Reverse Faber-Krahn inequality for a truncated laplacian operator
Enea Parini Julio Rossi Ariel Salort
math.AP


 In this paper we prove a reverse Faber-Krahn inequality for the principal eigenvalue μ_1(Ω) of the fully nonlinear eigenvalue problem 
    {[ -λ_N(D^2 u)           =         μ u       in Ω,;           u           =           0      on ∂Ω. ].

Here λ_N(D^2 u) stands for the largest eigenvalue of the Hessian matrix of u. More precisely, we prove that, for an open, bounded, convex domain Ω⊂^N, the inequality

    μ_1(Ω) ≤π^2/[diam(Ω)]^2 = μ_1(B_diam(Ω)/2),

where diam(Ω) is the diameter of Ω, holds true. 
The inequality actually implies a stronger result, namely, the maximality of the ball under a diameter constraint. 

Furthermore, we discuss the minimization of μ_1(Ω) under different kinds of constraints.


Correlation between UV resilience and wavelength of narrow diffuse interstellar bands
    A. Omont1
H. F. Bettinger2

    December 30, 2023
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§ INTRODUCTION

Given a bounded domain Ω⊂^N, we consider the fully nonlinear eigenvalue problem 

    {[ -λ_N(D^2 u)  =  μ u               in Ω,;             u  =  0              on ∂Ω, ].

where μ∈ is the eigenvalue, D^2 u is the Hessian matrix of a function u: Ω→, and λ_1(D^2 u),…,λ_N(D^2 u) are the ordered eigenvalues of D^2 u, so that

    λ_1(D^2 u) ≤…≤λ_N(D^2 u).

 
Following the celebrated approach by Berestycki, Varadhan, and Nirenberg <cit.>, it is possible to define the first eigenvalue μ_1(Ω) as

    μ_1(Ω) := sup{μ∈∃φ∈ LSC(Ω), φ > 0  in Ω s.t.  -λ_N(D^2 φ) ≥μφ},

where LSC(Ω) is the set of lower semicontinuous functions in Ω, and the inequality -λ_N(D^2 φ) ≥μφ is understood in the viscosity sense.
When the domain Ω is assumed to be strictly convex, the existence of a strictly positive eigenfunction associated to μ_1(Ω) 
has been proven in <cit.>. It is still unknown whether the assumptions on the domain can be weakened to (not strict) convexity; however, it should be observed that, if u is a positive eigenfunction, then it holds, in the viscosity sense,
 
    λ_1(D^2u) ≤…≤λ_N(D^2 u) ≤ 0,

which implies the negative semidefiniteness of D^2 u, and hence the concavity of u. The fact that Ω = {u > 0} implies the convexity of Ω. For this reason, unless otherwise stated, we will always suppose that Ω is a convex domain.


The operator 𝒫_N^+ (D^2 u):= _N(D^2 u) belongs to the class of so-called truncated Laplacian operators, 
which are defined as the sum of a finite number of consecutive eigenvalues of D^2 u. More precisely, this class of operators is defined, 
for k=1,…,N, as

    𝒫_k^+ (D^2 u) := ∑_i=N-k+1^Nλ_i(D^2 u),    𝒫_k^- (D^2u) := ∑_i=1^kλ_i(D^2 u)

and received some attention recently, see <cit.>.
These highly degenerate elliptic operators appeared first in the context of differential geometry, see <cit.>

A natural question arising in spectral theory is the following: which domain minimizes or maximizes the first eigenvalue of a differential operator, under some geometrical constraint on the domain? A classical example is given by the Laplacian operator

    Δ u = tr  (D^2 u)= ∑_i=1^N _i(D^2 u).

In this case, the well-known Rayleigh-Faber-Krahn inequality states that the first eigenvalue μ_1^Δ of -Δ under Dirichlet boundary conditions is uniquely minimized, among domains with fixed volume, by the ball. More precisely, if Ω⊂^N, and B_r is a ball of radius r such that |Ω|=|B_r|, then

    μ_1^Δ(B_r) ≤μ_1^Δ(Ω),

and equality holds if and only if Ω=B_r.

However, if we consider the Monge-Ampère operator, which is the fully nonlinear operator defined as

    det(D^2 u) = ∏_i=1^N λ_i(D^2 u),

the situation is completely different. Indeed, among all bounded, convex domains with fixed volume, the (unique) eigenvalue is maximized by the ball <cit.>, while it is conjectured that the N-dimensional regular simplex is a minimizer. 

Concerning the truncated Laplacian operator 𝒫_N^+, Birindelli, Galise and Ishii conjectured in <cit.> the validity of a reverse Faber-Krahn inequality, namely, that the ball maximizes μ_1(Ω) among convex domains with fixed volume. The conjecture was supported by some partial results proven in <cit.>: 


  * The hypercube has the largest first eigenvalue among hyperrectangles of given measure; 

  * The ball has a larger first eigenvalue than the hypercube with the same volume.


In this paper we prove the validity of the conjectured reverse Faber-Krahn inequality. 
More precisely, we first prove 
the following theorem. 

 
For an open, bounded, convex domain Ω⊂^N, the inequality

    μ_1(Ω) ≤π^2/[diam(Ω)]^2,

where diam(Ω) is the diameter of Ω, holds true.


Theorem <ref> readily implies that the ball maximizes μ_1(Ω) under a diameter constraint, and, as a consequence, we obtain the reverse Faber-Krahn inequality.


 
Let Ω⊂^N be an open, bounded, convex set. Let B_r be a ball such that |Ω|=|B_r|. Then,

    μ_1(Ω) ≤μ_1(B_r),

and equality holds if and only if Ω = B_r.


Although the techniques we employed to prove these theorems are of rather elementary nature, we believe that the results are among the first examples of isoperimetric inequalities for fully nonlinear operators, apart from the case of the well-known Monge-Ampère operator.

Finally, let us briefly discuss the minimization problem for μ_1(Ω). We will see in Section <ref> that the minimization problem for μ_1(Ω) under a volume constraint does not admit a solution, since there exists a sequence of hyperrectangles {R_n}_n ∈ 
that degenerate to a line keeping fixed volume such that μ_1(R_n) → 0 as n → +∞. Nevertheless, it is possible to consider the same minimization problem under a perimeter or a diameter constraint. In Section <ref> we provide some preliminary results and we state some conjectures.






 §.§ Acknowledgments
 
This article was started during a visit of the third author to Aix Marseille University in the context of the MSCA Project GHAIA (ref:777822).




§ PRELIMINARY RESULTS
 



 §.§ Notations


For a measurable set Ω⊂^N, we will denote by |Ω| its volume, which corresponds to its N-dimensional Lebesgue measure. P(Ω) will stand for the perimeter of Ω; since we will be mainly dealing with convex sets, we can define P(Ω) as

    P(Ω) := ℋ^N-1(∂Ω),

where ℋ^N-1 is the (N-1)-dimensional Hausdorff measure. The diameter of Ω will be denoted by diam(Ω),

    diam(Ω):= sup{ |x-y| | x,y ∈Ω}.

For r>0, the symbol B_r will stand for an open ball of radius r.



 §.§ The eigenvalue problem


Let Ω⊂^N be an open, bounded domain. It is clear from the definition of μ_1(Ω) in (<ref>) that the function

    Ω↦μ_1(Ω)

is monotone decreasing with respect to set inclusion, that is,

    Ω_1 ⊂Ω_2 ⇒μ_1(Ω_1) ≥μ_1(Ω_2).

Moreover, μ_1(Ω) enjoys the following scaling property: for a given open, bounded set Ω⊂^N, and t>0, define

    tΩ := { x ∈^N  | x/t ∈Ω}.

Then,

    μ_1(tΩ) = 1/t^2μ_1(Ω).

When Ω⊂^N is a strictly convex domain, it is possible to show the existence of a strictly positive viscosity solution u ∈ C(Ω) to the eigenvalue problem

    {[ -λ_N(D^2 u)  =  μ u               in Ω,;             u  =  0              on ∂Ω, ].

for μ = μ_1(Ω) (see <cit.>). 
It is still an open problem to determine whether also (not strictly) convex domains admit a positive first eigenfunction. 
However, convexity turns out to be a necessary condition. If u ∈ C(Ω) is a positive eigenfunction, then it satisfies, in viscosity sense,
λ_1(D^2u) ≤…≤λ_N(D^2 u) ≤ 0 and hence u is concave. 
Hence, the fact that Ω = {u > 0} implies the convexity of Ω. 

For some particular domains, it is possible to compute the eigenvalue μ_1(Ω) and its associated eigenfunction (see <cit.>):


  * When Ω=B_r⊂^N is the ball of radius r, it holds

    μ_1(B_r) = π^2/4r^2

and the associated eigenfunction is given by

    u(x)=cos(π/2r|x|).

It is worth noticing that the eigenvalue and the eigenfunction do not depend on the space dimension N.

  * In the hyperrectangle R = ∏_i=1^N (-α_i,α_i)⊂^N the first eigenvalue is given by

    μ_1(R) = π^2/4(α_1^2 + … + α_N^2)

and the associated eigenfunction has the form

    u(x) = ∏_i=1^N [cos( π/2α_ix_i)]^1/p_i + 1

for some adequate values of p_i > -1.


These examples have some significant consequences. First of all, it follows by monotonicity that the quantity μ_1(Ω) is well defined and finite for every open, bounded set, since there exists a sufficiently small ball B_r ⊂Ω, so that

    μ_1(Ω) ≤μ_1(B_r) < + ∞.


Moreover, the case of the hyperrectangle shows that the problem of minimizing μ_1(Ω) under a volume constraint 
does not have a solution: Fix c>0, and take {R_n}_n ∈ is a sequence of hyperrectangles defined as

    R_n = (0,n) × (0,1) ×…× (0,1) _(N-2)  times×(0,c/n).

Then, it is straightforward to verify that |R_n|=c for every n and 

    lim_n → +∞μ_1(R_n) = 0.





§ MAXIMIZATION OF THE FIRST EIGENVALUE UNDER CONSTRAINT


In this section we will deal with the problem of maximizing the first eigenvalue μ_1(Ω) among convex sets, under various 
different constraints.  



The key result is the following estimate for μ_1(Ω), which is reminiscent of a similar result (with reversed inequality) obtained by Payne and Weinberger for the first nontrivial eigenvalue of the Laplacian under Neumann boundary conditions, see <cit.>.

 
Let Ω⊂^N be an open, bounded, convex domain. Then,

    μ_1(Ω) ≤π^2/[diam(Ω)]^2,

where diam (Ω) is the diameter of Ω.


Let {x_n}_n∈ℕ and {y_n}_n∈ℕ be two sequences of points in Ω such that

    d_n:= dist(x_n,y_n) →diam (Ω)     as n → +∞.

By convexity, Ω contains, for ε_n ∈(0, 1/n) sufficiently small, a hyperrectangle R of sides 
of length ε_n (N-1 times) and d_n (one time). By monotonicity of μ_1, we obtain
 
    μ_1(Ω) ≤μ_1(R) = π^2/(N-1)ε_n^2 + d_n^2.

Passing to the limit n → +∞, we obtain

    μ_1(Ω) ≤π^2/[diam (Ω)]^2

as we wanted to show.


As an immediate corollary we have the following.

 
Let Ω⊂^N be a convex set which is contained in a ball of radius diam (Ω)/2. Then,

    μ_1(Ω) = π^2/[diam (Ω)]^2.



The result follows by the monotonicity of μ_1(Ω) with respect to set inclusion, Proposition <ref> and the explicit expression 
of μ_1(Ω) for the ball.


Examples of convex sets satisfying the conditions of Corollary <ref> include regular polygons in ^2 with an even number of sides, or hypercubes in ^N. 
However, not every convex set enjoys this property: for instance, an equilateral triangle can not be contained in a disk of radius smaller than diam (Ω)/√(3). Nevertheless, by Jung's Theorem <cit.> we obtain the following lower bound:

    μ_1(Ω) ≥N+1/2N·π^2/[diam (Ω)]^2,

for any open, bounded, convex set Ω⊂^N.

In view of these considerations, one might wonder whether there actually exist domains such that   inequality (<ref>) is strict. In the next proposition we show, under a smoothness assumption on the eigenfunction, that this is indeed the case for the Reuleaux triangle. We observe that, if ℛ⊂^2 is the Reuleaux triangle generated by the open equilateral triangle T ⊂ℛ of unit side length, then it holds diam(ℛ)=1.






 
 Let ℛ⊂^2 be the Reuleaux triangle generated by the open equilateral triangle T ⊂ℛ of unit side length. Suppose that the first eigenfunction is in C^2(ℛ)∩ C(ℛ). Then,
 
    μ_1(ℛ) < π^2.



Let u ∈ C^2(ℛ)∩ C(ℛ) be a positive eigenfunction associated to the first eigenvalue μ_1(ℛ). We know, by Proposition <ref>, that 
    μ_1(ℛ)≤π^2.
 
Suppose by contradiction that μ_1(ℛ)=π^2. Let A, B and C be the vertices of T, and let P ∈ T be a point. 
Let D ∈∂ℛ be the intersection of the line passing through A and P with ∂ℛ. Let AD be the segment, of unit length, parametrized as

    AD= { x ∈ℛ | x= tD + (1-t)A, t ∈ [0,1]}.

Set e_1 := AD ∈ S^1. Let v be the restriction of u to the segment AD, namely,

    v(t) := u(tD+(1-t)A).

The function v satisfies v(0)=v(1)=0, and

    -v”(t)     = -⟨ D^2 u(tD+(1-t)A) · e_1, e_1 ⟩
       ≥ -λ_2(D^2 u)(tD+(1-t)A) 
        = π^2 u(tD+(1-t)A) = π^2 v(t).

From <cit.> (applied to the operator F=Δ + λ_1), v coincides with a multiple of the eigenfunction in (0,1), namely v(t)=c sin(π t) for some c>0. Therefore, equality holds in (<ref>), which implies, in particular,

    -⟨ D^2 u(P) · e_1, e_1 ⟩ = π^2 u(P),

e_1 being an eigenvector associated to λ_2(D^2 u(P)). Repeating the same reasoning with the line passing through B and P, we would obtain that, for a unit vector e_2 ∈ S^1 different from e_1, it holds

    -⟨ D^2 u(P) · e_2, e_2 ⟩ = π^2 u(P).

Therefore,

    -D^2 u(P) = π^2 u(P) I

for every P ∈ T. The conditions 
    -u_xx=-u_yy=π^2 u
 imply that u is of the form

    u(x,y) = (a_1 sinπ x+b_1cosπ x)(a_2 sinπ y+b_2cosπ y).

On the other hand, the condition 
    u_xy=0
 implies that 
    a_1=a_2=b_1=b_2=0,
 a contradiction to the fact that u>0 in T.



In the proof of Proposition <ref>, we made the assumption that the eigenfunction belongs to C^2(ℛ)∩ C(ℛ). While the few explicit examples currently known (balls and hyperrectangles) support the conjecture that eigenfunctions belong to C^∞(Ω) ∩ C(Ω), the optimal regularity of eigenfunctions of the truncated Laplacian is far from being understood, due to the high degeneracy of the differential operator. It is worth mentioning that, in the case of the Monge-Ampère operator, which is also fully nonlinear, but enjoys particular structural properties, eigenfunctions belong to C^∞(Ω) ∩ C^0,β(Ω) for every β∈ (0,1) (see <cit.>). 



Proposition <ref> readily implies that the ball of radius d/2 maximizes μ_1(Ω) among convex domains with fixed diameter diam(Ω) = d>0. The maximizer is not unique, since every other convex set with diameter d, contained in B_d/2, has the same eigenvalue.

We will denote by 𝒦_N the set 

    𝒦_N:= {Ω⊂^N | Ω open, bounded and convex}.


  Fix d>0. Then, 
the ball is a solution of the maximization problem

    sup{μ_1(Ω) | Ω∈𝒦_N, diam(Ω)=d }.




By Proposition <ref> and (<ref>) we have

    μ(Ω) ≤π^2/d^2 = μ(B_d/2)

for every convex domain Ω with diam(Ω)=d.


Proposition <ref> directly implies that the ball maximizes μ_1(Ω) under a perimeter or a volume constraint.

 
The ball is a solution of the maximization problem

    sup{μ_1(Ω) | Ω∈𝒦_N, P(Ω)=c},

where c>0 is fixed. Moreover, if N ≥ 3, then the ball is the unique maximizer.


The proof is a consequence of Proposition <ref>, and the fact that the ball maximizes perimeter under a diameter constraint among convex sets, being the unique maximizer if N ≥ 3 (see <cit.>).


 
For c>0
the ball is the unique maximizer of the maximization problem

    sup{μ_1(Ω) | Ω∈𝒦_N, |Ω|=c }.
 


The proof follows from Proposition <ref> and the well-known isoperimetric property of the ball (see, for instance, <cit.>). 







§ MINIMIZATION OF THE FIRST EIGENVALUE UNDER CONSTRAINTS
 

In Section <ref> we have seen that the problem of minimizing μ_1(Ω) 
among bounded, convex sets of fixed volume 
does not have a solution. This holds since a minimizing sequence is given by a sequence of hyperrectangles whose diameter tends to infinity. Then, it is natural to wonder whether it makes sense to consider the minimization problem under other kinds of constraints. In this section we will mainly restrict to planar, convex sets, Ω⊂ℝ^2, and we will provide some results which support the following conjectures.


The minimization problem

    inf{μ_1(Ω) | Ω∈𝒦_2, P(Ω)=c}     (c > 0)

does not admit a solution. A minimizing sequence is given by a sequence of rectangles of constant perimeter, with one side length tending to zero.



The minimization problem

    inf{μ_1(Ω) | Ω∈𝒦_2, diam(Ω)=d }     (d > 0)

admits a solution.



The Reuleaux triangle is a minimizer for Problem (<ref>) 




We refer to <cit.> for the basic definitions and notions that we will use in this section. Our first result is the continuity of Ω↦μ_1 (Ω) with respect to the Hausdorff convergence of open sets.

 
Let {Ω_n} be a sequence of nonempty, open convex sets which converge, with respect to the Hausdorff convergence of open sets, to the nonempty, open convex set Ω∈𝒦_N. Then,

    lim_n → +∞μ_1(Ω_n) = μ_1(Ω).




Let {Ω_n}_n∈ be a sequence in 𝒦_N which converges, with respect to the Hausdorff convergence of open sets, to Ω∈𝒦_N. Then, there exists a sequence {ε_n} with ε_n → 0 such that

    (1-ε_n)Ω⊂Ω_n ⊂ (1+ε_n)Ω.

(see, for instance, <cit.>). Due to the monotonicity and the scaling properties of μ_1, we have

    1/(1+ε_n)^2μ_1(Ω) ≤μ_1(Ω_n) ≤1/(1-ε_n)^2μ_1(Ω).

This implies

    lim_n → +∞μ_1(Ω_n) = μ_1(Ω),

which is the claim.


The next result concerns the asymptotic behaviour of a shrinking sequence of convex, planar sets.

 
Let {Ω_n}_n ∈ be a sequence of convex, open sets in ^2 such that diam(Ω_n) → d > 0 and |Ω_n| → 0 as n → +∞. Then,

    lim_n → +∞μ_1(Ω_n) = π^2/d^2.



Set d_n := diam(Ω_n). Without loss of generality, we can translate and rotate the sets Ω_n in such a way that the points (0,0) and (d_n,0) are on ∂Ω_n. Let

    ε_n^(1) := max{ y ≥ 0 | (x,y) ∈∂Ω_n},


    ε_n^(2) := min{ y ≤ 0 | (x,y) ∈∂Ω_n}.

Since the sets Ω_n are convex, and |Ω_n|→ 0 as n → +∞, it must hold ε_n^(1), ε_n^(2)→ 0 as n → +∞. Moreover, Ω_n is contained in a rectangle R_n of sides d_n and ε_n^(1) +ε_n^(2). By monotonicity,

    μ_1(Ω_n) ≥μ_1(R_n) = π^2/d_n^2 + (ε_n^(1) +ε_n^(2))^2⇒lim inf_n → +∞μ_1(Ω_n) ≥π^2/d^2.

On the other hand, by Proposition <ref>,

    μ_1(Ω_n) ≤π^2/d_n^2⇒lim sup_n → +∞μ_1(Ω_n) ≤π^2/d^2.

Hence, we have obtained that

    lim_n → +∞μ_1(Ω_n) = π^2/d^2,

as we wanted to show.


Let us now discuss the minimization problem in ^2

    inf{μ_1(Ω) | Ω∈𝒦_2, diam(Ω)=d } =: m.

First, we observe that

    m ≥3/4·π^2/d^2 > 0

by (<ref>). Let {Ω_n}_n ∈ be a minimizing sequence. Since the functional Ω↦μ_1(Ω) is translation invariant, we can suppose, without loss of generality, that all the sequence is contained in a fixed ball B_r ⊂^N. By Blaschke's selection principle, there exists Ω∈𝒦_N and a subsequence (still denoted by {Ω_n}_n ∈) such that

    Ω_n ℋ→Ω

in the Hausdorff distance. If |Ω_n| → 0, by Lemma <ref> it would hold 

    m = π^2/d^2,
 which would contradict Proposition <ref> if we knew that the eigenfunction in the Reuleaux triangle is smooth. Therefore, we would have |Ω|>0. Now, Proposition <ref> implies that,

    lim_n → +∞μ_1(Ω_n) = μ_1(Ω)=m.

Moreover, by continuity of the diameter with respect to the Hausdorff convergence of convex sets (see, for instance, <cit.>), it holds

    diam(Ω) =d

so that Ω would be a minimizer. In this case, by arguing as in <cit.>, it is possible to show the existence of a minimizer which is a body of convex width. We conjecture that the Reuleaux triangle is a minimizer, since it is generated from a regular polygon, and among all Reuleaux polygons it is the "farthest" from being a ball.

It is possible to perform similar reasonings for the minimization problem

    inf{μ_1(Ω) | Ω∈𝒦_N, P(Ω)=c }.

In this case, we conjecture that the infimum is not attained, and that a minimizing sequence is given by any sequence of rectangles as in Proposition <ref>.



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