EQUAZIONI A DERIVATE PARZIALI. STEKLOV EIGENVALUES FOR THE -LAPLACIAN

EQUAZIONI A DERIVATE PARZIALI. STEKLOV EIGENVALUES FOR THE -LAPLACIAN J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J. D. ROSSI Abstract. We study the Steklov eigenvalue problem for the - laplacian. To this end we consider the limit as p of solutions of p u p = 0 in a domain Ω with u p p 2 u p / ν = λ u p 2 u on. We obtain a limit problem that is satisfied in the viscosity sense and a geometric characterization of the second eigenvalue. Riasunto. Si studiano gli autovalori di tipo Steklov per el - Laplaciano come limite per p di soluzioni di p u p = 0 in a dominio Ω e u p p 2 u p / ν = λ u p 2 u su la frontiera. Il problema limite qui is satisfato in le senso de viscosita. 1. Introduction. Let p u = div u p 2 u) be the p laplacian. The limit operator lim p p = is the -Laplacian given by N u 2 u u u = x j x j x i x i i,j=1 in the viscosity sense see [BBM], [CIL] and [ILi].) This operator appears naturally when one considers absolutely minimizing Lipschitz extensions of a boundary function f, see [A], [ACJ], and [J]. Our concern in this paper is the study of the Steklov eigenvalue problem for the Laplacian. To this end we consider the Laplacian Key words and phrases. Quasilinear elliptic equations, viscosity solutions, Neumann boundary conditions, eigenvalue problems. 2000 Mathematics Subject Classification. 35J65, 35J50, 35J55. JGA and IP supported by project MTM2004-02223, M.C.Y.T. Spain. JJM supported in part by NSF award DMS-0500983. JDR supported by Fundacion Antorchas, CONICET and ANPCyT PICT 05009. 1

2 J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J.D. ROSSI in a bounded smooth domain as limit of the p laplacian as p. Therefore our aim is to analyze the limit as p for the Steklov eigenvalue problem { p u = 0 in Ω, 1.1) p 2 u u = ν λ u p 2 u on,. Here Ω is a bounded domain in R N with smooth boundary and ν is the outer normal derivative. Steklov eigenvalues have been introduced in [S] for p = 2. For the existence of a sequence of variational eigenvalues see [S] for p = 2 and [FBR1] for general p. As happens for the eigenvalues for the Dirichlet problem for the p Laplacian, in general, it is not known if this sequence constitutes the whole spectrum. Note that the first eigenvalue of 1.1) is λ 1,p = 0 with eigenfunction u 1,p 1. Hence we can trivially pass to the limit and obtain λ 1, = 0 with eigenfunction u 1, 1. Our main result in this paper shows that we can pass to the limit in the variational eigenvalues defined in [FBR1]. Since the first eigenvalue is isolated, [MR], there exists a second eigenvalue that has a variational characterization, [FBR2]. We can pass to the limit in this second eigenvalue and obtain a geometric characterization of the second Steklov eigenvalue for the Laplacian. Moreover we obtain a uniform limit of the sequence of eigenfunctions along subsequences) and we find a limit eigenvalue problem that is satisfied in a viscosity sense which involves the Laplacian together with a boundary condition with the normal derivative u ν. Theorem 1.1. For the first eigenvalue of 1.1) we have, lim p λ1/p 1,p = λ 1, = 0, with eigenfunction given by u 1, = 1. For the second eigenvalue, it holds lim p λ1/p 2,p = λ 2, = 2 diamω). Moreover, given u 2,p eigenfunctions of 1.1) of eigenvalues λ 2,p normalized by u 2,p L ) = 1, there exits a sequence p i such that u 2,pi u 2,, in C α Ω). The limit u 2, is a solution of { u = 0 in Ω, 1.2) Λx, u, u) = 0, on,

STEKLOV EIGENVALUES FOR THE -LAPLACIAN 3 in the viscosity sense, where min { } u u λ 2, u, if u > 0, ν u Λx, u, u) max{λ 2, u u, } if u < 0, ν u if u = 0. ν For the k-th eigenvalue we have that if λ k,p is the k-th variational eigenvalue of 1.1) with eigenfunction u k,p normalized by u k,p L ) = 1, then for every sequence p i there exists a subsequence such that lim p i λ1/p k,p = λ, and u k,pi u, in C α Ω), where u, and λ, is a solution of 1.2). We have a simple geometrical characterization of λ 2, as 2/diamΩ). From this characterization and the convergence of the eigenfunctions we conclude that the second Steklov eigenfunction in an annulus or a ball is not radial. Also we have that the domain that maximizes λ 2, among domains with fixed volume is a ball. We end the introduction with a brief comment on the Dirichlet case. Eigenvalues of the p Laplacian, p u = λ u p 2 u, with Dirichlet boundary conditions, u = 0 on, have been extensively studied since [GAP]. The limit as p was studied in [JL], [JLM]. In these papers the authors prove results similar to ours. however our proofs are necessarily different due to the presence of the Neumann boundary condition. An anisotropic version of the Dirichlet problem was studied in [BK]. 2. The Steklov eigenvalue problem First, let us recall some well known results concerning the Steklov eigenvalue problem for the p laplacian. To this end, we introduce a topological tool, the genus, see [K]. Definition 2.1. Given a Banach Space X, we consider the class Σ = {A X : A is closed, A = A}. Over this class we define the genus, γ : Σ N { }, as γa) = min{k N : there exists ϕ CA, R k {0}), ϕx) = ϕ x)}. We have the following result whose proof can be obtained following [FBR1], we do not provide the details.

4 J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J.D. ROSSI Theorem 2.1. There exists a sequence of eigenvalues λ n of 1.1) such that λ n + as n +. The so-called variational eigenvalues λ k can be characterized by 2.1) 1 = sup min λ k C C k u C u p L p ) u p W 1,p Ω) where C k = {C W 1,p Ω); C is compact, symmetric and γc) k} and γ is the genus. There exists a second eigenvalue for 1.1) and it coincides with the second variational eigenvalue λ 2,p, see [FBR2]. Moreover, the following characterization of the second eigenvalue λ 2,p holds u p dx Ω λ 2,p = inf, u A u p dσ where A = {C W 1,p Ω); C is compact, symmetric and γc) 2}. Observe that every eigenfunction associated with λ 2 changes sign on, see [MR]. Following [B] let us recall the definition of viscosity solution taking into account general boundary conditions. Definition 2.2. Consider the boundary value problem { Fx, u, D 2.2) 2 u) = 0 in Ω, Bx, u, u) = 0 on. Ω 1) A lower semi-continuous function u is a viscosity supersolution if for every φ C 2 Ω) such that u φ has a strict minimum at the point x 0 Ω with ux 0 ) = φx 0 ) we have: If x 0 the inequality max{bx 0, φx 0 ), φx 0 )), Fx 0, φx 0 ), D 2 φx 0 ))} 0 holds, and if x 0 Ω then we require Fx 0, φx 0 ), D 2 φx 0 )) 0. 2) An upper semi-continuous function u is a subsolution if for every φ C 2 Ω) such that u φ has a strict maximum at the,

STEKLOV EIGENVALUES FOR THE -LAPLACIAN 5 point x 0 Ω with ux 0 ) = φx 0 ) we have: If x 0 the inequality min{bx 0, φx 0 ), φx 0 )), Fx 0, φx 0 ), D 2 φx 0 ))} 0 holds, and if x 0 Ω then we require Fx 0, φx 0 ), D 2 φx 0 )) 0. 3) Finally, u is a viscosity solution if it is a super and a subsolution. In our case for the Steklov problem for the p Laplacian we have where and F p η, X) TraceA p η)x), A p η) = Id + p 2) η η η 2, if η 0, A p0) = I N, 2.3) B p x, u, η) η p 2 < η, νx) > λ u p 2 u. With this notation we have, { p u = F p u, D 2 φx0 ) 2 φx 0 ) u) p 2 } + φx 0 ). Remark 2.1. If B p is monotone in the variable u Definition 2.2 ν takes a simpler form, see [B]. This is indeed the case for 2.3). More concretely, if u is a supersolution of 1.1) and φ C 2 Ω) is such that u φ has a strict minimum at x 0 with ux 0 ) = φx 0 ), then 1) if x 0 Ω, then { } φx0 ) 2 φx 0 ) + φx 0 ) 0, p 2 and 2) if x 0, then φx 0 ) p 2 φx 0 ), νx 0 ) λ φx 0 ) p 2 φx 0 ). Let us state a lemma that says that weak solutions of 1.1) are viscosity solutions. Lemma 2.1. A continuous weak solution of 1.1) is a viscosity solution.

6 J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J.D. ROSSI Proof. Let x 0 Ω and a test function φ such that ux 0 ) = φx 0 ) and u φ has a strict minimum at x 0. We want to show that p 2) φ p 4 φx 0 ) φ p 2 φx 0 ) 0. Assume that this is not the case, then there exists a radius r > 0 such that p 2) φ p 4 φx) φ p 2 φx) < 0, for every x Bx 0, r). Set m = inf x x0 =ru φ)x) and let ψx) = φx) + m/2. This function ψ verifies ψx 0 ) > ux 0 ) and div ψ p 2 ψ) < 0. Multiplying by ψ u) + extended by zero outside Bx 0, r) we get ψ p 2 ψ ψ u) < 0. {ψ>u} Taking ψ u) + as test function in the weak form we get u p 2 u ψ u) = 0. Hence, CN, p) {ψ>u} {ψ>u} {ψ>u} ψ u p a contradiction. If x 0 we want to prove ψ p 2 ψ u p 2 u, ψ u) < 0, max { φx 0 ) p 2 < φx 0 ), νx 0 ) > λ φx 0 ) p 2 φx 0 ), p 2) φ p 4 φx 0 ) φ p 2 φx 0 )} 0. Assume that this is not the case. We proceed as before and we obtain ψ p 2 ψ ψ u) < λ u p 2 uψ u), and {ψ>u} {ψ>u} u p 2 u ψ u) {ψ>u} {ψ>u} λ u p 2 uψ u).

Therefore, CN, p) STEKLOV EIGENVALUES FOR THE -LAPLACIAN 7 {ψ>u} {ψ>u} ψ u p ψ p 2 ψ u p 2 u, ψ u) < 0, again a contradiction. This proves that u is a viscosity supersolution. The proof of the fact that u is a viscosity subsolution runs as above, we omit the details. With all these preliminaries we are ready to pass to the limit as p in the eigenvalue problem. Since u 1,p 1 is the first eigenfunction of 1.1) associated to λ 1,p = 0 we can trivially pass to the limit and obtain and lim p λ1/p 1,p = 0 = λ 1, lim u 1,p = 1 = u 1,. p Now let us prove a geometrical characterization of the second Steklov eigenvalue for the laplacian, defined by, 2.4) { } u L λ 2, = inf Ω) : u C W 1, Ω) with γc) 2. We have u L ) Lemma 2.2. λ 2, has the following geometrical characterization Proof. Let λ 2, = 2 diamω). R = sup { r : x 0, x 1 Ω with Bx 0, r) Bx 1, r) = } = diamω). 2 We can take as a test function in 2.4) the combination of two cones centered at x 0 and x 1 with radius R, that is, if C 0 x) = 1 x x ) 0, C 1 x) = 1 x x ) 1, R R + +

8 J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J.D. ROSSI we consider φx) = C 0 x) C 1 x). We obtain λ 2, 1 R = 2 diamω). To prove the reverse inequality, let us take a function u in W 1, Ω) that changes sign and such that λ 2, u L Ω) u L ) ε. Now u + and u have disjoint supports and we may normalize with u + L ) = u L ) = 1 then Therefore, u + L Ω) 1/R, or u L Ω) 1/R. λ 2, 1 R ε = 2 diamω) ε. Since this holds for every ε, the proof is complete. Lemma 2.3. lim sup λ 1/p 2,p λ 2,. p Proof. As above, let C 0 x) and C 1 x) two cones centered at x 0 and x 1 of radius R as above, that is C 0 x) = 1 x x ) 0, C 1 x) = 1 x x ) 1. R + R + Let us normalize a function v = ac 0 bc 1 a, b > 0) by v L ) = 1, then ) 1/p v p λ 1/p Ω 2,p ) 1/p. v p Hence as we wanted to prove. lim sup λ 1/p 2,p 1 R = λ 2,,

STEKLOV EIGENVALUES FOR THE -LAPLACIAN 9 Lemma 2.4. Given u 2,p eigenfunctions of 1.1) of eigenvalues λ 2,p normalized by u 2,p L ) = 1, there exits a sequence p i such that u 2,pi u 2,, in C α Ω). The limit u 2, verifies u 2, L ) = 1 and it changes sign in. Moreover it is a minimizer of 2.4) and lim p λ1/p 2,p = λ 2,. Proof. If q < p, 2.5) ) 1/q ) 1/p u 2,p q Ω 1/q) 1/p) u 2,p p Ω Ω = λ 1/p 2,p Ω 1/q) 1/p) u 2,p p ) 1/p λ 1/p 2,p Ω 1/q) 1/p) 1/p. Therefore, by Lemma 2.3, we get that there exists a constant C independent of p such that ) 1/q 2.6) u 2,p q C. Ω Hence, as u 2,p are uniformly bounded in W 1,q Ω) we can take a subsequence such that it converges weakly in W 1,q Ω) and hence in C α Ω) if q > N) to a limit u 2,. Since this can be done for any q we obtain that u 2, W 1, Ω). Indeed, from 2.5), we get ) 1/q ) 1/q 2.7) u 2, q lim sup u 2,pi q λ 2, Ω 1/q. Ω p i Hence, taking limit as q in 2.7) we get 2.8) u 2, L Ω) lim inf p Ω λ1/p 2,p λ 2,. From the convergence in C α Ω) of the sequence u 2,pi and the normalization u 2,pi L ) = 1 we obtain that 2.9) u 2, L ) = 1.

10 J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J.D. ROSSI To end the proof we need to check that u 2, changes sign. Assume that u 2, 0. Hence u 2,p i converges uniformly to zero in Ω. From 2.9) there exists a point x 0 such that u 2, x 0 ) = 1. At level p we have, u p 2 u = 0, then Therefore, 2.10) 1/r) 1/p i 1)) = u 2,p i p i 1 u + ) p 1 = ) 1/pi 1) u ) p 1. ) 1/r u + 2,p i r ) 1/pi 1) u + 2,p i p i 1 1/p i 1) u 2,p i L ). From the uniform convergence of u 2,pi and 2.8) taking limit as p i we get that ) 1/r 2.11) 1/r u + 2, r 0. A contradiction. This proves that u 2, changes sign and verifies 2.8) and 2.9), hence, from the definition of λ 2, we obtain λ 2, lim inf p This fact and Lemma 2.3 end the proof. λ1/p 2,p. Now let us analyze the equation satisfied by u 2,. Let min { η λ 2, u, < η, νx) >} if u > 0, Λx, u, η) max{λ 2, u η, < η, νx) >} if u > 0, < η, νx) > if u = 0, Lemma 2.5. The limit u 2, is a viscosity solution of 2.12) { u 2, = 0 in Ω, Λx, u, u) = 0 on.

STEKLOV EIGENVALUES FOR THE -LAPLACIAN 11 Proof. First, let us check that u 2, = 0 in the viscosity sense in Ω. Let us recall the standard proof. Let φ be a smooth test function such that v φ has a strict maximum at x 0 Ω. Since u pi converges uniformly to v we get that u pi φ has a maximum at some point x i Ω with x i x 0. Now we use the fact that u pi is a viscosity solution of p u p = 0 and we obtain 2.13) p i 2) φ p i 4 φx i ) φ p i 2 φx i ) 0. If φx 0 ) = 0 we get φx 0 ) 0. If this is not the case, we have that φx i ) 0 for large i and then We conclude that φx i ) 1 p i 2 φ 2 φx i ) 0, as i. φx 0 ) 0. That is v is a viscosity subsolution of u = 0. Now we check the boundary condition. Assume that u 2, φ has a strict minimum at x 0 such that u 2, x 0 ) = φx 0 ) > 0. Using the uniform convergence of u 2,pi to u 2, we obtain that u 2,pi φ has a minimum at some point x i Ω with x i x 0. If x i Ω for infinitely many i, we can argue as before and obtain φx 0 ) 0. On the other hand if x i we have φ pi 2 x i ) ν x i) λ 2,pi φ pi 2 x i )φx i ). If φx 0 ) = 0, then ν x 0) = 0. If φx 0 ) 0 we obtain λ 1/pi 1) ν x i) λ 1/p i 1) 2,p i φ 2,p i φ x i ) ) pi 2 φx i ).

12 J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J.D. ROSSI Using that λ 1/p 1) 2,p λ 2, as p we conclude that λ 2, φ x 0 ) 1. φ Moreover, ν x 0) 0. Hence, if u 2, φ has a strict minimum at x 0 with φx 0 ) = u 2, x 0 ) > 0, we have { 2.14) max min{ λ 2, φ + φ x 0 ), } ν x 0)}, φx 0 ) 0. Now assume that u 2, φ has a strict maximum at x 0 with u 2, x 0 ) = φx 0 ) > 0. Using the uniform convergence of u 2,pi to u 2, we obtain that u 2,pi φ has a maximum at some point x i Ω with x i x 0. If x i Ω for infinitely many i, we can argue as before and obtain u 2, x 0 ) 0. On the other hand if x i we have φ pi 2 x i ) ν x i) λ 2,pi φ pi 2 x i )φx i ). If φx 0 ) = 0, then ν x 0) = 0. If φx 0 ) 0 we obtain λ 1/pi 1) ν x i) λ 1/p i 1) 2,p i φ If λ 2, φ x 0 ) < φ x 0 ), then Hence, 2.15) min 2,p i φ ν x 0) 0. x i ) ) pi 2 φx i ). { min{ λ 2, φ + φ x 0 ), } ν x 0)}, φx 0 ) 0. Now assume that u 2, φ has a strict maximum at x 0 such that u 2, x 0 ) = φx 0 ) < 0 Using the uniform convergence of u 2,pi to u 2,

STEKLOV EIGENVALUES FOR THE -LAPLACIAN 13 we obtain that u 2,pi φ has a maximum at some point x i Ω with x i x 0. If x i Ω for infinitely many i, we can argue as before and obtain u 2, x 0 ) 0. On the other hand if x i we have φ pi 2 x i ) ν x i) λ 2,pi φ pi 2 x i )φx i ). If φx 0 ) = 0, then ν x 0) = 0. If φx 0 ) 0 we obtain λ 1/pi 1) ν x i) λ 1/p i 1) 2,p i φ 2,p i φ x i ) ) pi 2 φx i ). Using that λ 1/p 1) 2,p λ 2, as p we conclude that Moreover, Hence, 2.16) min λ 2, φ x 0 ) 1. φ ν x 0) 0. { max{λ 2, φ φ x 0 ), } ν x 0)}, φx 0 ) 0. Now assume that u 2, φ has a strict minimum at x 0 with u 2, x 0 ) = φx 0 ) < 0. Using the uniform convergence of u 2,pi to u 2, we obtain that u 2,pi φ has a minimum at some point x i Ω with x i x 0. If x i Ω for infinitely many i, we can argue as before and obtain φx 0 ) 0. On the other hand if x i we have φ p i 2 x i ) ν x i) λ 2,pi φ p i 2 x i )φx i ). If φx 0 ) = 0, then ν x 0) = 0.

14 J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J.D. ROSSI If φx 0 ) 0 we obtain λ 1/pi 1) ν x i) λ 1/p i 1) 2,p i φ If λ 2, φ x 0 ) < φ x 0 ), then Hence, 2.17) max 2,p i φ ν x 0) 0. x i ) ) pi 2 φx i ). { max{λ 2, φ φ x 0 ), } ν x 0)}, φx 0 ) 0. Assume that u 2, φ has a strict minimum at x 0 such that u 2, x 0 ) = φx 0 ) = 0. Using the uniform convergence of u 2,pi to u 2, we obtain that u 2,pi φ has a minimum at some point x i Ω with x i x 0. If x i Ω for infinitely many i, we can argue as before and obtain φx 0 ) 0. On the other hand if x i we have φ pi 2 x i ) ν x i) λ 2,pi φ pi 2 x i )φx i ). If φx 0 ) = 0, then ν x 0) = 0. If φx 0 ) 0 we obtain λ 1/pi 1) ν x i) λ 1/p i 1) 2,p i φ As 0 = λ 2, φ x 0 ) < φ x 0 ), then Hence, 2.18) max 2,p i φ ν x 0) 0. x i ) ) pi 2 { } ν x 0), φx 0 ) 0. φx i ). Finally, assume that u 2, φ has a strict maximum at x 0 with u 2, x 0 ) = φx 0 ) = 0 Using the uniform convergence of u 2,pi to u 2,

STEKLOV EIGENVALUES FOR THE -LAPLACIAN 15 we obtain that u 2,pi φ has a maximum at some point x i Ω with x i x 0. If x i Ω for infinitely many i, we can argue as before and obtain u 2, x 0 ) 0. On the other hand if x i we have If φx 0 ) = 0, then If φx 0 ) 0 we obtain φ p i 2 x i ) ν x i) λ 2,pi φ p i 2 x i )φx i ). ν x 0) = 0. λ 1/pi 1) ν x i) λ 1/p i 1) 2,p i φ 2,p i φ x i ) ) pi 2 φx i ). As 0 = λ 2, φ x 0 ) < φ x 0 ), then Hence, 2.19) min ν x 0) 0. { } ν x 0), φx 0 ) 0. Inequalities 2.14)-2.19) prove the result. With the same ideas used to deal with the second eigenvalue we can prove the following lema. Lemma 2.6. Let λ k,p be the k-th variational eigenvalue of 1.1) with eigenfunction u k,p normalized by u k,p L ) = 1. Then for every sequence p i there exists a subsequence such that lim p i λ1/p k,p = λ,, u k,pi u,, where u,, λ, ) is a solution of 1.2). in C α Ω),

16 J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J.D. ROSSI Acknowledgements. This research was started while the fourth author JDR) was a visitor at Universidad Autónoma de Madrid. He is grateful to this institution for its hospitality and stimulating atmosphere. References [A] G. Aronsson, Extensions of functions satisfiying Lipschitz conditions. Ark. Math., 6 1967), 551-561. [ACJ] G. Aronsson, M.G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc., 41 2004), 439-505. [B] G. Barles, Fully nonlinear Neumann type conditions for second-order ellip- [BK] tic and parabolic equations. J. Differential Equations, 106 1993), 90-106. M. Belloni and B. Kawohl, The pseudo-p Laplace eigenvalue problem and viscosity solutions as p. ESAIM Control Optim. Calc. Var., 10 2004), 28-52. [BBM] T. Bhattacharya, E. Di Benedetto and J. Manfredi. Limits as p of p u p = f and related extremal problems. Rend. Sem. Mat. Univ. Politec. Torino, 1991), 15-68. [CIL] M.G. Crandall, H. Ishii and P.L. Lions. User s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 27 1992), 1-67. [FBR1] J. Fernández Bonder and J.D. Rossi, Existence results for the p-laplacian with nonlinear boundary conditions. J. Math. Anal. Appl., 263 2001), 195-223. [FBR2] J. Fernández Bonder and J.D. Rossi, A nonlinear eigenvalue problem with indefinite weights related to the Sobolev trace embedding. Publicacions Mathematiques, 46, 2002), 221-235. [GAP] J. García-Azorero and I. Peral, Existence and non-uniqueness for the p Laplacian: nonlinear eigenvalues. Comm. Partial Differential Equations, 12 1987), 1389-1430. [ILi] H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential euqations. J. Differential Equations, 83 1990), 26-78. [J] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Rational Mech. Anal. 123 1993), 51-74. [JLM] P. Juutinen, P. Lindqvist and J. J. Manfredi, The eigenvalue problem. [JL] [K] Arch. Rational Mech. Anal., 148 1999), 89-105. P. Juutinen and P. Lindqvist, On the higher eigenvalues for the eigenvalue problem. To appear in Calc. Var. M.A. Krasnoselski, Topological methods in the theory of nonlinear integral equations, Macmillan, New York 1964).

STEKLOV EIGENVALUES FOR THE -LAPLACIAN 17 [MR] [S] S. Martinez and J.D. Rossi, Isolation and simplicity for the first eigenvalue of the p-laplacian with a nonlinear boundary condition. Abst. Appl. Anal., 7 5), 2002), 287-293. M. W. Steklov, Sur les problèmes fondamentaux en physique mathématique, Ann. Sci. Ecole Norm. Sup., 19 1902), 455 490. Departamento de Matemáticas, U. Autonoma de Madrid, 28049 Madrid, Spain. E-mail address: jesus.azorero@uam.es, ireneo.peral@uam.es Department of Mathematics, University of Pittsburgh. Pittsburgh, Pennsylvania 15260. E-mail address: manfredi@math.pitt.edu Consejo Superior de Investigaciones Científicas CSIC), Serrano 123, Madrid, Spain, on leave from Departamento de Matemática, FCEyN UBA 1428) Buenos Aires, Argentina. E-mail address: jrossi@dm.uba.ar