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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle

Francesco Della PietraORCID iD: https://orcid.org/0000-0003-0324-2745 / Giuseppina di Blasio / Nunzia Gavitone
  • Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli studi di Napoli Federico II, Via Cintia, Monte S. Angelo – 80126 Napoli, Italy
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Published Online: 2018-09-20 | DOI: https://doi.org/10.1515/anona-2017-0281

Abstract

In this paper, we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue λF(p,Ω) of the anisotropic p-Laplacian, 1<p<+. Our aim is to enhance, by means of the 𝒫-function method, how it is possible to get several sharp estimates for λF(p,Ω) in terms of several geometric quantities associated to the domain. The 𝒫-function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient.

Keywords: Dirichlet eigenvalues; anisotropic operators; optimal estimates

MSC 2010: 35P30; 49Q10

1 Introduction

Given a bounded domain ΩN and p]1,+[, let us consider the first Dirichlet eigenvalue of the anisotropic p-Laplacian, that is,

λF(p,Ω)=minψW01,p(Ω){0}ΩF(ψ)pdxΩ|ψ|pdx,

where F:N[0,+[, N2, is a convex, even, 1-homogeneous C3,β(N{0})-function such that [Fp]ξξ is positive definite in N{0}, 1<p<+. We are interested in the study of optimal lower and upper bounds for functionals involving λF(p,Ω). In this order of ideas, our aim is to enhance how these estimates may be obtained as a consequence of a maximum principle for a function which involves an eigenfunction and its gradient, namely, the so-called 𝒫-function, introduced by L. E. Payne in the case of the classical Euclidean Laplace operator. We refer the reader to the book by Sperb [29] and the references contained therein for a survey on the 𝒫-function method in the Laplacian case and its applications. More precisely, if u is a positive eigenfunction associated to λF(p,Ω), we introduce the function

𝒫:=(p-1)Fp(u)+λF(p,Ω)(up-Mp),(1.1)

where M is the maximum value of u. We show that the function 𝒫 verifies a maximum principle in Ω¯ in order to get a pointwise estimate for the gradient in terms of u. This is the starting point to prove several useful bounds, involving quantities which depend on the domain Ω. As a matter of fact, the use of the 𝒫-function method in the anisotropic setting has been studied in the recent paper [14]. Here, the authors consider the p-anisotropic torsional rigidity

TFp-1(p,Ω)=supψW01,p(Ω){0}(Ω|ψ|dx)pΩF(ψ)pdx(1.2)

and show optimal bounds for two functionals involving TF(p,Ω) and some geometric quantities related to the domain. In this spirit, we aim to analyze the case of the eigenvalue problem. Given a convex, bounded domain ΩN, our main results can be summarized as follows: We prove the anisotropic version of the Hersch inequality for λF(p,Ω), namely, that

λF(p,Ω)(πp2)p1RF(Ω)p,(1.3)

where RF(Ω) is the anisotropic inradius defined in Section 2, and

πp:=20(p-1)1pdt[1-tpp-1]1p=2π(p-1)1ppsinπp.(1.4)

Regarding the Euclidean setting, for p=2, inequality (1.3) has been proved by Hersch [18], improved in [27] and generalized for any p in [19] (see also [26]). In the general anisotropic case or p=2, it has been studied in [6, 33]. Another consequence of the maximum principle for 𝒫 that we obtain is the inequality

(p-1p)p-1(πp2)pλF(p,Ω)MvΩp-1,(1.5)

where vΩ is the positive maximizer of (1.2) such that

TF(p,Ω)=ΩvΩdx,

and MvΩ is the maximum of vΩ. Inequality (1.5), in the Euclidean case (p=2), has been first proved in [24] and then studied also, for instance, in [29, 31, 17].

The last main result we show is the following: Let u be a first eigenfunction relative to λF(p,Ω), and consider the so-called anisotropic “efficiency ratio”

EF(p,Ω):=up-1|Ω|1p-1u.

Then we prove that

EF(p,Ω)1(p-1)1p(2πp)1p-1,(1.6)

where πp is defined in (1.4). In the Euclidean case and p=2, this inequality is due to Payne and Stakgold, who proved it in [25].

Finally, we show the optimality in (1.3) and (1.5), while the optimality of (1.6) in the class of convex sets is still an open problem.

As a matter of fact, the convexity assumption in (1.3), (1.5) and (1.6) can be weakened, since they are also valid in the case of smooth domains with anisotropic nonnegative mean curvature (see Section 2 for the definition).

In the present paper, we also emphasize the relation of λF(p,Ω) to the so-called anisotropic Cheeger constant hF(Ω) (see Section 3 for the definition). Indeed, in the class of convex sets, we prove the validity of a Cheeger-type inequality for λF, as well as a reverse Cheeger inequality.

The paper is organized as follows: In Section 2, we fix the notation and recall some basic facts regarding the eigenvalue problem for the anisotropic p-Laplacian and the torsional rigidity TF(p,Ω). Section 3 is devoted to the study of hF(Ω). More precisely, we recall the definition and the main properties, and we prove optimal lower and upper bounds for hF(Ω) in terms of the anisotropic inradius RF(Ω) of a convex set Ω. In Section 4, we prove that the 𝒫-function in (1.1) verifies a maximum principle. Finally, in Section 5, we prove the quoted results (1.3), (1.5), (1.6) and a reverse Cheeger inequality investigating also the optimality issue.

2 Notation and preliminaries

Throughout the paper, we will consider a convex, even, 1-homogeneous function

ξNF(ξ)[0,+[,

that is, a convex function such that

F(tξ)=|t|F(ξ),t,ξN,(2.1)

and such that

a|ξ|F(ξ),ξN,(2.2)

for some constant a>0. The hypotheses on F imply that there exists ba such that

F(ξ)b|ξ|,ξN.

Moreover, throughout the paper, we will assume that FC3,β(N{0}), and

[Fp]ξξ(ξ)is positive definite inN{0}(2.3)

with 1<p<+.

The hypothesis (2.3) on F ensures that the operator

𝒬pu:=div(1pξ[Fp](u))

is elliptic, hence there exists a positive constant γ such that

1pi,j=1nξiξj2[Fp](η)ξiξjγ|η|p-2|ξ|2

for any ηn{0} and for any ξn. The polar function Fo:N[0,+[ of F is defined as

Fo(v)=supξ0ξ,vF(ξ).

It is easy to verify that also Fo is a convex function which satisfies properties (2.1) and (2.2). Furthermore,

F(v)=supξ0ξ,vFo(ξ).

From the above property, it holds that

|ξ,η|F(ξ)Fo(η)for allξ,ηN.(2.4)

The set

𝒲={ξN:Fo(ξ)<1}

is the so-called Wulff shape centered at the origin. We put κN=|𝒲|, where |𝒲| denotes the Lebesgue measure of 𝒲. More generally, we denote by 𝒲r(x0) the set r𝒲+x0, that is, the Wulff shape centered at x0 with measure κNrN and 𝒲r(0)=𝒲r.

The following properties of F and Fo hold true:

Fξ(ξ),ξ=F(ξ),Fξo(ξ),ξ=Fo(ξ)for allξN{0}F(Fξo(ξ))=Fo(Fξ(ξ))=1for allξN{0},Fo(ξ)Fξ(Fξo(ξ))=F(ξ)Fξo(Fξ(ξ))=ξfor allξN{0}.

2.1 Anisotropic mean curvature

Let Ω be a C2-bounded domain, let n(x) be the unit outer normal at xΩ, and let uC2(Ω¯) such that Ωt={u>t}, Ωt={u=t} and u0 on Ωt. The anisotropic outer normal nF to Ωt is given by

nF(x)=Fξ(n(x))=Fξ(-u),xΩ.

It holds

Fo(nF)=1.

The anisotropic mean curvature of Ωt is defined as

F(x)=div(nF(x))=div[ξF(-u(x))],xΩt.

It holds that

unF=unF=uFξ(-u)=-F(u).(2.5)

In [14], it has been proved that, for a smooth function u on its level sets {u=t}, it holds

𝒬pu=Fp-2(u)(unFF+(p-1)2unF2).(2.6)

Finally, we recall the definition of the anisotropic distance from the boundary and the anisotropic inradius. Let us consider a bounded domain Ω, that is a connected, open set of N with nonempty boundary. The anisotropic distance of xΩ¯ to the boundary of Ω is the function

dF(x)=infyΩFo(x-y),xΩ¯.

We stress that when F=||, then dF=d, the Euclidean distance function from the boundary. It is not difficult to prove that dFW01,(Ω), and using the property of F, we have

F(dF(x))=1a.e. inΩ.(2.7)

Moreover, we recall that Ω is convex, and the anisotropic distance function is concave. The quantity

RF(Ω)=sup{dF(x),xΩ}(2.8)

is called the anisotropic inradius of Ω. For further properties of the anisotropic distance function, we refer the reader to [10].

2.2 The first Dirichlet eigenvalue for 𝒬p

Let Ω be a bounded, open set in N, N2, 1<p<+, and consider the eigenvalue problem

{-𝒬pu=λ|u|p-2uinΩ,u=0onΩ.(2.9)

The smallest eigenvalue, denoted by λF(p,Ω), has the following well-known variational characterization:

λF(p,Ω)=minφW01,p(Ω){0}ΩFp(φ)dxΩ|φ|pdx.(2.10)

The following two results which enclose the main properties of λF(p,Ω) hold true. We refer the reader, for example, to [4, 15].

Theorem 2.1.

If Ω is a bounded, open set in RN, N2, there exists a function u1C1,α(Ω)C(Ω¯) which achieves the minimum in (2.10) and satisfies the problem (2.9) with λ=λF(p,Ω). Moreover, if Ω is connected, then λF(p,Ω) is simple, that is, the corresponding eigenfunctions are unique up to a multiplicative constant, and the first eigenfunctions have constant sign in Ω.

In the following proposition, the scaling and monotonicity properties of λF(p,Ω) are recalled.

Proposition 2.2.

Let Ω be a bounded, open set in RN, N2. Then the following properties hold:

  • (1)

    For t>0 , it holds λF(p,tΩ)=t-pλF(p,Ω).

  • (2)

    If Ω1Ω2Ω , then λF(p,Ω1)λF(p,Ω2).

  • (3)

    For all 1<p<s<+ , we have p[λF(p,Ω)]1p<s[λF(s,Ω)]1s.

2.3 Anisotropic p-torsional rigidity

In this subsection, we summarize some properties of the anisotropic p-torsional rigidity. We refer the reader to [12] for further details.

Let Ω be a bounded domain in N, and let 1<p<+. Throughout the paper, we will denote by q the Hölder conjugate of p,

q:=pp-1.

Let us consider the torsion problem for the anisotropic p-Laplacian

{-𝒬pv:=-div(Fp-1(v)Fξ(v))=1inΩ,v=0onΩ.(2.11)

By classical result, there exists a unique solution of (2.11), that we will always denote by vΩ, which is positive in Ω. Moreover, by (2.3) and letting FC3(n{0}), then vΩC1,α(Ω)C3({vΩ0}) (see [21, 30]).

The anisotropic p-torsional rigidity of Ω is

TF(p,Ω)=ΩF(vΩ)pdx=ΩvΩdx.

The following variational characterization for TF(p,Ω) holds:

TF(p,Ω)p-1=maxψW01,p(Ω){0}(Ω|ψ|dx)pΩF(ψ)pdx,(2.12)

and the solution vΩ of (2.11) realizes the maximum in (2.12).

By the maximum principle, MvΩMvΩ~ holds, where MvΩ is the maximum of the torsion function in Ω. Finally, we recall the following estimates for MvΩ contained in [14].

Theorem 2.3.

Let Ω be a bounded, convex open set in RN, and RF the anisotropic inradius defined in (2.8). Then

RFq(Ω)qNq-1MvΩRFq(Ω)q.(2.13)

3 Anisotropic Cheeger constant

Let Ω be an open subset of N. The total variation of a function uBV(Ω) with respect to F is (see [3])

Ω|u|F=sup{Ωudivσdx:σC01(Ω;N),Fo(σ)1}.

This yields the following definition of anisotropic perimeter of KN in Ω:

PF(K)=N|χK|F=sup{Kdivσdx:σC01(N;N),Fo(σ)1}.(3.1)

It holds that

PF(K)=*KF(n)dN-1,

where N-1 is the (N-1)-dimensional Hausdorff measure in N, *K is the reduced boundary of F and n is the Euclidean unit outer normal to K (see [3]).

An isoperimetric inequality for the anisotropic perimeter holds, namely, 𝒲R is the Wulff shape such that |𝒲R|=|K|, then

PF(K)PF(𝒲R)=NκN1N|K|1-1N,(3.2)

and the equality holds if and only if Ω is a Wulff shape (see for example [5, 16, 2]). The following lemma will play a key role in order to investigate on optimality issue of the quoted results.

Lemma 3.1.

Let Ωa,k=]-a,a[×]-k,k[N-1 be an N-rectangle in RN, and suppose that RF(Ωa,k)=aFo(e1). Then

limk+PF(Ωa,k)|Ωa,k|=1aFo(e1).(3.3)

Proof.

First observe that (see [14])

Fo(e1)F(e1)=1.(3.4)

By definition of anisotropic perimeter, we get

PF(Ωa,k)|Ωa,k|=2(2k)N-1F(e1)+O(kN-2)2NkN-1a,

hence, using (3.4) and passing to the limit, we get (3.3). ∎

The anisotropic Cheeger constant associated to an open, bounded set ΩN is defined as

hF(Ω)=infKΩPF(K)|K|.

We recall that for a given bounded, open set in N, the Cheeger inequality states that

λF(p,Ω)(hF(Ω)p)p.(3.5)

This inequality, well known in the Euclidean case after the paper by Cheeger ([8]) for p=2, has been proved in [20] in the anisotropic case. We refer the reader to [22] and the references contained therein for a survey on the properties of the Cheeger constant in the Euclidean case.

It is known (see [20] and the references therein) that if Ω is a Lipschitz bounded domain, there exists a Cheeger set, that is, a set KΩ for which

hF(Ω)=PF(KΩ)|KΩ|.

When Ω=𝒲R, we immediately get K𝒲R=𝒲R and

hF(𝒲R)=NR.(3.6)

We observe that usually the Cheeger set KΩ is not unique, nevertheless, Ω is convex (see, for instance, [1, 7, 20]).

Theorem 3.2.

If Ω is a bounded, convex domain, there exists a unique convex Cheeger set.

The next results give an upper bound for the Cheeger constant in terms of the anisotropic inradius of Ω.

Proposition 3.3.

If Ω is a bounded, open set in RN, then

hF(Ω)NRF(Ω).(3.7)

Moreover, the equality holds if Ω is a Wulff shape.

Proof.

By definition, the constant hF(Ω) is monotonically decreasing with respect to the set inclusion. Then, by (3.6) and the definition of anisotropic inradius, we get inequality (3.7). ∎

Regarding a lower bound for the anisotropic Cheeger constant in terms of the inradius of Ω, we have the following:

Proposition 3.4.

If Ω is a bounded, open, convex set in RN, then

1RF(Ω)hF(Ω).(3.8)

Moreover, the inequality is optimal for a suitable sequence of N-rectangular domains.

Proof.

Using (2.7), (2.1) and the coarea formula, we have, for a bounded, convex set KΩ, that

|K|=KF(dF)dx=0RF(K)dtdF=tF(n)dσ=0RF(K)PF({dFt})dtPF(K)RF(K).

Hence, since K is convex, RF(K)RF(Ω), then

PF(K)|K|1RF(Ω).

Passing to the infimum on K, we get (3.8). The optimality follows immediately from (3.3). ∎

More generally, for convex sets, the following result holds (see also, for instance, [13], where the case N=2 is given with a different proof):

Proposition 3.5.

If ΩRN is a bounded, open, convex set, then

PF(Ω)|Ω|NRF(Ω).

For the Wulff shape, the equality holds.

Proof.

Let x0Ω be such that RF(Ω)=dF(x0). By the concavity of dF, we have

dF(x0)-dF(x)-dF(x)(x-x0)=n(x)(x-x0)|dF(x)|.

Hence, for xΩ, it holds that RF(Ω)n(x)(x-x0)|dF(x)|. By the divergence theorem and observing also that F(n)=1|dF|, we have

|Ω|=1NΩdiv(x-x0)dx=1NΩ(x-x0)n(x)dσRF(Ω)NΩ1|dF(x)|dσ=RF(Ω)PF(Ω)N,

and this completes the proof. ∎

Remark 3.6.

We observe that the equality in the inequality of Proposition 3.5 holds, in general, also for other kinds of convex sets. For example, if N=2 and F=, the equality holds for circles with two symmetrical caps (see, for instance, [28]).

An immediate consequence of the anisotropic isoperimetric inequality is the following:

Theorem 3.7.

Let Ω be a bounded, open set. Then

hF(Ω)hF(𝒲R),(3.9)

where WR is the Wulff shape such that |Ω|=|WR|, and the equality holds if and only if Ω is a Wulff shape.

Proof.

Let KΩ. Then, if |𝒲r|=|K|, by (3.2), we have

PF(K)|K|PF(𝒲r)|𝒲r|PF(𝒲R)|𝒲R|=h(𝒲R).

Passing to the infimum on K, we get the result. ∎

Remark 3.8.

Let Ω be an open, bounded set of N. Then inequality (3.7) implies

hF(Ω)-hF(WR)N(1RF(Ω)-1R).(3.10)

When Ω is convex, (3.10) can be read as a stability result for (3.9).

In [12], the following upper bound for λF(p,Ω) is proved in terms of volume and anisotropic perimeter of Ω for convex domains.

Theorem 3.9.

Let ΩRN be a bounded, convex, open set. Then

λF(p,Ω)(πp2)p(PF(Ω)|Ω|)p,(3.11)

where πp is defined in (1.4) and PF(Ω) is the anisotropic perimeter of Ω defined in (3.1).

The following reverse anisotropic Cheeger inequality holds (see [23] for the Euclidean case with N=p=2).

Proposition 3.10.

Let ΩRN, N2, be a bounded, open, convex set. Then

λF(p,Ω)(πp2)phF(Ω)p,(3.12)

where πp is defined in (1.4).

Proof.

Let KΩΩ be the convex Cheeger set of Ω. Since λp() is monotonically decreasing by set inclusion, then by (3.11), we have

λF(p,Ω)λF(p,KΩ)(πp2)p(PF(KΩ)|KΩ|)p=(πp2)phF(Ω)p.

The equality sign holds in the limiting case when Ω approaches a slab. This will be shown in Theorem 5.8.

4 The 𝒫-function

In order to give some sharp lower bound for λF(p,Ω), we will use the so-called 𝒫-function method. Let us consider the general problem

{-𝒬pw=f(w)inΩ,w=0onΩ,(4.1)

where f is a nonnegative C1(0,+)C0([0,+[)-function, and define

𝒫(x):=p-1pFp(w(x))-w(x)maxΩ¯wf(s)ds.

The following result is proved in [9].

Proposition 4.1.

Let Ω be a domain in RN, N2, and let wW01,p(Ω) be a solution of (4.1). Set

dij:=1F(uΩ)ξiξj[Fpp](uΩ).

Then it holds that

(dij𝒫i)j-bk𝒫k0𝑖𝑛{uΩ0},

where

bk=p-2F3(w)Fξ(w)𝒫xFξk(w)+2p-3F2(w)(Fξkξ(w)𝒫xp-1-f(w)Fξk(w)).

As a consequence of the previous result, we get the following maximum principle for 𝒫.

Theorem 4.2.

Let Ω be a bounded C2,α-domain in RN, N2, with nonnegative anisotropic mean curvature HF0 on Ω, and let w>0 be a solution to the problem (4.1). Then

𝒫(x)=p-1pFp(w(x))-w(x)maxΩ¯wf(s)ds0𝑖𝑛Ω¯,(4.2)

that is, the function P achieves its maximum at the points xMΩ such that w(xM)=maxΩ¯w.

Proof.

Let us denote by 𝒞 the set of the critical points of w, that is, 𝒞={xΩ¯:w(x)=0}. Being ΩC2,α, by the Hopf lemma (see, for example, [11]), 𝒞Ω=.

Applying Proposition 4.1, the function 𝒫 verifies a maximum principle in the open set Ω𝒞. Then we have

maxΩ𝒞¯𝒫=max(Ω𝒞)𝒫.

Hence, one of the following three cases occurs:

  • (1)

    The maximum point of 𝒫 is on Ω.

  • (2)

    The maximum point of 𝒫 is on 𝒞.

  • (3)

    The function 𝒫 is constant in Ω¯.

In order to prove the theorem, we have to show that statement (1) cannot happen. Let us compute the derivative of 𝒫 in the direction of the anisotropic normal nF in the sense of (2.5). Hence, on Ω, we get

𝒫nF=p-1pnF(-wnF)p+f(w)wnF=-(p-1)(-wnF)p-12wnF2+f(w)wnF=-F(w)𝒬p[w]-Fp(w)F-f(w)F(w)=-Fp(w)F,

where last identity follows by (2.6). On the other hand, if a maximum point x¯ of 𝒫 is on Ω, by the Hopf lemma, either 𝒫 is constant in Ω¯, or 𝒫nF(x¯)>0. Hence, since F0, we have a contradiction. ∎

Remark 4.3.

Let Ω be a bounded, open, convex set, and let us consider u a positive eigenfunction relative to the first eigenvalue λF(p,Ω) of problem (2.9). Then, with M denoting maxΩ¯u, inequality (4.2) becomes

(p-1)Fp(u)λF(p,Ω)(Mp-up)inΩ¯.(4.3)

Integrating over Ω in both sides of (4.3) and recalling that u satisfies problem (2.9), we get

ΩupMp|Ω|p.

By the definition of πp in (1.4), we have

πp2=0(p-1)1p[1-tpp-1]-1pdt=0M(p-1)1p(Mp-tpp-1)-1pdt,

where u is a first positive eigenfunction of -𝒬p. Let us consider the function

Φ(s)=(πp2)pp-1-(s(p-1)1pM(p-1)1pdt(Mp-tpp-1)1p)pp-1,s[0,M].

Proposition 4.4.

Let Ω be a bounded C2-domain in RN, N2, with nonnegative anisotropic mean curvature HF0 on Ω. Then the following inequalities hold:

Φ(u(x))pp-1λF(p,Ω)1p-1vΩ(x),(4.4)Φ(u)F(u)pp-1λF(p,Ω)1p-1F(vΩ)𝑜𝑛Ω,(4.5)

where vΩ is the stress function of Ω.

Proof.

In order to prove (4.4), we will show that

-𝒬p[Φ]-𝒬p[pp-1λF(p,Ω)1p-1vΩ]=λF(p,Ω)(pp-1)p-1.(4.6)

By the comparison principle, being Φ(u)=vΩ=0 on Ω, then (4.4) holds.

With φ(u) denoting

u(p-1)1pM(p-1)1p(Mp-tpp-1)-1pdt,

we have

Φ(u)=q(p-1)1pφ(u)q-1(Mp-up)-1p,

and

Φ′′(u)=-q(q-1)(p-1)2pφ(u)q-2(Mp-up)-2p+q(p-1)1pφ(u)q-1(Mp-up)-1p-1up-1=q(p-1)1pφ(u)q-1(Mp-up)-1p[up-1Mp-up-(q-1)(p-1)1pφ(u)-11(Mp-up)1p]=Φ(u)[up-1Mp-up-(q-1)(p-1)1pφ(u)-11(Mp-up)1p]=Φ(u)Ψ(u),

where we denoted the last square bracket with Ψ(u). Then

QpΦ(u)=div[(Φ)p-1F(u)p-1Fξ(u)]=(Φ)p-1Qpu+(p-1)(Φ)p-2Φ′′(u)F(u)p=(Φ)p-1[-λF(p,Ω)up-1+(p-1)F(u)pΨ(u)].

To prove the claim, we need to show that (4.6) holds, that is,

(Φ)p-1[-λF(p,Ω)up-1+(p-1)F(u)pΨ(u)]+qp-1λF(p,Ω)0.

Substituting, we get

-λF(p,Ω)up-1+(p-1)F(u)p[up-1Mp-up-(q-1)(p-1)1pφ(u)-11(Mp-up)1p]+λF(p,Ω)[Mp-up]p-1p(p-1)1qφ(u)={(p-1)-1qφ(u)-1[Mp-up]1-1p-up-1}[λF(p,Ω)-(p-1)F(u)pMp-up].

The function in the last square brackets is nonnegative by (4.2). To conclude, we show that the function

B(u):=[Mp-up]1-1p-(p-1)1-1pup-1φ(u)

is nonnegative. This is true, since B(M)=0 and B0. This concludes the proof of (4.4). Finally, by computing the derivative of Φ with respect to the anisotropic normal nF on Ω={u=0}, we have

ΦnF=ΦFξ(-u)=-Φ(u)F(u)on Ω.

Recalling (4.4), by the Hopf lemma, we get

ΦnFpp-1λF(p,Ω)1p-1vΩnFonΩ.

Then

Φ(u)F(u)pp-1λF(p,Ω)1p-1F(vΩ)onΩ,

which is (4.5), and this concludes the proof of the theorem. ∎

5 Applications

Now we prove several inequalities involving λF(p,Ω), RF(Ω), hF(Ω), MvΩ, EF(p,Ω). The main estimates that we prove using the 𝒫-function method (Theorem 5.1, Proposition 5.2, Theorem 5.3 and Theorem 5.6) are stated for C2-bounded domains in N with nonnegative anisotropic mean curvature. Actually, for bounded, convex sets, the C2-regularity is not needed. This can be proved approximating Ω in the Hausdorff distance by an increasing sequence of strictly convex smooth domains contained in Ω. A similar argument has been used, for example, in [14].

5.1 Anisotropic Hersch inequality

Theorem 5.1.

Let Ω be a bounded C2-domain in RN, N2, with nonnegative anisotropic mean curvature HF0 on Ω. Then the following anisotropic Hersch inequality holds:

λF(p,Ω)(πp2)p1RF(Ω)p,(5.1)

where RF(Ω) is the anisotropic inradius defined in (2.8).

Proof.

Let u be a positive eigenfunction relative to λF(p,Ω), and let v be a direction of N. Let M=maxΩ¯u. Then by Theorem 4.2 with f(w)=λwp-1 and property (2.4), we have

uv=u,vF(u)Fo(v)(λF(p,Ω)p-1)1p(Mp-up)1pFo(v).(5.2)

Let us denote by xM the point of Ω such that M=u(xM), by x¯Ω the point such that Fo(xM-x¯)=dF(xM) and by v the direction of the straight line joining the points xM and x¯. Then by (5.2), since Fo(x¯-xM)RF(Ω), we get

0M(Ω)1(Mp(Ω)-up)1pdu(λF(p,Ω)p-1)1pFo(v)|x¯-xM|=(λF(p,Ω)p-1)1pFo(x¯-xM)(λF(p,Ω)p-1)1pRF(Ω).(5.3)

By definition of (1.4) and a change of variables, we get

0M(Ω)1(Mp(Ω)-up)1pdu=1(p-1)1pπp2.(5.4)

Finally, joining (5.3) and (5.4), we get inequality (5.1). ∎

The equality sign in (5.1) holds in the limiting case when Ω approaches a slab. This will be shown in Theorem 5.8.

From the Hersch inequality (5.1) and the bound (3.7), the following immediately holds:

Proposition 5.2.

Let Ω be a bounded C2-domain in RN, N2, with nonnegative anisotropic mean curvature HF0 on Ω. Then

λF(p,Ω)(πp2N)phFp(Ω).(5.5)

Hence, for p2Nπp, inequality (5.5) gives a better constant than (3.5).

5.2 An upper bound for the efficiency ratio

As a consequence of the Theorem 4.4, we obtain the following inequality:

Theorem 5.3.

Let Ω be a bounded C2-domain in RN, N2, with nonnegative anisotropic mean curvature HF0 on Ω. Then

(p-1p)p-1(πp2)pλF(p,Ω)MvΩp-1,(5.6)

where MvΩ=maxΩ¯vΩ.

Proof.

The proof is a direct consequence of Theorem 4.4 and of the definition (1.4) of πp. Indeed, by (4.4) and the explicit expression of Φ, evaluating both sides at the maximizer xm of u, we obtain

(p-1p)p-1(πp2)pvΩp-1(xm)λF(p,Ω)λF(p,Ω)MvΩp-1,

which is the desired inequality (5.6). ∎

The equality sign in (5.6) holds in the limiting case when Ω approaches a slab. This will be shown in Theorem 5.8.

Remark 5.4.

We observe that the functional involved in Theorem 5.3 is related to other functionals studied in literature. Indeed, it holds that

λF(p,Ω)(TF(p,Ω)|Ω|)p-1λF(p,Ω)MvΩp-1(|Ω|MvΩTF(p,Ω))p-1.(5.7)

The functional on the left-hand side of (5.7) has been studied, for example, in [32] for p=2 in the Euclidean case. The functional on the right-hand side of (5.7) has been investigated, for instance, in [17] for p=2 in the Euclidean case and in [14] for any p in the anisotropic setting.

Remark 5.5.

Using the upper bound in (2.13) and (5.6) we directly get the anisotropic Hersch inequality for λF(p,Ω):

λF(p,Ω)(πp2)p1RF(Ω)p.

Let u be the first eigenfunction relative to λF(p,Ω), and let us define the anisotropic efficiency ratio as

EF(p,Ω):=up-1|Ω|1p-1u.(5.8)

We stress that, by Remark 4.3 and the Hölder inequality, for open, bounded, convex sets, we obtain the following upper bound for (5.8):

EFp(p,Ω)1p.(5.9)

Actually, as a consequence of Theorem 4.4, we get the following upper bound for EF(p,Ω) which, in the Euclidean case, is due to Payne and Stakgold [25]:

Theorem 5.6.

Let Ω be a bounded C2-domain in RN, N2, with nonnegative anisotropic mean curvature HF0 on Ω. Then

EF(p,Ω)(p-1)-1p(2πp)1p-1.(5.10)

Proof.

Passing to the power p-1 in both sides of (4.5), integrating on Ω and using the equations, by the divergence theorem, we have

(p-1)1qπp2Ωup-1dxMp-1|Ω|.

That gives the following upper bound for the efficiency ratio Ep:

EF(p,Ω)=up-1|Ω|1p-1u1(p-1)1p(2πp)1p-1.

Remark 5.7.

We observe that the bound in (5.10) improves the one given in (5.9).

Finally, we are in the position to give the following optimality result.

Theorem 5.8.

The equality sign in (3.8), (3.11), (3.12), (5.1), (5.6) and in the upper bound of (2.13) holds in the limiting case when Ω approaches a suitable infinite slab.

Proof.

Let Ω be a bounded, open, convex set of N. Then by (5.1), (3.12) and the definition of hF, we get

(πp2)pRF(Ω)pλF(p,Ω)(πp2)p(hF(Ω)RF(Ω))p(πp2)p(PF(Ω)RF(Ω)|Ω|)p,(5.11)

and by (5.6), (3.11) and the upper bound in (2.13), we get

(p-1p)p-1(πp2)pλF(p,Ω)MvΩp-1(p-1p)p-1(πp2)p(PF(Ω)RF(Ω)|Ω|)p.(5.12)

Choosing Ω=Ωa,k in (5.11) and (5.12) as in Lemma 3.1 and passing to the limit, we get the required optimality. ∎

Remark 5.9.

For a general planar, open, convex set, in [31], in the Euclidean case, the author proves the following result:

λ(Ω)MvΩ(π28)(1+7323(W(Ω)d(Ω))23),

where λ(Ω) is the first Dirichlet eigenvalue of -Δ, d(Ω) denotes the Euclidean diameter and W(Ω) the width. Then, for a planar, open, convex set and p=2, in the Euclidean case, the equality in (5.6) holds for the sets such that W(Ω)d(Ω)0.

Remark 5.10.

The slab is not optimal for EF(p,Ω). Indeed, if, for example, N=p=2 and F==(ixi2)12, for any rectangle R, it holds that E(2,R)=(2π)2.

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About the article

Received: 2017-12-18

Accepted: 2018-09-05

Published Online: 2018-09-20

Published in Print: 2019-03-01


This work has been partially supported by GNAMPA of INdAM.


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 278–291, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0281.

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