Computing the first eigenpair of the p-laplacian in annuli

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1 J. Mth. Anl. Appl Contents lists vilble t ScienceDiect Jounl of Mthemticl Anlysis nd Applictions Computing the fist eigenpi of the p-lplcin in nnuli Gey Ecole,, Júlio Cés do Espíito Snto b, Ede Minho Mtins b Univesidde Fedel de Mins Geis, Deptmento de Mtemátic, ICEx, Av. Antônio Clos 6627, Cix Postl 702, , Belo Hoizonte, MG, Bzil b Univesidde Fedel de Ouo Peto, Deptmento de Mtemátic, ICEB, Cmpus Univesitáio Moo do Cuzeio, , Ouo Peto, MG, Bzil t i c l e i n f o b s t c t Aticle histoy: Received 19 Mch 2014 Avilble online 16 Septembe 2014 Submitted by B. Kltenbche Keywods: Annulus Fist eigenpi Invese itetion method p-lplcin We popose method fo computing the fist eigenpi of the Diichlet p-lplcin, p > 1, in the nnulus Ω,b = x R N : < x < b}, N >1. Fo ech t, b), we use n invese itetion method to solve two dil eigenvlue poblems: one in the nnulus Ω,t, with the coesponding eigenvlue λ t) nd boundy conditions u) = 0 = u t); nd the othe in the nnulus Ω t,b, with the coesponding eigenvlue λ + t) nd boundy conditions u t) = 0 = ub). Next, we djust the pmete t using mtching pocedue to mke λ t) coincide with λ + t), theeby obtining the fist eigenvlue λ p. Hence, by simple splicing gument, we obtin the positive, L -nomlized, dil fist eigenfunction u p. The mtching pmete is the mximum point ρ of u p. In ode to pply this method, we deive estimtes fo λ t) nd λ + t), nd we pove tht these functions e monotone nd loclly Lipschitz) continuous. Moeove, we deive uppe nd lowe estimtes fo the mximum point ρ, which we use in the mtching pocedue, nd we lso pesent diect poof tht u p conveges to the L -nomlized distnce function to the boundy s p. We lso pesent some numeicl esults obtined using this method Elsevie Inc. All ights eseved. 1. Intoduction In this wok, we conside the following eigenvlue poblem Δ p u = λ u p 2 u, in Ω,b u =0 on Ω,b, 1) * Coesponding utho. E-mil ddesses: gey@mt.ufmg.b G. Ecole), jceses@iceb.ufop.b J.C. do Espíito Snto), ede@iceb.ufop.b E.M. Mtins) X/ 2014 Elsevie Inc. All ights eseved.

2 1278 G. Ecole et l. / J. Mth. Anl. Appl whee Δ p u := div u p 2 u) is the p-lplcin opeto, p > 1, nd Ω,b is the nnulus Ω,b := x R N :0<< x <b }, N > 1. It is well known tht the fist eigenvlue of the Diichlet p-lplcin in bounded domin Ω R N is positive nd it is lso chcteized s the minimum of the Ryleigh quotient v p LpΩ) tken ove v p LpΩ) ll nontivil functions v W 1,p 0 Ω). Futhemoe, the coesponding exteml functions, i.e., the fist eigenfunctions, do not chnge sign in Ω, e scl multiples of ech othe i.e., the fist eigenvlue is simple), nd they belong to C 1,τ Ω) fo some 0 < τ<1. We denote λ p s the fist eigenvlue of 1) nd u p s the positive nd L -nomlized eigenfunction tht coesponds to λ p. Thus, v p } L λ p =min p Ω,b ) v p :0 v W 1,p 0 Ω,b ) L p Ω,b ) = u p p L p Ω,b ) u p p, L p Ω,b ) u p > 0in Ω,b nd u p L Ω,b ) := 1. When the domin Ω is bll o n nnulus, the fist eigenfunctions must be dilly symmetic see [21]). In pticul, fo the nnulus Ω,b,we hve u p = u p ), whee = x [, b], λ p =min N 1 v ) p d N 1 v) p d 1,p ) } :0 v W0, b) = N 1 u p) p d N 1 u p ) p d nd u p C 1,τ [, b]) W 1,p 0, b)) stisfies the boundy vlue poblem N 1 u p 2 u ) = λp N 1 u,, b) u) =0=ub). 2) It is esy to check tht u p hs unique citicl point ρ, b), whee it ttins its mximum vlue. Thus, u p ) is stictly incesing if [, ρ), nd stictly decesing if ρ, b] nd u p ρ) = 1 = u p L [,b]). The popeties of the fist eigenpi of the Diichlet p-lplcin in n N-dimensionl bounded domin Ω e well known, but the fist eigenpi itself is genelly difficult to compute, even fo simple domins such s bll, sque, o n nnulus, when 1 < p 2nd N >1. In the lst thee decdes, sevel studies hve imed to impove the estimtes of the fist eigenvlue fo genel bounded domins, o poposed methods fo the numeicl computtion of the fist eigenpi fo some domins see [2 7,11,14,17,19]). In the pesent wok, we popose new method fo computing the fist eigenpi λ p, u p )of the eigenvlue poblem 1). In [7], n itetion method ws developed bsed on the invese powe method of line lgeb to compute the fist eigenpi of the Diichlet p-lplcin in bll. By exploiting the dil fom of the fist eigenfunction, this method poduces sequences tht convege pidly nd monotoniclly to the fist eigenpi. In [6], nothe itetive method ws poposed bsed on the sub-supesolution ppoch, which ws shown to wok fo genel bounded domin nd it ws implemented fo some simple cses. Fo nondil domins, such s sque, cube, o tous, the implementtion of this method used eguliztion of the fom divɛ 2 + u 2 ) p p 2 )fo Δp u in ode to void the singulity o degenecy t u = 0. The fist step of both itetive pocesses uses the p-tosion function, whee the solution of the following poblem is known s the tosionl ceep poblem see [16]) Δp u =1, in Ω u =0 on Ω.

3 G. Ecole et l. / J. Mth. Anl. Appl In the cse whee Ω is bll, the p-tosion function is lso dilly symmetic nd the coesponding dil vesion of the tosionl ceep poblem is ewitten s boundy vlue poblem fo n odiny diffeentil eqution ODE) in the dil vible. A simple exmintion of this ODE llows us to see tht the p-tosion function is stictly decesing with espect to. Hence, fte two integtions of the ODE), we cn esily obtin n explicit expession fo the p-tosion function. A simil pocedue cn be employed to solve the fist eigenvlue poblem in the unidimensionl cse N =1) fo ny p > 1. In fct, in this cse, Ω is n open intevl, b) nd both the p-tosion function nd the fist eigenfunctions e symmetic with espect to thei mximum point, which necessily occu t the sme plce: the midpoint m := +b 2. Theefoe, s in the cse of the bll, n explicit expession cn be deived fo the p-tosion function, theeby mking the invese itetion method diectly pplicble see [8]) to solving the eigenvlue poblem in, m) o in m, b), which llows us to obtin one of the hlves of the fist eigenfunction. The othe hlf is obtined by eflecting tht found peviously ound m. Howeve, the p-tosion function is genelly not expessed by simple fomul. Moeove, its numeicl computtion is not simple if 1 < p 2nd N >1. In some nondil domins see [6]), its computtion demnds significntly moe intections thn the subsequent tems of the sequences tht convege to the fist eigenpi. It should be noted tht the numbe of intections tht e necessy to poduce good ppoximtions of the p-tosion function seems to decese with p, fo lge vlues of p, s mentioned in [6]. Even in the cse of the nnulus whee N>1nd 1 < p 2, n explicit expession fo the p-tosion function is not vilble. Indeed, its dil fom lone is not sufficient to poduce such n expession fte integting the coesponding ODE boundy vlue poblem. This is becuse the position of its mximum point is not known pioi. Moeove, the mximum points in the subsequent itetions chnge duing ech step, theeby mking the computtions longe nd moe difficult. Consequently, invese itetion stting fom the p-tosion function does not ppe to be suitble fo computing the fist eigenpi of the nnulus Ω,b if we wnt to exploe its dil symmety. We compute the fist eigenpi of Ω,b by exploiting its dil symmety to split the coesponding dil eigenvlue poblem into two simple ones. Howeve, this sttegy mkes the mximum point ρ n dditionl unknown beyond λ p nd u p ) in ou poblem. In pticul, ou method fo computing the fist eigenpi of 1) involves splitting the fist eigenvlue poblem fo Ω,b into two dil poblems tht e detemined by the splitting pmete t, b), which should convege to ρ. The left dil eigenvlue poblem is posed in Ω,t whee the fist eigenpi is denoted by λ t), u t, )) nd the ight dil eigenvlue poblem is posed in Ω t,b whee the fist eigenpi is denoted by λ + t), u + t, )). Ech of these dil eigenvlue poblems hs the sme stuctue s the dil eigenvlue poblem fo bll. Next, we pply the invese itetion method to compute the fist eigenpis λ t), u t, )) nd λ + t), u + t, )) nd we use mtching pocedue to djust the pmete t to the mximum point ρ, i.e., to tht which mkes λ t) = λ + t) = λ p. This llows us to splice u t, ) with u + t, ), theeby foming the fist eigenpi u p of the nnulus. The numeicl implementtion of this scipt does not equie ny eguliztion of the p-lplcin. In ode to povide theoeticl suppot fo the pocedue descibed bove, we exploe the vitionl chcteiztion of the fist eigenvlues to pove the continuity of the functions λ t) nd λ + t), thei behvio t the endpoints nd b, nd the following explicit bounds fo the unknown mximum point ρ, which is elevnt issue fo stting the mtching pocedue: b + [ b )N 1 + b )N +1] 1 p 1+[ b )N 1 + b )N +1] 1 p <ρ< + b 2. 3) As bypoduct, these bounds show tht ρ tends to the midpoint m = +b 2 s p, which then llows us to povide diect poof of the following convegence

4 1280 G. Ecole et l. / J. Mth. Anl. Appl lim u p = δ, unifomly in Ω,b, 4) p δ whee x ; δx) := b x ; if x m if m x b is the distnce function of x to the boundy of Ω,b. The convegence 4) cn lso be veified by combining some esults on the Diichlet eigenvlue poblem fo the -Lplcin opeto in bounded domin Ω R N. In fct, it ws poved in [15] tht t lest u one subsequence of the fmily p u p } L 1 p>1 exists tht conveges unifomly in Ω to function u nd Ω) tht ny such limit function is positive viscosity fist eigenfunction of the Diichlet -Lplcin. In ddition, fo some specil domins, such s blls, nnuli, nd stdiums, it ws shown in [23] tht the distnce function to the boundy is the only viscosity fist eigenfunction of the Diichlet -Lplcin in Ω, up to some constnt fcto. To the best of ou knowledge, thee hve been no pevious epots of explicit computtions of the fist eigenvlue fo multidimensionl nnuli when 1 <p 2. The only pevious wok to del with the computtion of the dil eigenvlues in multidimensionl nnuli is [9], whee the uthos consideed moe genel dil p-lplcin eigenvlue poblem in the fmewok of the Stum Liouville poblem. They developed numeicl method to compute the dil eigenpis by tnsfoming the second-ode ODE into fist-ode system using genelized Püfe tnsfomtion), whee they pplied shooting lgoithms nd Newton s method. Thei method lso depends getly on the clculus of the genelized sine function intoduced in [13] see lso [20]). The dvntge of ou method is tht it is stightfowd when pplied to the Diichlet eigenvlue poblem fo genuine nnulus 0 < < b), whee ou ppoch only depends on the integl fomule obtined by diect integtion of the dil boundy vlue poblems posed in Ω,t nd Ω t,b. Unfotuntely, we could not compe ou outputs with those epoted in [9] becuse the Diichlet eigenvlue poblem fo genuine nnulus ws not mong the numeicl esults tht the uthos used to illustte thei method. Howeve, it is inteesting to note tht the numeicl esults pesented in [7, Tble 1] fo the fist eigenvlue of the unit bll e highly compble with the coesponding esults pesented in [9, Tble 4], whee they gee up to the thid deciml digit. We should lso note tht ou method cn be teted s outine fo computing the fist eigenpis fo finite numbe of nnuli tht coespond to ptition of the intevl [, b], nd thus it cn be used to compute ny dil eigenpi of the nnulus Ω,b. Fo exmple, in ode to compute the second dil eigenpi, we cn pply ou method to ech t, b) to clculte the Diichlet fist eigenpis γ t), u t, )) nd γ + t), u + t, )) of the nnuli Ω,t nd Ω t,b, espectively. Hence, we djust c such tht γ c) = γ + c) nd we tke this vlue s the second dil Diichlet eigenvlue of Ω,b we know fom [12] tht the j th -eigenfunction hs exctly j 1 zeoes in, b)). The second dil eigenfunction is then obtined by splicing u c, ) with ku + c, ), whee k := u c,c) u + c,c) guntees C1 contct t the oot = c. The constnt k cn be computed esily fom the integl expessions of the deivtives u ±c, c). This scipt cn lso be used to compute ny dil eigenpi of bll by combining ou method with tht employed in [7]. At this point, we should emphsize tht n eigenfunction ssocited with the second eigenvlue of dilly symmetic domin cn be nondil. In fct, the second eigenfunctions of the p-lplcin in pln disc e not dil, ccoding to [22] fo p = 2nd [5] fo ll p > 1. The eminde of this ppe is ognized s follows. In Section 2, we specify the nottions used in this study. In Section 3, we combine scling guments with vitionl chcteiztions of λ p to deive the pioibounds 3) fo the mximum point ρ of u p. In this section, we lso use simil guments to pove tht the functions λ ) nd λ + ) e stictly monotone nd loclly Lipschitz) continuous, s well s to detemine thei behvio t the end points nd b, espectively. In Section 4, we dpt the invese powe

5 G. Ecole et l. / J. Mth. Anl. Appl method developed in [7] to compute the eigenpis λ t), u t, )) nd λ + t), u + t, )) fo ech t, b). In Section 5, we use L Hôspitl ule fo monotonicity see Lemm 13) to obtin the lowe nd uppe bounds fo the fist eigenfunction u p in tems of explicit functions tht depend on p, nd hence we pove the convegence 4). Finlly, in Section 6, fo sevel vlues of p nd N, we pesent numeicl ppoximtions of λ p nd the coesponding gphs of u p. These numeicl esults endose the well-known symptotic behvios of the fist eigenpi when p goes to 1nd when p goes to ). 2. Peliminies nd nottions We ecll tht ρ denotes the mximum point of the fist eigenfunction u p. It is simple to check tht λ p is lso the fist eigenvlue of both eigenvlue poblems N 1 w p 2 w ) = λ N 1 w, < < ρ w) =0=w ρ) 5) nd N 1 w p 2 w ) = λ N 1 w, ρ<<b w ρ) =0=wb). 6) In ddition, the estiction of u p to the intevl [, ρ] is n eigenfunction of 5) coesponding to λ p s well s u p esticted to [ρ, b] is n eigenfunction of 5) coesponding to λ p. In fct, both of these eigenvlues poblems hve the following popety: the only eigenfunction tht does not chnge its sign is the fist. Theefoe, λ p cn lso be witten s ρ λ p =min N 1 w ) p d ρ N 1 w) p d :0 w C1 [, ρ] ) } such tht w) =0=w ρ) 7) s well s ρ λ p =min sn 1 w s) p ds ρ sn 1 ws) p ds :0 w C1 [ρ, b] ) } such tht w ρ) =0=wb). 8) Afte integting 5) with λ = λ p nd w = u p, we lso obtin 0 <u p) = λ p 1 N ρ σ N 1 u p σ) dσ, <ρ 9) nd u p ) = λ p 1 N ρ σ N 1 u p σ) dσ d, ρ. 10) Anlogously, fte integting 6) with λ = λ p nd w = u p, we obtin 0 < u p) = λ p 1 N ρ σ N 1 u p σ) dσ, ρ < b 11)

6 1282 G. Ecole et l. / J. Mth. Anl. Appl nd u p ) = λ p 1 N ρ σ N 1 u p σ) dσ d, ρ b. 12) In ode to clify ou method fo computing the fist eigenpi λ p, u p ), we need to povide some definitions. Fo ech < t < b, we denote λ t) s the fist eigenvlue ssocited with the eigenvlue poblem N 1 w p 2 w ) = λ N 1 w, <<t w) =0=w t) 13) nd u t, ) is the coesponding positive fist eigenfunction such tht u t, ) := mx t u t, ) =1. Anlogously, we denote λ + t) s the fist eigenvlue ssocited with the eigenvlue poblem N 1 w p 2 w ) = λ N 1 w, t<<b w t) =0=wb) 14) nd u + t, ) is the coesponding positive fist eigenfunction such tht u+ t, ) := mx t b u +t, ) =1. The eigenfunctions u t, ) nd u + t, ) belong to the clss C 2 in the closed intevls [, t] nd [t, b] if 1 < p 2nd they belong to C 1, 1 in these intevls if p > 2. It is lso esy to check tht u t, ) is stictly incesing whees u + t, ) is stictly decesing. The eigenvlues λ t) nd λ + t) e positive nd they stisfy, espectively, λ t) = min 0 w C 1 [,t]) w)=0=w t) N 1 w ) p d N 1 w) p d = N 1 u t, ) p d N 1 u t, ) p d 15) nd λ + t) = min 0 w C 1 [t,b]) w t)=0=wb) t N 1 w ) p d t N 1 w) p d = t N 1 u +t, ) p d t N 1 u + t, ) p d. 16) Of couse, λ ρ) =λ p = λ + ρ) nd u p ) = u ρ, ) if ρ u + ρ, ) if ρ b. 17) Remk 1. As shown in Section 3, the function Λt) := λ t) λ + t) is loclly Lipschitz) continuous nd stictly decesing in the intevl, b), nd it stisfies lim Λt) =+ nd lim Λt) =. t + t b Thus, the equlity λ t) = λ + t) occus if nd only if, t = ρ.

7 G. Ecole et l. / J. Mth. Anl. Appl Estimtes nd popeties of the eigenpis λ ± t), u ± t, )) In this section we deive lowe nd uppe bounds fo ρ nd we study the behvio of the pis λ ± t), u ± t, )) with espect to t, b). Lemm 2. Suppose 0 R 1 <R 2 nd tht u C 2 R 1, R 2 )) stisfies τ N 1 u τ) p 2 u τ) ) = τ N 1 f uτ) ) ; R 1 <τ<r 2, whee the function f is continuous. Conside the following function whee v) :=u s) ) ; R 1 <<R 2, s) =α + β, R 1 <<R 2 fo the constnts α nd β such tht minαr 1 + β, αr 2 + β} 0. Then, N 1 v ) p 2 v ) ) N 1)β N 2 = v ) p 2 v )+ α p N 1 f v) ) ; R 1 <<R 2. s) Poof. We hve whee we hve used N 1 v ) p 2 v ) ) = α p 2 α N 1 u s) ) p 2 u s) )) = α p 2 α [ = α p 2 α ) N 1 s) N 1 u s) ) p 2 u s) )) s) ) N 1 ] s) N 1 u s) ) p 2 u s) ) s) ) N 1 [ α p 2 α α d s N 1 u s) p 2 u s) )] s) ds s=s) = α p 2 αn 1)β N 2 u s) ) p 2 u s) ) s) ) N 1 + α p 2 α 2 s) N 1 f u s) )) s) = N 1)βN 2 v ) p 2 v )+ α p N 1 f v) ), s) [ ) N 1 ] ) N 2 s) N 1 s) α =N 1) s) s) s) 2 s) N 1 = N 1)βN 2. s) In the next poposition, we show tht the middius of the nnulus Ω,b is stict uppe bound fo ρ.

8 1284 G. Ecole et l. / J. Mth. Anl. Appl Poposition 3. The following uppe bound fo ρ holds: ρ< + b 2. 18) Poof. We use Lemm 2 with R 1 =, R 2 = ρ, u = u p, fξ) = λ p ξ, nd the constnts α nd β e defined by Note tht the gph of the function α = ρ b ρ < 0 nd β = ρ b ρ > 0. s) =α + β [ρ, b], ρ is the stight line connecting the points, b) nd ρ, ρ) on the s-plne. Thus, the function v) :=u p s) ), ρ stisfies v) = 0, v ρ) = 0, v ) = u ps))α >0 wheneve <ρ. Moeove, v C 1 [, ρ]) C 2 [, ρ)). Theefoe, by Lemm 2, we hve N 1 v ) p 2 v ) ) N 1)β N 2 = v ) + α p N 1 λ p v), <<t. 19) s) Hence, fte multiplying this equlity by v nd integting, we obtin ρ N 1 v ) ρ p d = N 1)β <λ p α p Now, by consideing 15) nd 17), we obtin ρ N 2 s) N 1 v) p d. v ) ρ v)d + λ p α p N 1 v) p d λ p ρ N 1 v ) p d ρ N 1 v) p d ) p b ρ <λ p α p = λ p ρ nd hence we ive t 1 < b ρ ρ. 20) This leds diectly to 18). Remk 4. In the unidimensionl cse N =1), we cn see fom 19) tht v is positive eigenfunction tht coesponds to the eigenvlue λ p α p. This fct implies tht α = 1nd ρ = +b 2. Now, we deive stict lowe bound fo ρ by using 20).

9 G. Ecole et l. / J. Mth. Anl. Appl Poposition 5. The following lowe bound fo ρ holds: b + [ b )N 1 + b )N +1] 1 p 1+[ b )N 1 + b )N +1] 1 p <ρ. 21) Poof. Following fom the ide of the pevious poof, we define ) v) :=u p s), ρ b, whee s) =α + β [, ρ], ρ b, α = ρ < 0 nd β = ρb b ρ b ρ = ρ 1+ α ) > 0. The gph of s is the stight line connecting points b, ) nd ρ, ρ) on the s-plne. Thus, vb) =0, v ρ) = 0nd v ) = u s))α <0 wheneve ρ < b. Then, it follows fom Lemm 2 tht N 1 v ) p 2 v ) ) N 1)β N 2 = v ) + α p N 1 λ p v), ρ<<b. 22) s) Hence, fte multiplying this equlity by v nd integting, we obtin ρ N 1 v ) p d =N 1)β Now, ccoding to 8), we obtin ρ N 2 s) v ) b v)d + λ p α p N 1 v) p d. ρ λ p ρ N 1 v ) p d ρ N 1 v) p d = N 1)β N 2 ρ s) v ) v)d + λ p α p. 23) ρ N 1 v) p d In ddition, by combining 9) with the monotonicity of u p s) fo s [, ρ], we obtin v ) v) = α u ) up ) p s) s) = α u p s) ) λ p s N = α λ p λ p α ρ s) ρ s) = λ p α ρ ρ s) σ N 1 u p σ) dσ ) N 1 σ ) u p s) up σ) dσ) s) ) N 1 ρ u p σ) p dσ ) ρ N 1 s) u p σ) p dσ. )

10 1286 G. Ecole et l. / J. Mth. Anl. Appl Afte chnging the vibles σ = sτ) in the ltte integl, we obtin nd hence we obtin v ) ) N 1 ρ ρ v) λ p α vτ) p αdτ = λ p α p ρ λ p α p ρ λ p α p ρ = λ p α p 1 N N 1)β N 2 ρ s) v ) v)d ρ N 1 v) p d whee we used ρ < +b 2 in the ltte inequlity. By substituting this estimte into 23), we ive t whee k := α 1 = b ρ ρ. Accoding to 20), k >1. Hence, Thus, we hve shown tht k p ) p b ρ = k p ρ 1+ 1 k < 2 bn 1 N 1+ 1 ) b N 1 k N ) N 1 ρ ) b N 1 ρ ) b N 1 ρ ρ vτ) p dτ vτ) p dτ ) N 1 τ vτ) p dτ ρ τ N 1 vτ) p dτ N 1)βλ p α p 1 N βλ p α p 1 N N 1) ρ ρ N 2 s) d N 2 d = ρ 1+ α ) λ p α p N b N 1 ρ N 1) <λ p α p 1+ α ) b N 1 ) + b N, 2 ) b N 1 + b N + b 2 + b 2 ) +1= 2 ) +1 ) +1, ) N 1 b + ) N b +1.

11 G. Ecole et l. / J. Mth. Anl. Appl b ρ ρ < [ ) N 1 b + ) N 1 b p +1], fom which 21) follows. It is inteesting to note tht ou estimtes 21) nd 18) yield the following bounds fo the quotient ρ in tems of the quotient b : b )+[b )N 1 + b )N +1] 1 p 1+[ b )N 1 + b )N +1] 1 p ρ 1+b ). 2 Remk 6. It follows immeditely fom estimtes 18) nd 21) tht lim ρ = + b p 2. 24) As mentioned in the Intoduction, this symptotic behvio is to be expected fom the convegence 4), which cn be obtined by combining esult on the existence in [15] with esult on the uniqueness in [23]. We use 24) in Section 5 to obtin diect poof of 4). In the sequel, we pove some popeties fo the functions λ nd λ + defined in 15) nd 16), espectively. Fist, we pove monotonicity esult, which implies tht the function Λt) = λ t) λ + t) is stictly decesing in the intevl, b). Lemm 7. The function λ is stictly decesing in the intevl, b] nd the function λ + is stictly incesing in the intevl [, b). Poof. Suppose < t 1 <t 2 b nd define ũ) = Obviously, ũ C 1 [, t 2 ]), ũ) = 0 = ũ t 2 ). Thus, λ t 2 ) = N 1 ũ ) p d N 1 ũ) p d 1 u t 1,) if t 1 1 if t 1 t 2. N 1 u t 1,) p d N 1 u t 1,) p d + 2 t 1 N 1 d < 1 theeby showing tht λ is stictly decesing in the open intevl, b]. The poof of the monotonicity of λ + is nlogous. N 1 u t 1,) p d 1 N 1 u t 1,) p d = λ t 1 ), In the eminde of this section, we use the following fomule, which cn be deived esily by integting 13) nd 14): u t, ) = λ t) 1 N t σ N 1 u t, σ) ) dσ, t 25)

12 1288 G. Ecole et l. / J. Mth. Anl. Appl nd u t, ) = λ t) 1 N u + t, ) = λ + t) 1 N u + t, ) = λ + t) 1 N t σ N 1 u t, σ) ) dσ d, t 26) t σ N 1 u + t, σ) ) dσ, t b σ N 1 u + t, σ) ) dσ d, t b. 27) The next esult shows the behvio of the function Λt) = λ t) λ + t) when t tends to the endpoints of the intevl, b). Lemm 8. The following lowe estimtes hold nd N N 1 t N N )t ) λ t), < t b 28) Nt N 1 b N t N )b t) λ +t), t<b. 29) Poof. Since u t, ) = u t, t) = 1, it follows fom 26) tht 1= λ t) 1 N λ t N σ N 1 u t, σ) ) dσ d σ N 1 dσ λ t) t N N d = N N 1 ) t ), fom which 28) follows. Anlogously, we obtin 29) by using 27). Poposition 9. Let < x y<b. Then, whee 0 λ x) λ y) λ x) λ x) λ y) λ y) Cy x) x ) p+n y x ) p 1, 30) x C := N 1) bn 2 N 1 b )p+n 1. 31) Poof. The fist nd second inequlities deive fom the monotonicity of λ t). In ode to pove the thid inequlity, let us define the function v) =u y, s) ), x,

13 G. Ecole et l. / J. Mth. Anl. Appl whee s) = α + β [, y], with We note tht α = y y > 0 nd β = x x x < 0. s) = nd sx) =y, tht is, the gph of s) is the stight line connecting the points, ) nd x, y) on the s-plne. We hve v) = 0, v x) = 0nd v ) =u ) y, s) α>0, <x. Fom Lemm 2, we lso hve N 1 v ) ) N 1)β N 2 = v ) + α p N 1 λ y)v). s) By multiplying this eqution by v) nd integting we obtin x It follows tht x N 1 v ) p d = N 1)β N 2 s) v ) v)d + α p λ y) x N 1 v) p d. x λ y) λ x) N 1 v ) p d x N 1 v) p d = N 1) β x N 2 s) v ) v)d x + α p λ N 1 v) p y), 32) d whee the fist inequlity is due to the monotonicity of the function λ ) nd the second inequlity deives fom the minimizing popety of λ x). We emk fom 25), with t = y, tht y u ) y, s) = λ y)s N s) σ N 1 u y, σ) ) dσ. Hence, v ) v) =α u ) u ) y, s) y, s) α λ y) α λ y) N 1 1 N y y s) σ N 1 u y, s) ) u y, σ) dσ σ N 1 u y, σ) p dσ, whee we used u y, s)) u y, σ) in the second inequlity since s) σ y. )

14 1290 G. Ecole et l. / J. Mth. Anl. Appl Now, fte chnging σ = sτ) in the ltte integl nd noting tht β <0, we obtin Theefoe, x N 1) β theeby implying tht v ) v) α λ y) N 1 = αp λ y) N 1 = αp λ y) N 1 x x y αp+n 1 λ y) N 1 N 2 s) v ) N 1) β xn 2 v)d N 1) β x N 2 s) v ) v)d x N 1 v) p d σ N 1 u y, σ) p dσ sτ) N 1 vτ) p dτ ατ + β) N 1 vτ) p dτ x τ N 1 vτ) p dτ. x v ) v)d N 1) β xn 2 x ) αp+n 1 λ y) N 1 x N 1) β xn 2 x ) αp+n 1 λ y) N 1 N 1) β bn 2 N x )α p+n 1 λ y) y =N 1) x y x b N 2 x ) N x N 1) bn 2 N 1 b )p+n 1 λ y)y x) x ) p+n 1 = Cλ y)y x) x ) p+n 1, whee C is given by 31). Finlly, we combine this estimte with 32) to obtin λ y) λ x) x N 1 v ) p d x N 1 v) p d Cy x) x ) p+n 1 + τ N 1 vτ) p dτ, ) p+n 1 λ y) ) p ) y λ y) x nd hence 30) follows. Poposition 10. Let < x y<b. Then, 0 λ +y) λ + x) λ + y) λ +y) λ + x) λ + x) 1+ y x ) p 1. 33) b y

15 G. Ecole et l. / J. Mth. Anl. Appl Poof. The monotonicity of the function λ + ) yields both the fist nd second inequlities. The thid inequlity follows in simil mnne s in the pevious poof, lthough it is slightly simple. In fct, we define the function v) =u + x, s) ), y b with s) = α + β [x, b] nd α = b x b y > 0 nd β = bx y b y < 0. Hee, the gph of s) is the stight line connecting the points y, x) nd b, b) on the s-plne. By pplying Lemm 2, we obtin N 1 v ) p 2 v ) ) N 1) β N 2 = v ) + α p N 1 λ + x)v). s) Note tht v 0.) Afte multiplying the ltte eqution by v) nd integting, we obtin Theefoe, y N 1 v ) p d = N 1) β α p λ + x) y y N 2 v ) v)d + α p λ + x) N 1 v) p d s) y N 1 v) p d. λ + x) λ + y) y N 1 v ) p d y N 1 v) p d αp λ + x) = ) p b x λ + x) b y nd 33) follows. The following coolly is n immedite consequence of Popositions 9 nd 10. Coolly 11. The functions λ, λ + :, b) 0, ) e loclly Lipschitz) continuous. Theoem 12. The following clims hold tue: 1. λ ρ) = λ + ρ) = λ p. 2. λ t) > λ + t) if < t < ρ. 3. λ t) < λ + t) if ρ < t < b. 4. u p ) = u ρ,) if ρ u + ρ,) if ρ b. Poof. The function Λt) = λ t) λ + t)is continuous t ny t, b) ccoding to Coolly 11. Moeove, it follows fom Lemm 7 tht this function is stictly decesing nd fom Lemm 8 tht lim t t b Λt) =+ nd lim Λt) =. +

16 1292 G. Ecole et l. / J. Mth. Anl. Appl Theefoe, unique vlue of t exists such tht λ t) = λ + t). Since we ledy know tht λ ρ) = λ + ρ), we cn conclude tht such vlue must be ρ, theeby poving the fist clim. The second nd thid clims follow diectly fom the fist given the monotonicity of Λt). Since the positive function v) = u ρ, ) if ρ u + ρ, ) belongs to C 1 [, b]), nd it stisfies mx b v) = 1nd if ρ b N 1 v ) p ρ d N 1 v) p d = N 1 u ρ, ) p d + ρ N 1 u +ρ, ) p d N 1 v) p d = λ ρ) ρ N 1 u ρ, ) p d + λ + ρ) ρ N 1 u + ρ, ) p d N 1 v) p d ρ = λ N 1 u ρ, ) p d + ρ N 1 u + ρ, ) p d p = λ p, N 1 v) p d we must hve v = u p, which poves the fouth clim. 4. Computing the eigenpis λ ± t), u ± t, )) In this section, we show how to pply the invese itetion method to solve, fo ech t, b), the eigenvlue poblems N 1 u p 2 u ) = λ N 1 u p 2 u, < < t 0 <u) < 1=ut), <<t u) =0=u t) 34) nd N 1 u p 2 u ) = λ N 1 u p 2 u, t < < b 0 <u) < 1=ut), t < < b u t) =0=ub). 35) We pesent some esults fo poblem 34) in gete detil whees we simply summize the coesponding esults fo the poblem 35) becuse they e fily nlogous. The method elies on the following lemm, which hs been used fequently s technicl tool in diffeentil geomety see [1]). It is simple consequence of the Cuchy men vlue theoem nd it functions s type of L Hôspitl ule fo monotonicity. Lemm 13. Let g, h : [, b] R be continuously diffeentible with g ) 0fo ll, b). If f is incesing esp. decesing), then the functions f) f) g) g) nd f) fb) g) gb) g e lso incesing esp. decesing). We note tht this lemm is lso in Section 5 to deive estimtes fo the fist eigenfunction u p, which e combined with Theoem 24 to pove the convegence 4).

17 G. Ecole et l. / J. Mth. Anl. Appl Solving the fist eigenvlue poblem in the intevl [, t] Conside the sequence of functions φ n } defined by φ 0 := 1 nd N 1 φ n+1 p 2 φ n+1 ) = N 1 φ n, < < t φ n+1 ) =0=φ n+1t). It is esy to check tht ech φ n is positive nd stictly incesing, nd tht the following fomule hold: φ n+1) = 1 N s N 1 φ n s) ds 36) nd φ n+1 ) = 1 N s N 1 φ n s) ds d. 37) Lemm 14. Fo ll n 1, we hve φ n+1 φ 1 φ n, in [, t]. Poof. Fo n = 1, we hve φ 2 ) = φ 1 1 N = φ 1 φ 1 ). s N 1 φ 1 s) ds 1 N s N 1 ds If we ssume tht the esult is tue fo n = k>1, we obtin φ k+2 ) = 1 N 1 N = φ 1 φ k+1 ). s N 1 φ k+1 s) ds d d d s N 1 φ 1 φ k s) ) ds d Thus, we hve poved the lemm by induction. Lemm 15. Fo ech n 1, we hve lim φ n ) φ n+1 ) = sn 1 φ n s) ds > 0. sn 1 φ n+1 s) ds Poof. Since φ n ) = φ n+1 ) = 0, the lemm follows immeditely fom L Hôspitl s ule nd 36).

18 1294 G. Ecole et l. / J. Mth. Anl. Appl Accoding to this lemm, the function Theoem 16. Fo ech n 0, the function φ n φ n+1 becomes continuous t = if we define φ n ) φ n+1 ) := sn 1 φ n s) ds. sn 1 φ n+1 s) ds φ n φ n+1 is decesing with espect to. Poof. Agin we pply induction on n. The cse whee n = 0is obvious since φ 0 φ 1 = 1 φ 1 nd φ 1 is n incesing function of. Now, let us suppose tht is decesing fo ech k 1,..., n}. Then, the quotient of deivtives φ k φ k+1 sn 1 φ n s) ds) ) sn 1 φ n+1 s) ds) = φn ) φ n+1 ) is decesing function. It follows fom Lemm 13 tht the quotient of deivtives φ n+1) φ n+2 ) = sn 1 φ n s) ds sn 1 φ n+1 s) ds is lso decesing function. Theefoe, Lemm 13 gin implies tht the quotient φ n+1) φ n+2 ) is decesing function nd the poof is complete. Fo ech n 1, let us define the following el numbes: γ n := φ n = φ nt) φ n+1 φ n+1 t) nd Γ n := φ n) φ n+1 ). Coolly 17. We clim the following. 1. γ n } is n incesing sequence. 2. Γ n } is decesing sequence φ 1 γ n Γ n Γ 1 := φ 1) φ 2 ) <. 4. The limits γ = lim λ n nd Γ := lim Γ n exist, nd 1 φ 1 γ Γ sn 1 φ 1 s) ds. sn 1 φ 2 s) ds Poof. The finl clim follows diectly fom the othe thee, which themselves follow diectly fom the fct tht the function is decesing. In fct, this monotonicity implies tht φ n φ n+1 γ n = φ nt) φ n+1 t) lim φ n ) φ n+1 ) φ n) φ n+1 ) = Γ n, which is pt of the thid clim. The fist inequlity in this clim follows diectly fom Lemm 14.

19 G. Ecole et l. / J. Mth. Anl. Appl Moeove, φ n+1 t) = = 1 N 1 N φ nt) φ n+1 t) s N 1 φ n s) ds d 1 ) s N 1 φn s) φ n+1s) ds) d φ n+1 s) 1 N nd thus the fist clim follows. Anlogously, implies tht φ n+1 ) = = 1 N 1 N φ n) φ n+1 ) s N 1 φ n s) ds s N 1 φ n+1 s) ds d d = 1 ) s N 1 φn s) φ n+1s) ds) d φ n+1 s) 1 N s N 1 φ n+1 s) ds d = φ nt) φ n+1 t) φ n+2t), φ n) φ n+1 ) φ n+2) Γ n+1 = φ n+1) φ n+2 ) = lim φ n+1 ) + φ n+2 ) φ n) φ n+1 ) = Γ n. Now, fo ech n 1, we define the function It is esy to check tht u n ) =γ n 1 u n := 1 N φ n φ n. s N 1 u n 1 s) ds d. 38) Theoem 18. The sequence u n } is decesing with espect to n fo ech fixed [, t] nd it conveges unifomly to the fist eigenfunction u t, ). Moeove, λ t) = γ. φ n φ n+1 Poof. Agin, the monotonicity of is impotnt becuse it gives the monotonicity of the sequence of functions u n }. Indeed, since φ i = φ i t), we hve u n ) = φ n) = φ n) φ n+1 ) φ nt) φ n+1 ) = φ n+1) = u n+1 ). φ n φ n+1 ) φ n φ n+1 t) φ n φ n+1 Theefoe, the function u) := lim n u n ) is well defined. Now, by using the Lebesgue Dominted Convegence Theoem, we cn pss to the limit in 38) to conclude tht

20 1296 G. Ecole et l. / J. Mth. Anl. Appl u) =γ 1 N s N 1 us) ds Futhemoe, u n} is lso unifomly bounded since 0 u n) =γ n 1 1 N Γ 1 1 N d, t. 39) s N 1 u n 1 s) ds d s N 1 ds Γ1 tn N. Hence, the Azelà Ascoli theoem shows tht the convegence u n u is unifom in [, t] nd tht u is continuous in this intevl. Hence, the expession 39) implies tht u C 1 [, t]) nd u ) =γ 1 N s N 1 us) ds d, t. Then, fte stightfowd clcultion, we cn veify tht u stisfies N 1 u p 2 u ) = γ N 1 u, < < t, u) =0=u t), theeby implying tht u is n eigenfunction tht coesponds to the eigenvlue γ. Since u 0in [, t] nd ut) = 1, we must hve u = u t, ) nd γ = λ t). The next esult is somewht supising. Poposition 19. The sequence of functions φ n φ n+1 ) } conveges to the constnt function λ t) pointwise in [, t] nd unifomly in ech closed intevl contined in, t]. Theefoe, Γ = λ t). Poof. Fist, we emk tht 1 u n+1 ) = φ n+1 = 1 N 1 N As we know, the sequence φ n φ n+1 } is bounded since γ 1 γ n s N 1 φ n s) ds d ) s N 1 φn s) u n+1s) d. φ n+1 s) φ n) φ n+1 ) Γ n Γ 1, t. Now, we clim tht the sequence of deivtives φ n φ n+1 ) } is lso bounded in ech intevl of the fom [ +ɛ, t]. Fist, we note tht

21 0 G. Ecole et l. / J. Mth. Anl. Appl φn φ n+1 ) ) φn = = φ n φ n+1 φ n φ n+1 Γ 1. φ n+1 φ n+1 φ n+1 φ n+1 φ n+1 Hence, in ode to pove ou clim, it is sufficient to veify tht φ n+1 φ n+1 } is bounded sequence in [ + ɛ, t]. Indeed, fo [ + ɛ, t], we hve φ n+1 ) = = 1 N 1 N s N 1 φ n s) ds s N 1 φ n s) ds φ n+1)d = φ n+1) ) ɛφ n+1). Then, it follows fom the Azelà Ascoli theoem tht subsequence φ n k φ nk } exists tht conveges unifomly to function v in the intevl [ + ɛ, t] nd pointwise in the intevl, t]. Moeove, v is continuous +1 in, t] nd These fcts llow us to pss to the limit in u nk +1 = λ t = γ = vt) v) v) =Γ. 1 N d d ) s N 1 φnk s) u nk+1s) d φ nk +1s) nd to conclude tht u t, ) = 1 N s N 1 vs) u t, s) d. It follows tht u t, ) ) = 1 N Futhemoe, we hve ledy poved tht u t, ) ) = 1 N s N 1 vs) u t, s) ds. s N 1 λ t)u t, s) ds. Hence, by combining these expessions fo u t, )), we conclude tht s N 1 vs) λ t) ) u t, s) ds =0.

22 1298 G. Ecole et l. / J. Mth. Anl. Appl Given tht vs) λ t), this implies tht vs) = λ t) fo ll s, t]. Since the limit function v does not depend on pticul subsequences, we conclude tht the whole sequence φ n φ n+1 } conveges to λ t unifomly in ech closed intevl contined in, t] nd pointwise in this intevl nd, of couse, the sme convegence occus fo the sequence φ n φ n+1 ) }, the limit of which is λ t). φ n ) φ n+1 ) = We ecll tht ) Γ. Theefoe, in ode to complete the poof, we need to show tht Γ = λ t. Howeve, this follows fom Lebesgue s Dominted Convegence Theoem fo φ n) φ n+1 ) = φ n ) φ n+1 sn 1 φ n 1 s) ds sn 1 φ n s) ds = sn 1 φ n 1s) φ n s) ) φ ns) = λ t. φ n ) ds sn 1 φ ns) φ n ) ds 4.2. Solving the fist eigenvlue poblem in the intevl [t, b] t sn 1 λ t)u t, s) ds sn 1 u t, s) ds The esults given in this subsection e fily nlogous to those pesented in the pevious subsection. Thus, they e simply stted nd the coesponding poofs e omitted. The invese itetion method fo obtining the fist eigenpi λ + t), u + t, )) is bsed on the sequence of itetions φ n } defined by: φ 0 := 1 nd nd N 1 φ n+1 p 2 φ n+1) = N 1 φ n, t<<b φ n+1t) =0=φ n+1 b). These functions e stictly decesing, nd they stisfy φ n+1) = φ n+1 ) = t t 1 ) N 1 s φ n s) ds) 1 ) N 1 s φ n s) ds) d. 40) The next esult is lso consequence of Lemm 13 nd it compises the bsis fo the emining esults. Theoem 20. Fo ech n 0, the function φ n φ n+1 is incesing with espect to. We note tht the only diffeence comped with the coesponding esult in the pevious section is tht φ the monotonicity of the function n φ n+1 is evesed. Howeve, we show tht this fct llows us to mintin the sme type of monotonicity fo the othe coesponding sequences, which e defined in the sequel. Fo ech n 1, let us define the numbes nd γ n := φ n = φ nt) φ n+1 φ n+1 t)

23 G. Ecole et l. / J. Mth. Anl. Appl s well s the function Γ n := φ nb) φ n+1 b) = lim b Theoem 21. The following clims hold. φ b n) φ n+1 ) = u n := 1. γ n } is n incesing sequence. 2. Γ n } is decesing sequence φ 1 γ n Γ n Γ 1 := φ 1b) φ 2 b) <. 4. The limits γ = lim λ n nd Γ := lim Γ n exist, nd 1 φ 1 γ Γ φ n φ n. t sn 1 φ n 1 s) ds t sn 1 φ n s) ds t sn 1 φ 1 s) ds. t sn 1 φ 2 s) ds, 5. The sequence u n } is decesing with espect to n fo ech fixed [t, b] nd it conveges unifomly to the fist eigenfunction u + t, ). Moeove, λ + t) = γ. 6. The sequence of functions φ n φ n+1 ) } conveges to the constnt function λ + t) pointwise in [t, b] nd unifomly in ech closed intevl contined in [t, b). Theefoe, Γ = λ + t). 5. Asymptotics s p In this section, we use the symptotic behvio 24) nd Lemm 13 to pove tht u p conveges unifomly δ to δ, whee ; δ) := b ; if m if m b is the distnce function to the boundy of the nnulus Ω,b nd m := +b 2 is its middius. We lso use well-known esult see [15], o [11] fo diffeent poof) tht is vlid fo ny bounded domin Ω, which sttes tht [ lim λp Ω) ] 1 p 1 =, p d, Ω) whee λ p Ω) denotes the fist eigenvlue of the p-lplcin Diichlet of Ω nd dx, Ω) denotes the distnce of x Ω fom the boundy Ω. In ou cse, this mens tht lim λ 1 p = 1 = 1 p δ δm) = 2 b. 41) In the next lemm, we estimte u p in tems of the following functions φ ρ ) := ρ 1 N σ N 1 dσ d, ρ nd

24 1300 G. Ecole et l. / J. Mth. Anl. Appl nd φ + ρ ) := 1 N Afte consideing 24), it is esy to check tht Lemm 22. The following estimtes hold: ρ σ N 1 dσ d, ρ b. lim p φ ρ ) = = δ), <m 42) lim p φ ρ ) =b = δ), m b. 43) nd λ 1 p ρ σn 1 u p σ) dσ ρ φ ρ ) u p ) φ ρ ) σn 1 dσ φ ρ ρ), ρ 44) λ 1 ρ σn 1 u p σ) dσ p ρ σn 1 dσ Poof. Fist, we pove 44). Since the function is incesing, Lemm 13 shows tht the function φ + ρ ) u p ) φ+ ρ ) φ +, ρ b. 45) ρ ρ) [, ρ) ρ σn 1 u p σ) dσ) ρ = u σn 1 dσ) p ) [, ρ) is incesing in the intevl [, ρ). Thus, the function ρ σn 1 u p σ) dσ ρ σn 1 dσ u ρ p) φ p ) ) = λ 1 σn 1 u p σ) dσ p ρ σn 1 dσ is lso incesing in this intevl. Theefoe, by pplying Lemm 13, we gin conclude tht the function is incesing in the intevl [, ρ). u p ) φ p ) It follows tht u p ) φ p ) lim u p σ) σ + φ p σ) = lim u pσ) + φ p ) σ) = λ 1 p ρ σn 1 u p σ) dσ ρ, σn 1 dσ < < ρ nd u p ) φ p ) lim u p σ) σ ρ φ p σ) = u pρ) φ p ρ) = 1 φ p ρ), <<ρ, theeby poving 44). We cn pove 45) in n nlogous mnne.

25 G. Ecole et l. / J. Mth. Anl. Appl Theoem 23. The following unifom convegence in Ω,b holds: Poof. We know tht lim u p = δ. p δ u p ) 1 N λ 1 p = 1 N λ 1 p ρ σn 1 u p σ) dσ ρ σn 1 u p σ) dσ if ρ if ρ b nd tht 0 u p 1. Theefoe, which implies tht u p ) 1 N λ 1 p ρ σn 1 dσ ρ σn 1 dσ if ρ if ρ b, lim sup u p ) 1 N = 21 N p δ b, b. Then, we cn pply the Azelà Ascoli theoem to obtin sequence p n such tht u pn conveges unifomly in [, b] to continuous function u. Of couse, um) = 1since u pn ρ n ) = 1nd ρ n m hee, ρ n := ρp n ) < m). Let us tke < τ 1 <m < τ 2 <b. Then, fte using the monotonicity of ech function u pn nd 24), we obtin lim ρn σ N 1 u pn σ) p n 1 dσ pn 1 lim ρn τ 1 lim u pn τ 1 ) σ N 1 u pn σ) pn 1 dσ ρn τ 1 σ N 1 dσ pn 1 pn 1 = uτ 1 ). Anlogously, lim b ρ n σ N 1 u pn σ) pn 1 dσ Thus, by mking τ 1, τ 2 m, we conclude tht nd lim ρn pn 1 lim uτ 2 ) σ N 1 u pn σ) p n 1 dσ τ2 m σ N 1 u pn σ) p n 1 dσ lim pn 1 τ 2 m σ N 1 dσ pn 1 pn 1 = uτ 2 ). um) = 1 46)

26 1302 G. Ecole et l. / J. Mth. Anl. Appl lim b σ N 1 u pn σ) pn 1 dσ pn 1 um) =1. 47) ρ n By combining these two symptotics with 42), 43), nd 41), we cn conclude fom Lemm 22 tht δ) lim λ 1 p φ ρ δ n ) u) lim φ ρ n ) φ ρ n ρ n ) = δ) m = δ), δ <<ρ nd δ) lim λ 1 p φ + ρ δ n ) u) lim φ+ ρ n ) φ + ρ n ρ n ) = δ) b m = δ), δ ρ<<b. Thus, we hve veified tht u = δ δ δ in [, b]. Futhemoe, s p, the function δ is the limit of ny unifom convegent sequence extcted fom the fmily u p } p>1. This fct implies tht u p conveges to unifomly in [, b] nd tht δ δ lim p ρ 6. Numeicl esults σ N 1 u p σ) dσ = lim p ρ σ N 1 u p σ) dσ =1. In this section, we pesent some numeicl esults, which we obtined by implementing ou method fo the nnulus Ω 1,3 = 1 < x < 3} R N, fo N =2, 3, nd 4. Befoe poceeding, fo ech t, b), let us denote the sequences φ n t, )} nd φ n t +, )}, espectively, by φ 0 t,) 1; φ n+1 t,):= 1 N s N 1 φ n t,s) ds d; t t nd φ 0 t +,) 1; φ n+1 t +,)= 1 N t s N 1 φ n t +,) ds t b d; t b. Using this nottion, it follows fom Section 4 tht ) φn t, ) λ t) = lim nd u p t,) = lim φ nt,) ; φ n+1 t, ) φ n t, ) t

27 G. Ecole et l. / J. Mth. Anl. Appl Tble 1 Some fist eigenvlues λ p nd the coesponding mximum points ρ fo the nnulus Ω 1,3 in dimensions N =2, 3, nd 4. p N =2 N =3 N =4 λ p ρ λ p ρ λ p ρ nd tht ) φn t +, ) λ + t) = lim nd u p t +,) = lim φ nt +,) ; t b. φ n+1 t +, ) φ n t +, ) We lso ecll tht φ n t ±, ) = φ n t ±, t ± ).) Ou numeicl esults fo the nnulus Ω 1,3 wee obtined by implementing the following lgoithm fo = 1nd b = 3in C++ using the GCC compile. Algoithm Set N, p,, nd b. 2. Set t 1 := b+[ b )N 1 + b )N +1] 1 p 1+[ b )N 1 + b )N +1] 1 p nd t 2 := +b 2 ccoding to 3)). 3. Solve Λρ) = 0by using mtching pocedue with initil estimtes of t 1 nd t 2 ). 4. Retun ρ the oot of Λ, which is the ppoximtion fo the mximum point of u p ). 5. Compute the pis λ ρ), u ρ, )) nd λ + ρ), u + ρ, )) by invese itetion). 6. Retun λ := λ ρ)+λ + ρ) 2 the ppoximtion fo λ p ). u ρ,) if ρ 7. Retun u p ) := the ppoximtion fo u p). u + ρ,) if ρ< b To solve Λρ) := λ ρ) λ + ρ) =0in step 3, we used the secnt method with initil estimtes of t 1 nd t 2, s defined in step 2. Thus, we geneted finite sequence t 3, t 4,..., t n } t 1, t 2 )using the ecuence fomul t i = t i 2Λt i 1 ) t i 1 Λt i 2 ), i =3, 4,...,n Λt i 1 ) Λt i 2 ) with the stopping citeion t n t n 1 t n < Theefoe, we took ρ := t n in step 4. In ode to compute λ t i )nd λ + t i )in step 3 nd thus to compute Λt i )), we pefomed 10 itetions of the invese itetion method, i.e., we computed the functions φ 1 t i ) ±, ),..., φ 10 t i ) ±, ) nd, fo ech of these functions, we used 101 nodes in the -intevls [, t i ]nd [t i, b]. The coesponding integls wee computed using Simpson s ule. The sme pocedue ws employed in step 5 to compute λ ± ρ), u ± ρ, )) using ρ obtined in step 4. Some of the numeicl esults we obtined fo the nnulus Ω 1,3 = 1 < x < 3} R N e pesented in Tble 1, fo N =2, 3, nd 4. In the line cse, p = 2, the 3-dimensionl fist eigenpi is esie to obtin nlyticlly. In fct, fte mking convenient chnge of vible, we cn tnsfom the oiginl eigenpoblem into simple one to

28 1304 G. Ecole et l. / J. Mth. Anl. Appl Fig. 1. Gphs showing the L -nomlized fist eigenfunction fo N = 2 top), N = 3 middle), N =4bottom), nd some vlues of p. The mximum points nd thei espective lowe bounds given by 3) e highlighted. obtin both the fist eigenvlue λ = π b )2 nd the fist eigenfunction u) = b π π ) sin b ).

29 G. Ecole et l. / J. Mth. Anl. Appl Fig. 2. Gphs fo the function p λ 1 p when N =2, 3, nd 4, ll of which ppoch Λ = 1 when p is lge. By witing the mximum point of u s ρ = μ b π, we cn see tht μ must stisfy the eqution μ =tn μ π ). 48) b Thus, when =1nd b =3, the exct fist eigenvlue is λ = π nd the oot of 48) in the intevl π 2, 3π 2 )is μ Hence, ρ Fom Tble 1, we note tht these vlues fo λ nd ρ gee with those obtined in ou numeicl esults. Fig. 1 shows the gphs of some of the L -nomlized fist eigenfunctions fo the nnulus Ω 1,3. We note tht the gphs ppe to esemble the gph of the chcteistic function of the nnulus s p tends to 1. This suggests tht the Cheege set of n nnulus is the nnulus itself. In the 2-dimensionl cse, this follows fom the esults given in [18] the Cheege poblem is ssocited with the p-lplcin s p 1vi the fist eigenpoblem in [17] nd the p-tosion functions in [10]). In Fig. 1, we cn see tht s p, the L -nomlized fist eigenfunction ppoches the L -nomlized distnce function to the boundy, s descibed in Section 5 see the gphs fo p = 10 4 in Fig. 1). Moeove, the gphs in Fig. 1 show tht ou lowe bound fo the mximum point given in 3) impoves s p inceses nd it vitully coincides with the mximum point fo p = The symptotic behvio of λ 1 p s p is shown clely in Fig. 2, whee we plot sevel vlues of λ p fo p, which vy fom 1.1 to As mentioned in Section 5, the p th oot of the fist eigenvlue of the p-lplcin in bounded domin Ω conveges to Λ := R 1, whee R is the indius of Ω the dius of the lgest bll contined in Ω). Fo the nnulus Ω 1,3, we hve Λ =1. In Fig. 3, we finish this section by pesenting the gphs fo the fouth, fifth, sixth, nd seventh itetions n = 3, 4, 5, nd 6, esp.) of the functions φ nρ,) φ n+1 ρ,) ) nd φ nρ +,) φ n+1 ρ +,) ) fo p = 1.5 left) nd p = 4 ight). We cn obseve the inteesting fcts poved in Section 4: the convegence of the fist sequence to λ p is unifom with espect to in ny closed intevl contined in, ρ] nd pointwise in the whole intevl [, ρ], nd the convegence of the second sequence is unifom with espect to in ny closed intevl contined in [ρ, b) nd pointwise in the whole intevl [ρ, b]. In ech gph, we cn lso see tht the convegence of the ight-hnd side section which coesponds to φ nρ +,) φ n+1 ρ +,) ) ) is slowe thn the left-hnd side section. This

30 1306 G. Ecole et l. / J. Mth. Anl. Appl Fig. 3. Gphs of the fouth, fifth, sixth, nd seventh itetions n = 3, 4, 5, nd 6, esp.) of the functions φ nρ,) φ n+1 ρ,) ) nd φ nρ +,) φ n+1 ρ +,) ) fo p = 1.5 left) nd p = 4 ight) in dimensions N =2, 3, nd 4. The contct points t = ρ e highlighted. is pobbly becuse which cn be checked esily) φ n ρ, ) is concve in the -vible whees φ n ρ +, ) chnges the concvity t point close to b. Acknowledgments The uthos e gteful fo the suppot of CNPq /2013-1) nd FAPEMIG CEX-PPM ), Bzil. Refeences [1] G.D. Andeson, M.K. Vmnmuthy, M. Vuoinen, Inequlities fo qusiconfoml mppings in spce, Pcific J. Mth [2] B. Andeinov, F. Boye, F. Hubet, Finite volume schemes fo the p-lplcin on Ctesin meshes, ESAIM Mth. Model. Nume. Anl. 38 6) 2004) [3] J. Benedikt, P. Dábek, Estimtes of the pincipl eigenvlue of the p-lplcin, J. Mth. Anl. Appl ) [4] J. Benedikt, P. Dábek, Asymptotics fo the pincipl eigenvlue of the p-lplcin on the bll s p ppoches 1, Nonline Anl ) [5] J. Benedikt, P. Dábek, P. Gig, The second eigenfunction of the p-lplcin on the disk is not dil, Nonline Anl ) [6] R.J. Biezune, J. Bown, G. Ecole, E.M. Mtins, Computing the fist eigenpi of the p-lplcin vi invese itetion of subline supesolutions, J. Sci. Comput [7] R.J. Biezune, G. Ecole, E.M. Mtins, Computing the fist eigenvlue of the p-lplcin vi the invese powe method, J. Funct. Anl ) [8] R.J. Biezune, G. Ecole, E.M. Mtins, Computing the sin p function vi the invese powe method, Comput. Methods Appl. Mth [9] B.M. Bown, W. Reichel, Computing eigenvlues nd Fučik-spectum of the dilly symmetic p-lplcin, J. Comput. Appl. Mth [10] H. Bueno, G. Ecole, Solutions of the Cheege poblem vi tosion functions, J. Mth. Anl. Appl ) [11] H. Bueno, G. Ecole, A. Zumpno, Positive solutions fo the p-lplcin nd bounds fo its fist eigenvlue, Adv. Nonline Stud ) [12] M. DelPino, R. Mnásevich, Globl bifuction fom the eigenvlues of the p-lplcin, J. Diffeentil Equtions ) [13] Á. Elbet, A hlf-line second ode diffeentil eqution, Colloq. Mth. Soc. János Bolyi [14] X. Feng, Y. He, High ode itetive methods without deivtives fo solving nonline equtions, Appl. Mth. Comput [15] J. Juutinen, P. Lindqvist, J. Mnfedi, The -eigenvlue poblem, Ach. Rtion. Mech. Anl )

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