Corrigendum to “An isoperimetric inequality for a nonlinear eigenvalue problem” [Ann. I. H. Poincaré – AN 29 (1) (2012) 21–34]

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Ann. I. H. Poincaré – AN 32 (2015) 485–487

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Corrigendum

Corrigendum to “An isoperimetric inequality for a nonlinear eigenvalue problem” [Ann. I. H. Poincaré – AN 29 (1) (2012) 21–34]

Gisella Croce

^a,^∗

, Antoine Henrot

^b

, Giovanni Pisante

^c

aLaboratoiredeMathématiquesAppliquéesduHavre,UniversitéduHavre,25,ruePhilippeLebon,76063LeHavre,France bInstitutÉlieCartanNancyUMR7502,UniversitédeLorraine-CNRS-INRIA,B.P.70239,54506VandoeuvrelesNancy,France

cDipartimentodiMatematicaeFisica,SecondaUniversitàdeglistudidiNapoli,vialeLincoln,5,81100,Caserta,Italy

Received 16February2014;accepted 18April2014 Availableonline 14June2014

MSC:35J60;35P30;47A75;49R50;52A40

Keywords:Shapeoptimization;Eigenvalues;Symmetrization;Eulerequation;Shapederivative

Recently it came to our attention that our proof proposed in[1]about the minimizing set of the eigenvalue

λ^p,q(Ω)=inf

∇vL^p(Ω)

vL^q(Ω)

, v=0, v∈W₀^1,p(Ω),

Ω

|v|^q⁻²v dx=0

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among the bounded open sets Ω⊂R^Nof given volume is not correct. Indeed the argument that we used to write the Euler–Lagrange equation in Theorem 6cannot be applied. More precisely in the last paragraph of the proof we need to test the minimality with the function u +t_n(ϕ+c_t_n)which is not in W₀^1,pbut merely constant on the boundary.

The statement of Theorem 1 in[1]is not true for every p, q satisfying

1< p <∞ and

2≤q < p^∗, if 1< p < N,

2≤q <∞, ifp≥N (2)

DOIoforiginalarticle:http://dx.doi.org/10.1016/j.anihpc.2011.08.001.

* Correspondingauthor.

E-mailaddresses:[email protected](G. Croce),[email protected](A. Henrot),[email protected] (G. Pisante).

http://dx.doi.org/10.1016/j.anihpc.2014.04.006

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486 G. Croce et al. / Ann. I. H. Poincaré – AN 32 (2015) 485–487

as remarked also by Nazarov in[3].¹For a correct statement we should replace λ^p,q(Ω)in[1]by λ^p,q_per(Ω)=inf

∇vL^p(Ω)

vL^q(Ω)

, v=0, v∈W_per^1,p(Ω),

Ω

|v|^q⁻²v dx=0

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whereW_per^1,p(Ω)stands for the set of W^1,p(Ω)functions with constant boundary value. As in[2]for the case p= q=2, λ^p,qper(Ω)is minimized by the union of two equal balls for every p, qsatisfying (2).

The proof goes exactly as in[1], up to these modifications:

(1) In Theorem 5one needs to use the Schwarz symmetrization on Ω₊= {x∈Ω: u(x) > c}and Ω₋= {x∈Ω: u(x) < c}, where cis the value of the eigenfunction uat ∂Ω.

(2) Theorem 7is no longer true in general for this eigenvalue, nevertheless we can prove Theorem 8in a similar way.

There are two differences in the proof:

(a) we have to deal with the boundary condition which is not standard when performing the domain derivative;

(b) even when the optimal domain has a multiple first eigenvalue, we can still deduce that the derivative of any eigenfunction has to vanish. For this we use the existence of directional derivatives along with the minimality of the eigenvalue.

(3) The ordinary differential equation satisfied by the radial component wof u1, that is,

−

(p−1)z(t)+N−1

t z(t)z(t)^p⁻²=

λ^p,q_per(Ω) ^pu^p_L⁻q(Ω)^q z(t )^q⁻²z(t ), implies that if ^∂u_∂ν¹

1 =0 then u₁=0 on the boundary. Therefore the proof of Theorem 9is valid.

We end this erratum with a remark about λ^p,q(Ω). For q small enough, it turns out that the eigenfunction of λ^p,q_per(Ω)seems to be symmetric and satisfies u =0 on the boundary (at this time, we are only able to prove it for q=p).²In this case, our result about λ^p,q(Ω)still holds, since

λ^p,q(Ω)≥λ^p,q_per(Ω)≥λ^p,q_per(B∪B)=λ^p,q(B∪B)

where the last equality is due to the fact that if uis an eigenfunction of λ^p,qper(Ω)satisfying u =0 on ∂Ω, thenuis an eigenfunction of λ^p,q(Ω).

Conflict of interest statement

There are no known conflicts of interest associated to this publication.

1 NazarovconsidersthefollowingtestfunctionsondomainsΩgivenbyunionofballsB₊,B₋ofradiiR₊,R₋.Letvbetherestrictiontoone balloftheeigenfunctionrealizingλ^p,q(B₊∪B₊).Forx→c₊v(^R_R^1/2^x

+ )χ_B₊−c₋v(^R_R^1/2^x

− )χ_B₋(wherec₊,c₋≥0,c^q−1₊ R₊^N=c^q−1₋ R₋^Nand 2R^N_1/2=^|ω^Ω_N^|)onehas

λ^p,q(B₊∪B₋)≤∇vL^p(B_1/2) vL^q(B_1/2)

R¹⁻

N p+^N_q 1/2

[c₊^pR₊^N⁻^p+c^p₋R₋^N⁻^p]^p¹ [c^q₊R₊^N+c₋^qR₋^N]¹^q

.

Thenhefindsanexplicitvalueqˆsuchthat,forq >q,ˆ asimplefirstorderanalysisshowsthatthelastfunctionof(R₊,R₋)doesnotattainits minimumvaluein(R1/2,R1/2).Thisprovesthattheminimizingsetisnotgivenbyapairoftwoequalballs.

2 Indeed,letu=u₁(|x−x₁|)χ_B₁+u₂(|x−x₂|)χ_B₂beaneigenfunctionofλ=λ^p,q_per(B₁∪B₂)andwbethesolutionto

−

(p−1)w(t)+N−1

t w(t)w(t)^p⁻²=w(t )^q⁻²w(t), w(0)=1, w(0)=0.

Letαi=u(xi)andwα_i(t)=αiw(c(αi)t),wherec(α)=^λu

1−q/p Lq

|α|¹⁻^q^p ,forα=0.Thenitisnotdifficulttoprovethatui=wα_i(|x−xi|).Since uisconstantontheboundary,inthecaseq=ponehasα₁w(λR₁)=α₂w(λR₂).Assumingw.l.o.g.thatu=c≥0 ontheboundary,weget w(λR₁)=0=w(λR₂)sinceα₁>0> α₂.

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G. Croce et al. / Ann. I. H. Poincaré – AN 32 (2015) 485–487 487

References

[1]G.Croce,A.Henrot,G.Pisante,Anisoperimetricinequalityforanonlineareigenvalueproblem,Ann.Inst.HenriPoincaré,Anal.NonLinéaire 29 (1)(2012)21–34.

[2]A.Greco,M.Lucia,LaplacianeigenvaluesformeanzerofunctionswithconstantDirichletdata,ForumMath.20 (5)(2008)763–782.

[3]A.I.Nazarov,Onsymmetryandasymmetry inaproblemofshapeoptimization,arXiv:1208.3640,2012.