G. Barles

Annales de la Faculté des Sciences de Toulouse

ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSEG. BARLESRemarks on uniqueness results of the firsteigenvalue of the p-laplacianAnnales de la faculté des sciences de Toulouse 5e série, tome 9, no 1(1988), p. 65-75<http://www.numdam.org/item?id=AFST_1988_5_9_1_65_0>© Université Paul Sabatier, 1988, tous droits réservés.L’accès aux archives de la revue « Annales de la faculté des sciences deToulouse » (http://picard.ups-tlse.fr/~annales/) implique l’accord avec lesconditions générales d’utilisation (http://www.numdam.org/conditions).Toute utilisation commerciale ou impression systématique est constitu-tive d’une infraction pénale. Toute copie ou impression de ce fichierdoit contenir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/- 65 -Remarks on uniqueness results of thefirst eigenvalue of the p-LaplacianG. BARLES (1)Annales Faculte des Sciences de Toulouse Vol. IX, n°1 1988RSUM.- Nous prouvons que la premiere valeur propre du laplacien-pest associee a une fonction propre qui est strictement positive etunique a une constante multiplicative pres si ~ est connexe. De plus, nousmontrons que c’est la seule valeur propre associee a une fonction proprepositive.ABSTRACT. 2014 We prove that the first eigenvalue of the p-Laplacian isassociated to a positive eigenfunction, which is unique up to amultiplicative constant if 8Q is connected. Moreover, we show that it is theunique eigenvalue associated to a non negative eigenfunction.IntroductionIn this work, we are interested in the properties of the "first eigenva,lue"of the p-laplacian. Let us recall that the typical eigenvalue problem for thep-laplacian is to find A E R and u( 0) in weak solution of :where H is a bounded domain of RN. In all the following, we will alwaysassume that the boundary 8 S2 is of class C’2~~. The problem of the existenceof such À and u has been studied recently by J.P. GARCIA- AzORERO andI. PERAL-ALONZO [5] : They show the existence of an increasing sequenceof eigenvalues such that a k -~ (1) Ceremade - Université de Paris IX-Dauphine. Place de Lattre de Tassigny. 75775Paris Cedex 16.We prove here that, roughly speaking, the first eigenvalue Ai, which isdefined bypossesses all the properties of the first eigenvalue of a second -order nondegenerate elliptic operator as soon as aSZ is connected. Ai is associatedto a eigenfunction which is positive in H and is unique (up to amultiplicative constant). Moreover, Ai is the unique eigenvalue associatedto a nonnegative eigenfunction. This kind of properties has been studied forN = 1 by M. OTANI [13] and in a ball of RN by F. de THELIN [4].Of course, the main difficulty to prove these results relies on the - a priori- lack of regularity of the eigenfunctions. For example, in the unit ball, theexistence of a radially symmetric eigenfunction, for which it is easy to showthat it is smooth except perhaps in zero, simplifies all the arguments below.Therefore, the first part of this paper is devoted to the study of the regularityof a solution u of (1) for A = À1. It would be too long to mention here allthe works (that we know!) concerning the regularity for similar degenerateelliptic equations; we only give here those we have directly used and referthe reader to the references in these works. Essentially, these results areof two types: the first one concerns the minima (and quasi-minima ) in thecalculus of variations (cf. E. de GIORGI [3], M. GIAQUINTA and E. GIUSTI [6],E. Di BENEDETTO and N.S. TRUDINGER [2]).These results give the local Holder continuity and provide Hanack in-equalities. Let us just recall that, in our case, the eigenfunctions for Ai areminima ofThe second ones concerns degenerate elliptic equations on divergence form(cf. E. Di BENEDETTO [I], D.A. LADYZENSKAYA and N.N. URALTSEVA [9],P. TOLKSDORF [14], K. UHLENBECK [16]). Very often, as in our case, itis the Euler equation of a variational problem. These results give the regularity of the weak solutions. Finally, in P. TOLKSDORF [15], an argument,which allows to use the Schwarz reflection principle, gives the globalregularity in certain cases. Using these results and these methods, we showthat every eigenfunction for Ai is positive in Q and belongs to forsome 0  a  1 and is of class in a neighbourhood of a Q. These resultsdo not use the connectedness of 8 S~.In the second part, we are interested in the uniqueness results. Let us firstmention that similar uinqueness problems for nonlinear "first eigenvalue "have been considered in P.L. LIONS [11,12] and that we use here the samegeneral ideas as in [11,12]. The main point is to have a strong comparisonprinciple which allows to use a method due to T. LAETSCH [10]. Therefore,we first prove a strong comparison property between a weak subsolution uand a weak supersolution v, both in n C(H) ofwhere / ~ Lq(O) (1 p + 1 q = 1). We show that if we assume that there existsa neighbourhood of 8SZ in which u is C2, v is C’1 and > 0 then eitheru and v coincide in this neighbourhood or u  v in H. Moreover, if weassume that u, v are positive in Q, then either u - v in a neighbourhoodof ~03A9 or there exists 8 > 1 such that 8u  v. To do so, we need theconnectedness of This results complements the strong comparisonprinciple of P. TOLKSFORF [15]. Then, we prove the uniqueness resultsannounced above by the Laetsch’s method and we conclude this paperby mentionning a bifurcation results which can be treated by the samemethods.N.B.. - When this manuscript was typed, we have heard from J.I. DIAZand J.E. SAA that they have obtained very general uniqueness results forequations with the p-Laplacian. (See [17,18]). We have also learned thatS. SAKAGUCHI has obtained independently some results similar to ours.I. On the eigenfunction for AiIn this section, we investigate the properties of a non-zero solution u inofLet us remark that every solution u of (4) satisfiesand therefore, the variational problem (5) is equivalent to (4), which isits associated Euler Equation. This property will allow us to use resultsconcerning both minima in the calculus of variations and degenerate ellipticequations on divergence form. Our results are t.he following..PROPOSITION I.I.- let u be a weak solution of (/ ) different from 0,then u e C§’"(Q) for some 0  a  I and satisfiesocPROPOSITION L2. - Let u be a weak solution of (.~~, thenTHEOREM 1.3.- Let u be a weak solution of (l~, then there existsa E (0,1) and e > 0 such thatwhereBefore proving these results, let us precise that the local regularityis a consequence of the results of E. de GIORGI [3], O.A. LADYZENSKAYA andN.N. URALT’SEVA [9] or M. GIAQUINTA and E. GIUSTI [6]. Moreover, the localbounds in Lloo depend only on the LP norm of u and the distance to an(cf. E. Di BENEDETTO and N.S. TRUDINGER [2], via Harnack inequalities).Then, one gets easily estimates by the results of E. Di BENEDETTO[1] or P. TOLKSDORF [14]. So, in theorem 1.3, only the boundary estimateshave to be shown. To do so, we adapt a method of P. TOLKSDORF [15] whichreplaces the boundary estimates by an interior estimates via the Schwarzreflection principle.so u+ is a solution of (4) and (5). But, by a result of E. Di BENEDETTO andN.S. TRUDINGER [2], the nonnegative solutions of (5) satisfy an Harnackinequality. Then either u+ > 0 and so u > 0, or u-t- = 0. In this last case,the same argument for -u yields the result.Proof of Proposition L2. Of course, this result is interesting only in thecase when p  N since if p~> N, Wo’p(S2) ~ for some ’0  a  1 ~ ~and if p = N, Wo’p(~) ~ Lq(O) for all q E f 1, ~-oo~. So, we assume thatp  N. Let u be a solution of (4). Changing u in -u, we may assume thatu > 0 in H, by Proposition 1.1. Now, we define v byand we take c,p = (k > 0) in (4); we obtainThe left-hand side is equal toand with q = Np. ~ (N - p)-1. ThereforeFinally, if u E we choose k such thatand the right-hand side is estimated byWe conclude that u E L’’ (Q), where r’ = (k + l)q = qr . But q > p, so anpeasy bootstrap argument gives the result.Proof of Theorem L 3 . The proof consists essentially in obtainingboundary estimates in by using the method of P. TOLKSDORF [15]and the estimates of E. Di BENEDETTO [1]. Since the arguments are routineadaptations of those of [15] and ~1~, we only sketch the proof. Let :co E 80;since ~03A9 is a C2,03B2- surface, there exists a C2,03B2 diffeomorphism 5p froma neighbourhood of 0 in RN-l into a neighbourhood of :ro in 80. Weextend it to a C1,03B2 diffeomorphism 03C8 from a neighbourhood of 0 inH+ = {(y’, yn) ~ RN-1 R+} in a neighbourhood of xo in S2 by settingwhere n(x) is the outward unit vector at x E 8 SZ. This transformation hasthe following property: if we define v bythenwhereand A( y) is, for all y, an invertible (N - 1) x ( N - 1) matrix. So, thistransformation decouples the (N -1) first derivatives, and the ~ derivative.A similar idea was used by R. JENSEN [8]. Finally, if u is solution of (5), vis solution of ,,wherefor R, r~ small enough, andNow, we use the Schwarz reflection principle i.e., we extend v to U =I _ ~~ Iyn ’?} by settingand the other functions are extended by parity on yn . The particular formof J implies that v is solution of an analogous variational problem in ’Then, if we denote byfor y E !7, p’ E pn E R, v is a weak solution ofin U+ and U- But, since v and its derivatives are continuous on yn = 0,we have (6) in U and v E Now, we use the C1,03B1loc-estimates of E. Di BENEDETTO [1]. Let us justrecall that these estimates depend only on N, p, 03C8, Ai and the Lq-norm offor a q > Np’, ( " + 1 - , = 1). Finally, let u be a solution of (4) and(5) different from zero; we consider the approximated problemsBy standard arguments, ue exists and belongs to for all q E (1, -~oo( [since E for all q E (1, -i-oo(. Moreover, uf satisfies the a prioriestimates above which depend only on the constants of the problem and theLq norm of u for a q > Np’. So, by Ascoli theorem, there exists a such that ue converges in and the classical uniqueness result forthe Dirichlet problem (3) implies that the limit is necessarely u. Hence,u E for some 0  a  1. Moreover, since u > 0 in 0 or - u > 0 inH, by the strong maximum principle (cf. P. TOLKSDORF [15]), we havetherefore, since E there exists ~ > 0 such thatSo, in (4) becomes strictly elliptic and, by classical results, (seeD. GILBARG and N.S. TRUDINGER [7]), u E C2,03B2(03A9~), for all ~  ~.II. Uniqueness resultsThe aim of the section is to prove that the positive eigenfunctionassociated to Ai is unique (up to a multiplicative constant) and that Ai isthe unique eigenvalue associated to a nonnegative eigenfunction,in the casewhen ~03A9 is connected. We will assume, in all this part, that ~03A9 is connected.Before giving this result, we need the following strong comparison result.THEOREM 11.1.- Let u and v be respectively weak sub and supersolutionof (3) in n ~’(S~). We assume in addition that there exists E > 0such that u E with > 0 in SZE and v E Then, eitheru = v in n or u  v in Q. Moreover, if u and v are positive in SZ, theneither u - v in SZ~ or there exists 8 > 1 such that 8u  v.Remark 1. Of course, the same result holds if v E with|~v| > 0 in 03A9~ and u ~ Remark 2. - In order to compare the result of Theorem II. 1 with thestrong comparison result of [15], we remark that we only need u and v tobe smooth in a neighbourhood of a S2; but with this weaker assumption, ourresult is weaker than the result of [15] since if u( x) = v( x) for some x E Q,we can only conclude that u - v in Nevertheless, we are more interested- for application to the eigenvalue problem - in the case when u, v > 0 andthis type of result was not considered in [15].Now, we can state theTHEOREM 11.2.2014 Let v E be a weak solution of- div in n (7)with v > 0 in Q, then u a positive eigenfunction for then(z) À = ÀI(ii) ~~c E R such that v = ~cu.Remark 3. Let us just remark that weak solutions of (7) in Wo’p(S2)are Holder continuous: this is a consequence of the fact that such solutionsare in the De GIORGI classes (cf. [2], [3], [6]...). Therefore, the Harnackinequality of [2] concludes that v > 0 in SZ.We first prove Theorem II.2, using Theorem 11.1. .Proof of Theorem Let Xo E SZ~ being defined in Theorem 1.3.Let = and = u. We are going to prove that ~ == v. Inu(x° ) ~  fact, first, we want to prove that 03C9 ~ 0. To do so, we use a method due toTh. LAETSCH [10]. We define to byIf to = 1, the result is proved. If to  1, tocv is subsolution ofbecause 03BB ~ Ai and tocv  v. By Theorem II. 1, either t003C9 - v in 03A9~ orthere exists 0 > 1 such that v. But, at xo E  v(xo ); so,we are in the second case and the inequality 03B8t003C9  v with 8 > 1 contradictsthe definition of to. Hence, to - 1 and the result is proved. Now arguingexactly as before with to = 1, we find that 03C9 ~ v in therefore A = Aiand the same argument as above proves the opposite equality > v.Now, we turn to the proof of Theorem 11.1.Proof of Theorem II.1. - We only treat the case when u and v are positivein H, the other result being obtained in the same way. Since v E C1 (U ~~,where U’1 = {x E  d(x, 8 ~)  E~, we can use the strong comparisonpriciple of P. TOLKSDORF [15] in U,~ since we already know by the weakmaximum principle that u  v in H. Hence, if u is not equal to v in SZE thenfor ~ small enough U" is connected and this result givesIn particular, on {x E = ~/2}, u  v. Let b be defined on 03A9~/2byd( ~, a SZ~ is on 5~~~2 and for C and a small enough, it is easy to checkthat u + b is still a subsolution ofand that u + b  v on So, by the weak maximum principleNow, we can make two remarks. First, since Vu is bounded in andsince > aC > 0 in S2E~2, there exists r~l > 0 such thatThe second remark is that, on c~(S2 - where 6 = C exp a~ - 1 . So, by the weak maximum principle inS~E~2, we haveDenoting by r~2 = inf ~S(u(x))-1, x E S~ - SZE~2 ~, we haveIf we set ~ = we getand the proof is complete.Let us conclude by giving a bifurcation result which can be proved bythe same methods as above.THEOREM 11.3. 2014 Let f : R ~ R be a nondecreasing function satisfyingf (o) = 0. Y~e assume that t -~ is decreasing for t > 0 and that ~1. . Then+00(g) I 03BB1, 0 ia- the unique nonnegative solution ofIf > 03BB1, there exists exactly two nonnegative solutions ofo~’8~ : 0 and a solution which is positive i~a ~.Réfereuces[1] DI BENEDETTO (E.).2014 C1+03B1 local regularity of weak solutions of degenerateelliptic Equations. Non linear Anal. TMA, Vol 7, N°8, 1983.- 75 -[2] DI BENEDETTO (E.) and TRUDINGER (N.S.).2014 Harnack inequalities for quasi-minima of variational integrals . Anna. Inst. H. Poincaré. Anal. Non Lin., Vol 1,N°4, 1984.[3] DI GIORGI (E.). 2014 Sulla differenziabilita e l’analitica delle estremali degli integralimultipli regolari. Mem. Acad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) t.3, 1957.[4] DE THELIN (F.).2014 Sur l’espace propre associé à la première valeur propre dupseudo Laplacien dans la boule unité. CRAS, Paris, série 198.[5] GARCIA AZORERO (J.P.) and PERAL ALONSO (I.) .2014 Existence and nonunique-ness for the p-Laplacian : non linear eigenvalues. Preprint.[6] GIAQUINTA (M.) and GIUSTI (E.).2014 Quasi minima. Annal. Inst. H. Poincaré.Anal. Non Lin. Vol. 1, N°2, 1984.[7] GILBARG (D.) and TRUDINGER (N.S.). 2014 Elliptic partial differential equations ofsecond order. 2nd édition . Ed. Springer-Verlag (New-York), 1983.[8] JENSEN (R.). 2014 Boundary regularity for variational inequalities. Indiana Univ.Math. J. 29 1980.[9] LADYZENSKAYA (O.A.) and URALTSEVA (N.N.). 2014 Linear and quasi linear ellipticequations. Academic Press (New-York), 1968.[10] LAETSCH (Th.). - A uniqueness theorem for elliptic quasivariational inequalities.J. Funct. Anal. 12, 1979.[11] LIONS (P.L.).2014 Two remarks on Monge Ampère Equations. Annali di Matematicapura ed applicata (IV). vol. CXLII.[12] LIONS (P.L.). - Bifurcation and optimal stochastic control. Non linear Anal. TMA,7, 1983.[13] OTANI (M.). 2014 Proceed, Fac. Sci. Tokai Univ. 19, 1984.[14] TOLKSDORF (P.).- Regularity for a more general class of quasilinear ellipticequations. J. Dif. Equ. 51, 1984.[15] TOLKSDORF (P.) .2014 On the Dirichlet problem for quasilinear elliptic equationswith conical boundary points.[16] UHLENBECK (K.).- Regularity for a class of non linear elliptic systems. Acta.Math. 138, 1977.[17] DIAZ (J.I.) and SAA (J.E.).- Uniqueness of nonnegative solutions for ellipticnonlinear diffusion equations with a general perturbation term. Proceedings theVIII CEDYA, Santander, 1985.[18] DIAZ (J.I.) and SAA (J.E.).- Uniqueness of nonnegative solutions for second orderquasilinear equations with a possible source term. (To appear).(Manuscrit reçu le 7 avril 1987)

Remarks on uniqueness results of the first eigenvalue of the $p$-laplacian

https://afst.centre-mersenne.org/item/AFST_1988_5_9_1_65_0.pdf

Remarks on uniqueness results of the first eigenvalue of the $p$ -laplacian

Abstract

Similar works

Full text

Available Versions

Annales de la Faculté des Sciences de Toulouse

Remarks on uniqueness results of the first eigenvalue of the ppp-laplacian

Abstract

Similar works

Full text

Available Versions

Annales de la Faculté des Sciences de Toulouse

Remarks on uniqueness results of the first eigenvalue of the $p$ -laplacian