i≠j, then x∈Ei∖N⇒χEi(x)=1, which implies χi,n(x)=1 definitely. This means χj,n(x)=0 definitely, hence χEj(x)=0, that is, x∉Ej∖N. If x∈N, we can assign arbitrary values to the characteristic functions (this does not affect the total variation). Hence we obtain the claim.Definition 3.2.
Anyk-tuple of sets E1,…,Ek as in Theorem 3.1 will be called a k-tuple of multiple Cheeger sets. If k=2, we will also speak of coupled Cheeger sets.Remark 3.3.
The proof of the theorem shows that we can always consider a minimizing sequence ofk-tuples of sets for hk(Ω), where the volumes of the sets are uniformly bounded from below.Remark 3.4.
Proceeding as in Proposition2.3, one can show that at least one of the minimizing sets must touch the boundary.In the following we will give a different characterization of the higher Cheeger constants.Proposition 3.5.
Let𝒫k be the set of all partitions of Ω with k subsets E1,…,Ek. Then
(3.7)hk(Ω)=inf𝒫kmaxi=1,…,kh1(Ei).Proof.
Setĥk(Ω)=inf𝒫kmaxi=1,…,kh1(Ei). Let us suppose ĥk(Ω)<hk(Ω); then there exists a partition E1,…,Ek of Ω such that
(3.8)maxi=1,…,kh1(Ei)<hk(Ω)
which is a contradiction. Thus ĥk(Ω)≥hk(Ω). On the other hand, if C1,…,Ck are a k-tuple of multiple Cheeger sets (which exist by Theorem 3.1), we can find a partition E1,…,Ek of Ω with the property that Ci⊂Ei for every i=1,…,k. Hence, for every i,
(3.9)h1(Ei)≤Per(Ci)V(Ci)≤hk(Ω),
and consequently
(3.10)maxi=1,…,kh1(Ei)≤hk(Ω),
that is,
(3.11)ĥk(Ω)≤hk(Ω)
which finally yields
(3.12)ĥk(Ω)=hk(Ω).Remark 3.6.
The sets realizinghk(Ω) can be supposed to be connected. Indeed, if E=E1∪E2, with E¯1∩E¯2=∅, we have
(3.13)Per(E)V(E)=Per(E1)+Per(E2)V(E1)+V(E2)≥min{Per(E1)V(E1),Per(E2)V(E2)}.
This follows from the fact that, for any a, b, c, d>0,
(3.14)ac≤a+bc+d≤bd⇔ac≤bd.
So one connected component of E has a lower or equal ratio perimeter/area. If E1∩E2=∅, but E¯1∩E¯2≠∅, we modify E on a set of measure zero (this does not affect the total variation) to get a connected set E′ defined as
(3.15)E′=E1∪E2∪(∂E1∩∂E2).Theorem 3.7.
It is possible to find multiple Cheeger sets such that the part of their boundary contained inΩ is a piecewise smooth hypersurface of piecewise constant mean curvature.Proof.
We will give the proof for the casek=2. Let E1 and E2 be two coupled Cheeger sets, which exist according to Theorem 3.1. Since E1 minimizes perimeter (measured in ℝn) in Ω∖E2¯ with a volume constraint, it will have interior regularity according to [5]. More precisely, ∂E1∩(Ω∖E2¯) is an analytic hypersurface up to a singular set with Hausdorff dimension n-8, whose regular points have constant mean curvature. The same can be stated for E2. Then we have to consider the possibly nonempty contact surface; also in this case, [5, Theorem 2] can be applied to state that the contact surface (if it exists) enjoys the same regularity as the interior boundary of the two sets and has constant mean curvature.Definition 3.8.
LetE1 and E2 be a pair of coupled Cheeger sets. The free boundary of E1 is defined as ∂E1∩(Ω∖E2¯) (analogously for E2). The contact surface between E1 and E2 is ∂E1∩∂E2∩Ω.Theorem 3.9.
It is possible to find two coupled Cheeger setsE1 and E2 such that the following holds. Suppose that ∂E1∩∂E2≠∅. Let us denote by c1 the mean curvature of the free boundary of E1, by c2 the mean curvature of the free boundary of E2, and by c3 the mean curvature of the contact surface, measured from E1. Then the relation
(3.16)c1-c2-2c3=0
holds.Proof.
We follow [11, pages 10-11]. Take x1∈(∂E1∖∂E2)∩Ω, x2∈(∂E2∖∂E1)∩Ω, and x3∈∂E1∩∂E2∩Ω. Suppose that the boundaries of E1 and E2 can be locally described by the graph of a function u defined in an open subset ω=ω1∪ω2∪ω3 of ℝn-1, where ω1, ω2, and ω3 are disjoint open neighborhoods of x1, x2, and x3, respectively. For i=1,2,3, let vi be a function defined in ωi with compact support and such that the following conditions are satisfied:
(3.17)∫ω1v1+∫ω3v3=0,∫ω2v2-∫ω3v3=0.
Since E1 and E2 are coupled Cheeger sets, we can suppose that u is such that
(3.18)∫ω1∪ω31+|∇u|2≤∫ω1∪ω31+|∇u+ε∇(v1+v3)|2,∫ω2∪ω31+|∇u|2≤∫ω2∪ω31+|∇u+ε∇(v2+v3)|2
for small ε>0. It follows that
(3.19)0≤∫ω1∇u∇v11+|∇u|2+∫ω2∇u∇v21+|∇u|2+2∫ω3∇u∇v31+|∇u|2=-∫ω1div(∇u1+|∇u|2)v1-∫ω2div(∇u1+|∇u|2)v2-2∫ω3div(∇u1+|∇u|2)v3=-c1∫ω1v1-c2∫ω2v2-2c3∫ω3v3.
Since also the functions -v1, -v2, and -v3 are admissible, it follows that
(3.20)c1∫ω1v1+c2∫ω2v2+2c3∫ω3v3=0
for arbitrary v1, v2, v3 satisfying conditions (3.17); hence we obtain
(3.21)c1-c2-2c3=0.Remark 3.10.
The condition on the mean curvatures is similar to the one given in [12] for the double bubble problem: find two regions in ℝn which enclose two given amounts of volume, such that they minimize the sum of the surface measures. However, in that problem the quantity to minimize is slightly different, so also the condition on the mean curvatures differs and reads c1-c2-c3=0.Proposition 3.11.
LetΩ⊂ℝ2 be a convex planar domain; then it is possible to find two coupled Cheeger sets E1, E2 such that they satisfy condition (3.16) in Theorem 3.9 and such that, if ∂E1∩∂E2≠∅, then their boundaries meet tangentially.Proof.
We can suppose thatc1, c2≥0; otherwise, since Ω is convex, it would be possible to modify the sets suitably in order to decrease their perimeter and increase their volume. As a consequence, at least one of the two sets (say E1) is convex. Let us suppose that ∂E1 and ∂E2 meet each other in a nonsmooth way. Then one could consider the Cheeger set C1 of E1, which is convex and has a 𝒞1 boundary, and then find a perimeter-minimizing set C2 in Ω∖C¯1 under the volume constraint |C2|=|E2|. The boundaries ∂C1 and ∂C2 will meet tangentially as proved in [5]. Then one can apply again Theorem 3.9 to get the condition on the curvatures.Proposition 3.12.
LetΩ⊂ℝn admit a unique Cheeger set. Then
(3.22)h1(Ω)<h2(Ω).Proof.
Let us suppose thath1(Ω)=h2(Ω); then there exist two disjoint subsets C1, C2⊂Ω such that
(3.23)max{Per(C1)V(C1),Per(C2)V(C2)}=h1(Ω)
which means, by definition of h1(Ω),
(3.24)Per(C1)V(C1)=Per(C2)V(C2)=h1(Ω).
This is a contradiction to the uniqueness of the Cheeger set for Ω.Remark 3.13.
It is worth noting that there exist nonconvex domains for whichh1(Ω)=h2(Ω); think, for example, of a “barbell domain” made of two identical rectangles connected by a thin rope. For instance, we could consider the planar set
(3.25)Ω={(0,1)×(0,2)}∪{[1,2]×(0,ε)}∪{(2,3)×(0,2)},
where ε>0 is small enough.Proposition 3.14.
Let us denote byωn the volume of the unit ball in ℝn. Then
(3.26)hk(Ω)≥n(kωn|Ω|)1/n.Proof.
LetE1,…,Ek be a family of multiple Cheeger sets, so that
(3.27)maxi=1,…,kh1(Ei)≤hk(Ω).
The volume of each Ei cannot be smaller than the volume of a ball with Cheeger constant hk(Ω), which is exactly ωn(n/hk(Ω))n. In fact, let B̃ be a ball such that |Ei|=|B̃|, and B a ball such that h1(B)=hk(Ω); if |B̃|<|B| we would have, applying Proposition 2.7,
(3.28)hk(Ω)=h1(B)<h1(B̃)≤h1(Ei)≤hk(Ω)
which is a contradiction. So we obtain
(3.29)kωn(nhk(Ω))n≤|Ω|⇒hk(Ω)≥n(kωn|Ω|)1/n.Corollary 3.15.
It holds(3.30)hk(Ω)→+∞
as k→+∞.Remark 3.16.
The lower bound in Proposition3.14 for k=1 follows directly from Proposition 2.7, and is obviously optimal if Ω is a ball. For the higher Cheeger constants, it can be easily seen that the estimate is optimal for the union of k balls with equal radii. If we try to minimize hk(Ω) among connected sets, it turns out that the infimum is the same (consider a family of k balls of equal radii connected by thin strips whose width goes to 0). An interesting question would be to minimize hk(Ω) among convex sets. If we focus on h2(Ω), it seems that a stadium (the convex hull of two tangent balls with both radii equal to R) is very near to be a minimizer; namely, it is possible to show that
(3.31)1.874R≤h2(Ω)≤1.912R.
The lower bound follows directly from Proposition 3.14. To obtain the upper bound, one can note that the common tangent divides Ω into two equal convex halves, whose Cheeger set E is given by the union of balls of constant radius x≤R. E satisfies then the conditions
(3.32)Per(E)=4R+πR-4x+πx,V(E)=12πR2+2R2-2x2+12πx2,
and since it must be
(3.33)Per(E)V(E)=1x,
we get x=0.523R. This yields the estimate from above. However, it should be mentioned that the stadium does not minimize the second eigenvalue of the Laplacian among convex planar domains, as proved in [13].


## 4. Coupled Cheeger Sets for a Disc

In this section we will determine the coupled Cheeger sets of a discΩ⊂ℝ2 with radius r. As a first step we will compute the Cheeger set E for a half-disc Ω′ of same radius. According to the results in Section 2, the Cheeger set must have the geometry shown in Figure 1.Figure 1
The candidate Cheeger set for a half-disc.We will denote byα the inner angle and by x the radius of the inner arc. Thus we have the relation (4.1)(r-x)sinα=x
which gives the existence condition 0≤x≤r/2. Then (4.2)Per(E)=2(r-x)cosα+2x(π2+α)+r(π-2α),V(E)=x(r-x)cosα+x2(π2+α)+r22(π-2α).
Remember that α=arcsin(x/(r-x)) and cosα=1-(x/(r-x))2, since we consider 0≤α≤π/2. Numerical resolution of the equation (4.3)Per(E)V(E)=1x(=possibleh1(Ω′))
gives, for r=1, (4.4)x=0.317028…
which means (4.5)h1(Ω′)=3.15429…This is the best configuration with convex subsets to computeh2(Ω); indeed, a convex partition of a convex set can be obtained only cutting the set with hyperplanes (otherwise we would have a point of nonzero curvature which gives convexity from one side but concavity from the other one). The Cheeger sets of each of the two partitioning subsets are then convex. Conversely, two convex subsets can be separated by a hyperplane thanks to the Hahn-Banach theorem. The Cheeger constant of a circular segment strictly contained in a half-disc is then strictly higher, due to uniqueness reasons. So the above configuration is the best among convex subsets of the disc.Observe that the two coupled Cheeger setsE1 and E2 must have a contact surface. If it was not the case, we can suppose without loss of generality that (4.6)Per(E1)V(E1)≤Per(E2)V(E2),
and that E1 is a Cheeger set for Ω∖E2. Notice that E2 is automatically a Cheeger set for Ω∖E1. Due to the properties of Cheeger sets, the free boundaries of E1 and E2 must be circular arcs which meet ∂Ω tangentially. The only possibility is that E1 and E2 are discs, and the best configuration is given by to equal discs with radius r/2, which is clearly not optimal for Ω.We are now going to prove that the contact surface cannot be closed; if it was the case, then one of the two coupled Cheeger sets, sayE1, would be a disc of radius r′<r, as in Figure 2. The other set E2 will be then contained in Ω∖E1. Suppose that E2 has a free boundary consisting of arcs with constant curvature c2≥0. An easy computation shows that the case c2=0 is never optimal; so we can suppose that the arcs have constant curvature c2>0. Due to the fact that ∂E1 is the contact surface, these arcs cannot start on ∂Ω and end on ∂E1; the only possibility is that the free boundary “encloses” E1 as the dashed line in Figure 2. But in this case, the choice E2=Ω∖E1 would give a lower ratio perimeter/area. So the optimal choice is the pair consisting of E1 and its complement. By modifying r′ suitably, one can easily convince himself that the optimal configuration is achieved when the ratios perimeter/area of E1 and E2 are equal. This implies (4.7)Per(E1)V(E1)=Per(E2)V(E2)⇒2r′=2r-r′⇒r′=r2
which yields, for r=1, (4.8)h1(E1)=h1(E2)=4.
This gives a worse configuration than the one found before. As a consequence, the contact surface cannot be a closed line.Figure 2
The contact surface cannot be closed.We will now use the regularity results about the coupled Cheeger sets; in particular, by Proposition3.11 we can suppose that the boundary of each of the two sets meets the boundary of the other set tangentially. Suppose that the separating surface is an arc PQ with constant curvature c3. From the point P two arcs of curvature c1 and c2, respectively, will depart, in such a way that the centres of curvature lie on the chord AB orthogonal to PQ and such that P∈AB. Notice that we can suppose, without loss of generality, that c1, c2≥0.LetE1 be the “candidate” Cheeger set containing the segment AP and such that the curvature of its free boundary is c1; let E2 be the set containing the segment PB and with curvature of the free boundary equal to c2. Without loss of generality, we can suppose that AP¯≤PB¯. Let M be the middle point of the segment AB. If P≠M, it is impossible that c3≥0 (as in Figure 3); indeed, since E1 would be a subset of a circular segment strictly contained in a half-disc, this would contradict the fact that the configuration of the Cheeger sets of the two half-discs is better. So it must be c3<0.Figure 3
The casec3>0.LetC and D the centers of curvature of the free boundaries of E1 and E2, respectively, and E, F as in Figure 4 such that CP¯=EC¯ and PD¯=DF¯. Since c3<0, from Theorem 3.9 it must be c1<c2, that is PC¯>PD¯. This is impossible for geometrical reasons; indeed, take a point C′ on AB such that AC¯=C′B¯; it follows PC′¯>PC¯>PD¯, which means that the point D must lie between P and C′. If E′ is the intersection of the circle with the line OC′, it is clear that DF¯>C′E′¯. This is a contradiction because we would have C′E′¯=CE¯>DF¯>C′E′¯.Figure 4
The casec3<0.Figure 5
Second eigenfunction in a planar disc forp=1.1. The value of λ2(p;Ω) is about 4.199 (picture courtesy of Jiří Horák).It follows that necessarilyP=M. For symmetry reasons, this implies c1=c2 and hence, again from Theorem 3.9, c3=0. So we recover the optimal configuration consisting of the Cheeger sets of the two half-discs.


## 5. The Second Eigenvalue asp→1

Let us consider now the eigenvalue problem (1.1). The natural question which arises is how one can find eigenvalues of the p-Laplacian. A possibility is to use the direct method of Calculus of Variations by minimizing the so-called Rayleigh quotient; working this way we obtain the first eigenvalue (5.1)λ1(p;Ω)=infu∈W01,p(Ω)∖{0}∫Ω|∇u|p∫Ω|u|p.
One can prove (see e.g., [14]) that there exists, up to a nonzero multiplicative constant, one and only one eigenfunction e1,p associated to λ1(p;Ω). Moreover, e1,p is of only one sign.We will now describe how higher eigenvalues of thep-Laplacian can be obtained. Let A⊂W01,p(Ω) be a closed, symmetric subset. The Krasnoselskii genusγ(A) of A is defined as (5.2)γ(A)=min{m∈ℕ∣∃φ:A→ℝm∖{0},φiscontinuousandodd}.
Let us denote by Γk the set (5.3)Γk={A⊂W01,p(Ω)∖{0}∣A∩{∥u∥p=1}iscompact,Asymmetric,γ(A)≥k}.
Then, for every k∈ℕ, the following numbers are eigenvalues: (5.4)λk(p;Ω)=infA∈Γkmaxu∈A∫Ω|∇u|p∫Ω|u|p.
In the literature they are sometimes called variational eigenvalues. It can be easily seen that the two definitions of λ1(p;Ω) given so far coincide. It is still an open question whether other eigenvalues can exist. As shown, for instance, in [15, Lemma 3.1], eigenfunctions associated to higher eigenvalues of the p-Laplacian must be sign-changing. Moreover, there does not exist any eigenvalue between λ1(p;Ω) and λ2(p;Ω), which means that λ1(p;Ω) is isolated (see [16]).Anodal domain of a function u:Ω→ℝ is a connected component of the set {x∈Ω|u(x)≠0}. It is not known whether the zero set of an eigenfunction of the p-Laplacian has Hausdorff dimension n-1, or if it can be even an open subset. A generalization of Courant's nodal domain theorem states that every eigenfunction associated to λk(p;Ω) has at most 2k-2 nodal domains ([17,Theorem 3.3]). As an easy consequence it follows that any second eigenfunction has exactly two nodal domains.We are now ready to prove the main result of this paper.Lemma 5.1.
LetE⊂ℝn be a set with Lipschitz boundary, and let Eε be the ε-strip around E defined as
(5.5)Eε={x∈ℝn∖E∣dist(x,E)≤ε}.
Then
(5.6)V(Eε)=εPer(E)+o(ε),
where (o(ε))/ε→0 as ε→0.Proof.
The proof can be found in [18].Theorem 5.2.
It holds(5.7)lim supp→1λ2(p;Ω)≤h2(Ω).Proof.
LetC1, C2⊂⊂Ω be two subsets such that C1∩C2=∅, and
(5.8)max{Per(C1)V(C1),Per(C2)V(C2)}≤h2(Ω)+12k.
By Proposition 2.1 it is possible to find E1, E2 with the property that, for i=1,2, Ei⊂⊂Ci, ∂Ei is smooth, and
(5.9)Per(Ei)V(Ei)≤h2(Ω)+1k.
Let ε>0, and let vi (i=1,2) be two functions such that: vi=1 on Ei, vi=0 outside a ε-neighbourhood of Ei, and |∇vi|=ε-1 on the ε-layer Eiε outside Ei. ε should be chosen in a way that E1ε∩E2ε=∅. Set
(5.10)A0={αv1+βv2∣|α|p+|β|p=1}.
Then A0∈Γ2 (see also [19, Lemma 2.1]). Thus we have
(5.11)λ2(p;Ω)≤supu∈A0∫Ω|∇u|p∫Ω|u|p≤sup|α|p+|β|p=1ε-p|α|pV(E1ε)+ε-p|β|pV(E2ε)|α|pV(E1)+|β|pV(E2)=sup|α|p+|β|p=1ε1-p|α|pPer(E1)+ε1-p|β|pPer(E2)+ε-po(ε)|α|pV(E1)+|β|pV(E2)≤ε1-p(h2(Ω)+1k)+ε-po(ε)min{V(E1),V(E2)},
as we have from Lemma 5.1(5.12)V(Eiε)=εPer(Ei)+o(ε),
where (o(ε))/ε→0 as ε→0. The last inequality follows from (3.14). If we send p→1, we obtain
(5.13)lim supp→1λ2(p;Ω)≤h2(Ω)+1k+ε-1o(ε)min{V(E1),V(E2)},
and if ε→0(5.14)lim supp→1λ2(p;Ω)≤h2(Ω)+1k.
The claim follows if we send k→∞. The fact that E1 and E2 depend on k does not constitute a problem, since we can estimate V(Ei) uniformly from below (see Remark 3.3).Remark 5.3.
The theorem can be easily generalized to thekth variational eigenvalue obtaining
(5.15)lim supp→1λk(p;Ω)≤hk(Ω).The so-calledCheeger's inequality, whose proof can be found in [20], yields the following lower bound for the first eigenvalue:(5.16)λ1(p;Ω)≥(h1(Ω)p)p.In the following theorem we show that a similar estimate holds also for the second eigenvalue.Theorem 5.4.
The following Cheeger-type inequality holds:(5.17)λ2(p;Ω)≥(h2(Ω)p)p.Proof.
Lete2,p be a second eigenfunction of the p-Laplacian. From [17, Theorem 3.3], we know that e2,p has exactly two nodal domains N1,p, N2,p. e2,p is also a first eigenfunction on each of the two nodal domains; from Cheeger's inequality it follows, for i=1,2,
(5.18)λ2(p;Ω)=λ1(p;Ni,p)≥(h1(Ni,p)p)p.
But as N1,p∩N2,p=∅, we have
(5.19)max{h1(N1,p),h1(N2,p)}≥h2(Ω)
due to the definition of h2(Ω). So we obtain the claim.Theorem 5.5.
It holds(5.20)limp→1λ2(p;Ω)=h2(Ω).Proof.
The claim follows easily from Theorems5.2 and 5.4.The Second Eigenfunction asp→1
In the following we will focus on the asymptotic behaviour of the second eigenfunctions asp→1.Theorem 5.6.
LetN1,p, N2,p be the nodal domains of a second eigenfunction of the p-Laplacian. Then
(5.21)limp→1max{h1(N1,p),h1(N2,p)}=h2(Ω).Proof.
By definition ofh2(Ω) we have
(5.22)h2(Ω)≤max{h1(N1,p),h1(N2,p)}.
It remains to prove that for every ε>0, there exists p0>1 such that for every 1<p<p0,
(5.23)max{h1(N1,p),h1(N2,p)}≤h2(Ω)+ε.
Suppose that this is not the case; then there exists ε>0 such that, without loss of generality, h1(N1,pk)>h2(Ω)+ε for a subsequence pk→1. From Cheeger's inequality
(5.24)λ2(pk)≥(h1(N1,pk)pk)pk>(h2(Ω)+εpk)pk>h2(Ω)+ε2
for k large enough. But this contradicts the fact that limp→1λ2(p;Ω)=h2(Ω). Hence the claim follows.Corollary 5.7.
Forp→1, the volume of each of the nodal sets is uniformly bounded from below by ωn(n/2h2(Ω))n.Proof.
From the preceding theorem there existsp0>1 such that, for every 1<p<p0,
(5.25)max{h1(N1,p),h1(N2,p)}≤2h2(Ω).
Arguing as in Proposition 3.14, the volume of the nodal sets cannot be smaller than the volume of a ball with Cheeger constant 2h2(Ω), which is exactly ωn(n/2h2(Ω))n. Thus, for i=1,2,
(5.26)|Ni,p|≥|B|=ωn(n2h2(Ω))n
as claimed.We are now going to investigate the asymptotic behaviour of the second eigenfunctions asp→1. First, we state some technical lemmas.Lemma 5.8.
LetΩ⊂ℝn be a bounded set with Lipschitz boundary, pj→1 as j→∞ (pj≥1), uj∈W01,pj(Ω) for every j, uj→u in L1(Ω) as j→∞. Then
(5.27)∥Du∥(ℝn)≤lim infj→∞∫Ω|∇uj|pj.Proof.
Since∂Ω is Lipschitz, the functions uj are in particular in BV(ℝn). Let us denote by pj′ the exponent conjugate to pj; by (2.2), Hölder's inequality and Young's inequality we have
(5.28)∥Du∥(ℝn)≤lim infj→∞∥Duj∥(ℝn)=lim infj→∞∫Ω|∇uj|≤lim infj→∞(∫Ω|∇uj|pj)1/pj|Ω|1/pj′≤lim infj→∞[(∫Ω|∇uj|pj)+|Ω|·pj-pj′/pjpj′]≤lim infj→∞∫Ω|∇uj|pj+lim supj→∞|Ω|·pj-pj′/pjpj′=lim infj→∞∫Ω|∇uj|pj.Lemma 5.9.
LetΩ⊂ℝn be a bounded set, pj→1 as j→∞ (pj≥1), 0<∥uj∥L∞(Ω)≤c for every j (c>0), u∈L1(Ω), and uj→u in L1(Ω) as j→∞. Then
(5.29)limj→∞∫Ω|uj|pj=∫Ω|u|.Proof.
Let us denote bypj′ the exponent conjugate to pj. By Hölder's inequality and Young's inequality we have
(5.30)∫Ω|u|=limj→∞∫Ω|uj|≤lim infj→∞(∫Ω|uj|pj)1/pj|Ω|1/pj′≤lim infj→∞[(∫Ω|uj|pj)+|Ω|·pj-pj′/pjpj′]=lim infj→∞∫Ω|uj|pj+lim supj→∞|Ω|·pj-pj′/pjpj′=lim infj→∞∫Ω|uj|pj.
On the other hand from0<∥uj∥L∞(Ω)≤c and pj≥1 we have(5.31)∫Ω|uj|∥uj∥∞≥∫Ω(|uj|∥uj∥∞)pj,
so that
(5.32)∫Ω|u|=limj→∞∫Ω|uj|≥lim supj→∞∥uj∥∞1-pj·∫Ω|uj|pj≥lim supj→∞∫Ω|uj|pj.
The last equation and (5.30) end the proof.Lemma 5.10.
Lete2,p be a second eigenfunction of the p-Laplacian. Then
(5.33)∥e2,p∥∞≤4n·λ2(p;Ω)n/p·∥e2,p∥1.Proof.
The proof can be found in [21].Theorem 5.11.
Lete2,p be second eigenfunctions of the p-Laplacian such that ||e2,p||p=1. Then (after possibly passing to a subsequence) e2,p converge, as p→1, in L1(Ω) and hence pointwise a.e. to a function u∈BV(Ω) such that ||u||1=1 and ||Du||(ℝn)≤h2(Ω). Moreover, u cannot be strictly positive or strictly negative.Proof.
From Theorem5.5, Lemma 5.10, and Hölder's inequality, e2,p are uniformly bounded in L∞(Ω). Moreover, we have
(5.34)∥De2,p∥(ℝn)=∫Ω|∇e2,p|≤(∫Ω|∇e2,p|p)1/p|Ω|1/p′=λ2(p;Ω)1/p·|Ω|1/p′,
where p′ is the exponent conjugate to p. Since λ2(p;Ω)→h2(Ω), the functions are uniformly bounded in BV(Ω); hence there exists a subsequence converging in L1(Ω) to a function u∈BV(Ω). From Lemma 5.8 we have
(5.35)∥Du∥(ℝn)≤lim infp→1λ2(p;Ω)=h2(Ω).
Finally, Lemma 5.9 yields ||u||1=1.
The fact thatu cannot be strictly positive or strictly negative is a consequence of Corollary 5.7.


## 6. Nonradiality of the Second Eigenfunction in a Planar Disc

In this section we will apply the previously found results to the particular case where the domainΩ⊂ℝ2 is a disc, in order to establish whether a second eigenfunction can be radial or not. Let us recall that the existence of radial eigenfunctions was shown in [22]; in this case, one has to solve the ordinary differential equation(6.1)-(r|u′|p-2u′)′=λr|u|p-2uin(0,R),u′(0)=0,u(R)=0,The question seems to be still an open problem in its full generality, except for the casep=2 (see [23]) where the answer is negative. In the following theorem it is shown that the answer is also negative if p is sufficiently close to 1.Theorem 6.1.
LetΩ=B1(0)⊂ℝ2 be the unit disc. Then, for p close to 1, the second eigenfunction of the p-Laplacian in Ω cannot be radial.Proof.
From the results in Section4 and Theorem 5.5 we have
(6.2)limp→1λ2(p;Ω)=3.15429….
Then there exists p0>1 such that
(6.3)λ2(p;Ω)≤3.5
for 1<p<p0. Let us suppose that there exists p∈(1,p0) such that a second eigenfunction of the p-Laplacian is radial; this implies that its nodal domains are a disc Br(0) of radius r (0<r<1), compactly contained in Ω, and an annulus A=Ω∖Br(0). If we suppose w.l.o.g. p<1.1, Cheeger's inequality allows us to state that
(6.4)λ2(p;Ω)≥(h1(Br(0))p)p=(2rp)p≥(1.818r)p≥1.818r,λ2(p;Ω)≥(h1(A)p)p=(2(1-r)p)p≥(1.8181-r)p≥1.8181-r.
Indeed, since the Cheeger set of A is A itself (see Remark 2.8), one has
(6.5)h1(A)=Per(A)V(A)=2π(1+r)π(1-r2)=21-r.
Then we have the following compatibility conditions:
(6.6)1.818r≤3.5⇒r≥0.519,1.8181-r≤3.5⇒1-r≥0.519⇒r≤0.481
which are incompatible. Hence we obtain the claim.The following image was obtained by an implementation of the numerical method described in [24].



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*Source: 984671-2009-12-15.xml*