Neumann to Steklov eigenvalues: asymptotic and monotonicity results
aa r X i v : . [ m a t h . SP ] F e b NEUMANN TO STEKLOV EIGENVALUES: ASYMPTOTIC ANDMONOTONICITY RESULTS
PIER DOMENICO LAMBERTI AND LUIGI PROVENZANO
Abstract.
We consider the Steklov eigenvalues of the Laplace operator as lim-iting Neumann eigenvalues in a problem of mass concentration at the boundaryof a ball. We discuss the asymptotic behavior of the Neumann eigenvalues andfind explicit formulas for their derivatives at the limiting problem. We deducethat the Neumann eigenvalues have a monotone behavior in the limit and thatSteklov eigenvalues locally minimize the Neumann eigenvalues. Introduction
Let B be the unit ball in R N , N ≥
2, centered at zero. We consider the Stekloveigenvalue problem for the Laplace operator(1.1) (cid:26) ∆ u = 0 , in B, ∂u∂ν = λρu, on ∂B, in the unknowns λ (the eigenvalue) and u (the eigenfunction), where ρ = M/σ N , M > σ N denotes the surface measure of ∂B .As is well-known the eigenvalues of problem (1.1) are given explicitly by thesequence(1.2) λ l = lρ , l ∈ N , and the eigenfunctions corresponding to λ l are the homogeneous harmonic poly-nomials of degree l . In particular, the multiplicity of λ l is (2 l + N − l + N − / ( l !( N − λ is simple, the corresponding eigenfunctions being theconstant functions. See [7] for an introduction to the theory of harmonic polyno-mials.A classical reference for problem (1.1) is [18]. For a recent survey paper, we referto [8]; see also [11], [14] for related problems.It is well-known that for N = 2, problem (1.1) provides the vibration modes ofa free elastic membrane the total mass of which is M and is concentrated at theboundary with density ρ ; see e.g., [4]. As is pointed out in [14], such a boundaryconcentration phenomenon can be explained in any dimension N ≥ < ε <
1, we define a ‘mass density’ ρ ε in the whole of B by setting Mathematics Subject Classification.
Primary 35C20; Secondary 35P15, 35B25, 35J25,33C10.
Key words and phrases.
Steklov boundary conditions, eigenvalues, density perturbation, mono-tonicity, Bessel functions. (1.3) ρ ε ( x ) = ( ε, if | x | ≤ − ε, M − εω N (1 − ε ) N ω N (1 − (1 − ε ) N ) , if 1 − ε < | x | < , where ω N = σ N /N is the measure of the unit ball. Note that for any x ∈ B wehave ρ ε ( x ) → ε →
0, and R B ρ ε dx = M for all ε >
0, which means that the‘total mass’ M is fixed and concentrates at the boundary of B as ε →
0. Then weconsider the following eigenvalue problem for the Laplace operator with Neumannboundary conditions(1.4) (cid:26) − ∆ u = λρ ε u, in B, ∂u∂ν = 0 , on ∂B. We recall that for N = 2 problem (1.4) provides the vibration modes of a free elasticmembrane with mass density ρ ε and total mass M (see e.g., [6]). The eigenvaluesof (1.4) have finite multiplicity and form a sequence λ ( ε ) < λ ( ε ) ≤ λ ( ε ) ≤ · · · , depending on ε , with λ ( ε ) = 0.It is not difficult to prove that for any l ∈ N (1.5) λ l ( ε ) → λ l , as ε → , see [2], [14]. (See also [5] for a detailed analysis of the analogue problem for thebiharmonic operator.) Thus the Steklov problem can be considered as a limitingNeumann problem where the mass is concentrated at the boundary of the domain.In this paper we study the asymptotic behavior of λ l ( ε ) as ε →
0. Namely,we prove that such eigenvalues are continuously differentiable with respect to ε for ε ≥ λ ′ l (0) = 2 lλ l λ l N (2 l + N ) . In particular, for l = 0, λ ′ l (0) > λ l ( ε ) is strictly increasing and the Stekloveigenvalues λ l minimize the Neumann eigenvalues λ l ( ε ) for ε small enough.It is interesting to compare our results with those in [17], where authors considerthe Neumann Laplacian in the annulus 1 − ε < | x | < N = 2the first positive eigenvalue is a decreasing function of ε . We note that our analysisconcerns all eigenvalues λ l with arbitrary indexes and multiplicity, and that we donot prove global monotonocity of λ l ( ε ), which in fact does not hold for any l ; seeFigures 1, 2.The proof of our results relies on the use of Bessel functions which allows to recastproblem (1.4) in the form of an equation F ( λ, ε ) = 0 in the unknowns λ, ε . Then,after some preparatory work, it is possible to apply the Implicit Function Theoremand conclude. We note that, despite the idea of the proof is rather simple andused also in other contexts (see e.g., [13]), the rigorous application of this methodrequires lenghty computations, suitable Taylor’s expansions and estimates for thecorresponding remainders, as well as recursive formulas for the cross-products ofBessel functions and their derivatives.Importantly, the multiplicity of the eigenvalues which is often an obstruction inthe application of standard asymptotic analysis, does not affect our method.We note that if the ball B is replaced by a general bounded smooth domainΩ, the convergence of the Neumann eigenvalues to the Steklov eigenvalues when TEKLOV EIGENVALUES 3 the mass concentrates in a neighborhood of ∂ Ω still holds. However, the explicitcomputation of the appropriate formula generalizing (1.6) is not easy and requiresa completely different technique which will be discussed in a forthcoming paper.We also note that an asymptotic analysis of similar but different problems iscontained in [9, 10], where by the way explicit computations of the coefficients inthe asymptotic expansions of the eigenvalues are not provided.It would be interesting to investigate the monotonicity properties of the Neu-mann eigenvalues in the case of more general families of mass densities ρ ε . However,we believe that it would be difficult to adapt our method (which is based on explicitrepresentation formulas) even in the case of radial mass densities (note that if ρ ε is not radial one could obtain a limiting Steklov-type problem with non-constantmass density, see [2] for a general discussion).This paper is organized as follows. The proof of formula (1.6) is discussed inSection 2. In particular, Subsection 2.1 is devoted to certain technical estimateswhich are necessary for the rigorous justification of our arguments. In Subsection2.2 we consider also the case N = 1 and prove formula (1.6) for λ which, by theway, is the only non zero eigenvalue of the one dimensional Steklov problem. InAppendix we establish the required recursive formulas for the cross-products ofBessel functions and their derivatives which are deduced by the standard formulasavailable in the literature.2. Asymptotic behavior of Neumann eigenvalues
It is convenient to use the standard spherical coordinates ( r, θ ) in R N , where θ = ( θ , ...θ N − ). The corresponding trasformation of coordinates is x = r cos( θ ) ,x = r sin( θ ) cos( θ ) , ... x N − = r sin( θ ) sin( θ ) · · · sin( θ N − ) cos( θ N − ) ,x N = r sin( θ ) sin( θ ) · · · sin( θ N − ) sin( θ N − ) , with θ , ..., θ N − ∈ [0 , π ], θ N − ∈ [0 , π [ (here it is understood that θ ∈ [0 , π [ if N = 2). We denote by δ the Laplace-Beltrami operator on the unit sphere S N − of R N , which can be written in spherical coordinates as δ = N − X j =1 q j (sin θ j ) N − j − ∂∂θ j (cid:18) (sin θ j ) N − j − ∂∂θ j (cid:19) , where q = 1 , q j = (sin θ sin θ · · · sin θ j − ) , j = 2 , ..., N − , see e.g., [16, p. 40]. To shorten notation, in what follows we will denote by a and b the quantities defined by a = √ λε (1 − ε ) , and b = p λ ˜ ρ ε (1 − ε ) , where ˜ ρ ε = M − εω N (1 − ε ) N ω N (cid:16) − (1 − ε ) N (cid:17) . PIER DOMENICO LAMBERTI AND LUIGI PROVENZANO
As customary, we denote by J ν and Y ν the Bessel functions of the first and secondspecies and order ν respectively (recall that J ν and Y ν are solutions of the Besselequation z y ′′ ( z ) + zy ′ ( z ) + ( z − ν ) y ( z ) = 0).We begin with the following lemma. Lemma 2.1.
Given an eigenvalue λ of problem (1.4), a corresponding eigenfunc-tion u is of the form u ( r, θ ) = S l ( r ) H l ( θ ) where H l ( θ ) is a spherical harmonic ofsome order l ∈ N and (2.2) S l ( r ) = r − N J ν l ( √ λεr ) , if r < − ε,r − N (cid:0) αJ ν l ( √ λ ˜ ρ ε r ) + βY ν l ( √ λ ˜ ρ ε r ) (cid:1) , if 1 − ε < r < , where ν l = ( N +2 l − and α , β are given by α = πb (cid:16) J ν l ( a ) Y ′ ν l ( b ) − ab J ′ ν l ( a ) Y ν l ( b ) (cid:17) ,β = πb (cid:16) ab J ν l ( b ) J ′ ν l ( a ) − J ′ ν l ( b ) J ν l ( a ) (cid:17) . Proof.
Recall that the Laplace operator can be written in spherical coordinates as∆ = ∂ rr + N − r ∂ r + 1 r δ. In order to solve the equation − ∆ u = λρ ε u , we separate variables so that u ( r, θ ) = S ( r ) H ( θ ). Then using l ( l + N − l ∈ N , as separation constant, we obtain theequations(2.3) r S ′′ + r ( N − S ′ + r λρ ε S − l ( l + N − S = 0and(2.4) − δH = l ( l + N − H. By setting S ( r ) = r − N ˜ S ( r ) into (2.3), it follows that ˜ S ( r ) satisfies the Besselequation ˜ S ′′ + 1 r ˜ S ′ + (cid:18) λρ ε − ν l r (cid:19) ˜ S = 0 . Since solutions u of (1.4) are bounded on Ω and Y ν l ( z ) blows up at z = 0, itfollows that for r < − ε , S ( r ) is a multiple of the function r − N J ν l ( √ λεr ). For1 − ε < r < S ( r ) is a linear combination of the functions r − N J ν l ( √ λ ˜ ρ ε r )and r − N Y ν l ( √ λ ˜ ρ ε r ). On the other hand, the solutions of (2.4) are the sphericalharmonics of order l . Then u can be written as in (2.2) for suitable values of α, β ∈ R .Now we compute the coefficients α and β in (2.2). Since the right-hand side ofthe equation in (1.4) is a function in L (Ω) then by standard regularity theory asolution u of (1.4) belongs to the standard Sobolev space H (Ω), hence α and β must be chosen in such a way that u and ∂ r u are continuous at r = 1 − ε , that is ( αJ ν l ( √ λ ˜ ρ ε (1 − ε )) + βY ν l ( √ λ ˜ ρ ε (1 − ε )) = J ν l ( √ λε (1 − ε )) ,αJ ′ ν l ( √ λ ˜ ρ ε (1 − ε )) + βY ′ ν l ( √ λ ˜ ρ ε (1 − ε )) = q ε ˜ ρ ε J ′ ν l ( √ λε (1 − ε )) . Solving the system we obtain
TEKLOV EIGENVALUES 5 α = J ν l ( a ) Y ′ ν l ( b ) − ab J ′ ν l ( a ) Y ν l ( b ) J ν l ( b ) Y ′ ν l ( b ) − J ′ ν l ( b ) Y ν l ( b ) , β = ab J ν l ( b ) J ′ ν l ( a ) − J ′ ν l ( b ) J ν l ( a ) J ν l ( b ) Y ′ ν l ( b ) − J ′ ν l ( b ) Y ν l ( b ) . Note that J ν l ( b ) Y ′ ν l ( b ) − J ′ ν l ( b ) Y ν l ( b ) is the Wronskian in b , which is known to be πb (see [1, § (cid:3) We are ready to establish an implicit characterization of the eigenvalues of (1.4).
Proposition 2.5.
The nonzero eigenvalues λ of problem (1.4) are given implicitlyas zeros of the equation (2.6) (cid:18) − N (cid:19) P ( a, b ) + b (1 − ε ) P ( a, b ) = 0 where P ( a, b ) = J ν l ( a ) (cid:18) Y ′ ν l ( b ) J ν l ( b − ε ) − J ′ ν l ( b ) Y ν l ( b − ε ) (cid:19) + ab J ′ ν l ( a ) (cid:18) J ν l ( b ) Y ν l ( b − ε ) − Y ν l ( b ) J ν l ( b − ε ) (cid:19) ,P ( a, b ) = J ν l ( a ) (cid:18) Y ′ ν l ( b ) J ′ ν l ( b − ε ) − J ′ ν l ( b ) Y ′ ν l ( b − ε ) (cid:19) + ab J ′ ν l ( a ) (cid:18) J ν l ( b ) Y ′ ν l ( b − ε ) − Y ν l ( b ) J ′ ν l ( b − ε ) (cid:19) . Proof.
By Lemma 2.1, an eigenfunction u associated with an eigenvalue λ is of theform u ( r, θ ) = S l ( r ) H l ( θ ) where for r > − εS l ( r ) = πb r − N (cid:20)(cid:16) J ν l ( a ) Y ′ ν l ( b ) − ab J ′ ν l ( a ) Y ν l ( b ) (cid:17) J ν l ( br − ε )+ (cid:16) ab J ν l ( b ) J ′ ν l ( a ) − J ′ ν l ( b ) J ν l ( a ) (cid:17) Y ν l ( br − ε ) (cid:21) . We require that ∂u∂ν = ∂u∂r | r =1 = 0, which is true if and only if πb (cid:18) − N (cid:19) (cid:20)(cid:16) J ν l ( a ) Y ′ ν l ( b ) − ab J ′ ν l ( a ) Y ν l ( b ) (cid:17) J ν l ( b − ε )+ (cid:16) ab J ν l ( b ) J ′ ν l ( a ) − J ′ ν l ( b ) J ν l ( a ) (cid:17) Y ν l ( b − ε ) (cid:21) + πb − ε ) (cid:20)(cid:16) J ν l ( a ) Y ′ ν l ( b ) − ab J ′ ν l ( a ) Y ν l ( b ) (cid:17) J ′ ν l ( b − ε )+ (cid:16) ab J ν l ( b ) J ′ ν l ( a ) − J ′ ν l ( b ) J ν l ( a ) (cid:17) Y ′ ν l ( b − ε ) (cid:21) = 0 . The previous equation can be clearly rewritten in the form (2.6). (cid:3)
We now prove the following.
PIER DOMENICO LAMBERTI AND LUIGI PROVENZANO
Lemma 2.7.
Equation (2.6) can be written in the form λ ε (cid:18) M N ω N − ν l (1 + ν l ) (cid:19) + λε (cid:18) N − ν l + (2 − N ) N ω N ν l (1 + ν l ) M (cid:19) − λ + 2 N ω N lM − N ω N lM (cid:18) N − − ω N M − ν l (cid:19) ε + R ( λ, ε ) = 0(2.8) where R ( λ, ε ) = O ( ε √ ε ) as ε → .Proof. We plan to divide the left-hand side of (2.6) by J ′ ν l ( a ) and to analyze theresulting terms using the known Taylor’s series for Bessel functions. Note that J ′ ν l ( a ) > ε small enough. We split our analysis into three steps. Step 1.
We consider the term P ( a,b ) J ′ νl ( a ) , that is (2.9) J ν l ( a ) J ′ ν l ( a ) (cid:20) Y ′ ν l ( b ) J ′ ν l ( b − ε ) − Y ′ ν l ( b − ε ) J ′ ν l ( b ) (cid:21) + ab (cid:20) Y ′ ν l ( b − ε ) J ν l ( b ) − Y ν l ( b ) J ′ ν l ( b − ε ) (cid:21) . Using Taylor’s formula, we write the derivatives of the Bessel functions in (2.9),call them C ′ ν l , as follows(2.10) C ′ ν l (cid:18) b − ε (cid:19) = C ′ ν l ( b ) + C ′′ ν l ( b ) εb − ε + · · · + C ( n ) ν l ( b )( n − (cid:18) εb − ε (cid:19) n − + o (cid:18) εb − ε (cid:19) n − . Then, using (2.10) with n = 4 for J ′ ν l and Y ′ ν l we get (2.11) J ν l ( a ) J ′ ν l ( a ) (cid:20) εb − ε (cid:0) Y ′ ν l ( b ) J ′′ ν l ( b ) − J ′ ν l ( b ) Y ′′ ν l ( b ) (cid:1) + ε b − ε ) (cid:0) Y ′ ν l ( b ) J ′′′ ν l ( b ) − J ′ ν l ( b ) Y ′′′ ν l ( b ) (cid:1) + ε b − ε ) (cid:0) Y ′ ν l ( b ) J ′′′′ ν l ( b ) − J ′ ν l ( b ) Y ′′′′ ν l ( b ) (cid:1) + R ( b ) (cid:21) + ab (cid:20)(cid:0) J ν l ( b ) Y ′ ν l ( b ) − Y ν l ( b ) J ′ ν l ( b ) (cid:1) + εb − ε (cid:0) J ν l ( b ) Y ′′ ν l ( b ) − Y ν l ( b ) J ′′ ν l ( b ) (cid:1) + ε b − ε ) (cid:0) J ν l ( b ) Y ′′′ ν l ( b ) − Y ν l ( b ) J ′′′ ν l ( b ) (cid:1) + R ( b ) (cid:21) , where(2.12) R ( b ) = + ∞ X k =4 ε k b k k !(1 − ε ) k (cid:18) Y ′ ν l ( b ) J ( k +1) ν l ( b ) − J ′ ν l ( b ) Y ( k +1) ν l ( b ) (cid:19) and(2.13) R ( b ) = + ∞ X k =3 ε k b k k !(1 − ε ) k (cid:18) J ν l ( b ) Y ( k +1) ν l ( b ) − Y ν l ( b ) J ( k +1) ν l ( b ) (cid:19) . TEKLOV EIGENVALUES 7
Let R be the remainder defined in Lemma 2.25. We set (2.14) R ( λ, ε ) = R ( a ) (cid:20) εb − ε (cid:0) Y ′ ν l ( b ) J ′′ ν l ( b ) − J ′ ν l ( b ) Y ′′ ν l ( b ) (cid:1) + ε b − ε ) (cid:0) Y ′ ν l ( b ) J ′′′ ν l ( b ) − J ′ ν l ( b ) Y ′′′ ν l ( b ) (cid:1) + ε b − ε ) (cid:0) Y ′ ν l ( b ) J ′′′′ ν l ( b ) − J ′ ν l ( b ) Y ′′′′ ν l ( b ) (cid:1)(cid:21) + R ( b ) (cid:20) aν l + a ν l (1 + ν l ) (cid:21) + R ( b ) ab + R ( a ) R ( b ) . By Lemma 2.30, it turns out that R ( λ, ε ) = O ( ε ) as ε → f ( ε ) = b ( ε ) a ( ε ) f ( ε ); g ( ε ) = b ( ε ) a ( ε ) g ( ε ) + a ( ε ) g ( ε ); h ( ε ) = a ( ε ) h ( ε ) + ε a ( ε ) b ( ε ) h ( ε ); k ( ε ) = a ( ε ) b ( ε ) k ( ε ) , where a ( ε ) = a √ λε = (1 − ε ); b ( ε ) = b r ελ ; f ( ε ) = 16 ν l (1 + ν l )(1 − ε ) ; g ( ε ) = 13 ν l (1 − ε ) ; g ( ε ) = − ν l (1 + ν l )(1 − ε ) + ε ν l (1 + ν l )(1 − ε ) − ε (3 + 2 ν l )6 ν l (1 + ν l )(1 − ε ) ; h ( ε ) = − ν l (1 − ε ) + εν l (1 − ε ) − ε (3 + 2 ν l )3 ν l (1 − ε ) − ε (1 − ε ) ; h ( ε ) = 1(1 + ν l )(1 − ε ) − ε ν l )(1 − ε ) + ε ( ν l + 11 ν l )6 ν l (1 + ν l )(1 − ε ) ; k ( ε ) = 2 + 2 εν l (1 − ε ) − ε ν l (1 − ε ) + ε ( ν l + 11 ν l )3 ν l (1 − ε ) − ε (1 − ε ) + ε (2 + ν l )(1 − ε ) . Note that functions f, g, h, k are continuous at ε = 0 and f (0) , g (0) , h (0) , k (0) = 0.Using the explicit formulas for the cross products of Bessel functions given byLemma 3.2 and Corollary 3.7 in (2.11), (2.9) can be written as(2.15) 1 √ λπ ε √ εk ( ε ) + √ λπ ε √ εh ( ε ) + λ √ λπ ε √ εg ( ε ) + λ √ λπ ε √ εf ( ε ) + R ( λ, ε ) . PIER DOMENICO LAMBERTI AND LUIGI PROVENZANO
Step 2.
We consider the quantity P ( a,b ) J ′ νl ( a ) , that is (2.16) J ν l ( a ) J ′ ν l ( a ) (cid:20) Y ′ ν l ( b ) J ν l ( b − ε ) − J ′ ν l ( b ) Y ν l ( b − ε ) (cid:21) + ab (cid:20) J ν l ( b ) Y ν l ( b − ε ) − Y ν l ( b ) J ν l ( b − ε ) (cid:21) . Proceeding as in Step 1 and setting ˜ f ( ε ) = − a ( ε ) b ( ε )2 πν l (1 + ν l )(1 − ε ) ;˜ g ( ε ) = a ( ε ) b ( ε ) (cid:18) πν l (1 + ν l ) + ε π (1 + ν l )(1 − ε ) (cid:19) − a ( ε ) b ( ε ) ν l π (1 − ε ) ;˜ h ( ε ) = a ( ε ) b ( ε ) (cid:18) ν l π + 2 επ (1 − ε ) + ( ν l − π (1 − ε ) ε (cid:19) , one can prove that (2.16) can be written as(2.17) ε ˜ h ( ε ) + λε ˜ g ( ε ) + λ ε ˜ f ( ε ) + ˆ R ( λ, ε ) , where ˆ R ( λ, ε ) = O ( ε √ ε ) as ε →
0; see Lemma 2.30.
Step 3.
We combine (2.15) and (2.17) and rewrite equation (2.6) in the form (2.18) ε (1 − N h ( ε ) + ε b ( ε ) k ( ε ) π (1 − ε ) + λε (1 − N g ( ε ) + λε b ( ε ) h ( ε ) π (1 − ε )+ λ ε (1 − N f ( ε ) + λ ε b ( ε ) g ( ε ) π (1 − ε ) + λ ε b ( ε ) f ( ε ) π (1 − ε ) + R ( λ, ε ) = 0 , where R ( λ, ε ) = √ λb ( ε )(1 − ε ) √ ε R ( λ, ε ) + (cid:18) − N (cid:19) ˆ R ( λ, ε ) . Note that R ( λ, ε ) = O ( ε √ ε ) as ε →
0. Dividing by ε in (2.18) and setting R ( λ, ε ) = R ( λ,ε ) ε , we obtain(1 − N h ( ε ) + b ( ε ) k ( ε ) π (1 − ε ) + λε (1 − N g ( ε ) + λ b ( ε ) h ( ε ) π (1 − ε )(2.19) + λ ε (1 − N f ( ε ) + λ ε b ( ε ) g ( ε ) π (1 − ε ) + λ ε b ( ε ) f ( ε ) π (1 − ε ) + R ( λ, ε ) = 0 . We now multiply in (2.19) by πν l (1 − ε ) b ( ε ) which is a positive quantity for all 0 <ε <
1. Taking into account the definitions of functions g, h, k, ˜ g, ˜ h , we can finallyrewrite (2.19) in the form λ ε (cid:18) ˆ ρ ( ε )3 − ν l (1 + ν l ) (cid:19) + λε (cid:18) N − ν l + 2 − N ν l (1 + ν l )ˆ ρ ( ε ) (cid:19) − λ + 2 l (1 + εν l )ˆ ρ ( ε ) + R ( λ, ε ) = 0 , (2.20)where ˆ ρ ( ε ) = ε ˜ ρ ( ε ) = M − ω N ε (1 − ε ) N ω N (cid:16) N − N ( N − ε − P Nk =3 (cid:0) Nk (cid:1) ( − k ε k − (cid:17) , TEKLOV EIGENVALUES 9 and R ( λ, ε ) = O ( ε √ ε ) as ε →
0. The formulation in (2.8) can be easily deducedby observing thatˆ ρ ε = MN ω N + 2 MN ω N (cid:18) N − − ω N M (cid:19) ε + O ( ε ) , as ε → . (cid:3) We are now ready to prove our main result
Theorem 2.21.
All eigenvalues of problem (1.4) have the following asymptoticbehavior (2.22) λ l ( ε ) = λ l + (cid:18) lλ l λ l N (2 l + N ) (cid:19) ε + o ( ε ) , as ε → , where λ l are the eigenvalues of problem (1.1).Moreover, for each l ∈ N the function defined by λ l ( ε ) for ε > and λ l (0) = λ l ,is continuous in the whole of [0 , and of class C in a neighborhood of ε = 0 .Proof. By using the Min-Max Principle and related standard arguments, one caneasily prove that λ l ( ε ) depends with continuity on ε > ε λ l ( ε ) can be extended by continuity at thepoint ε = 0 by setting λ l (0) = λ l .In order to prove differentiability of λ l ( ε ) around zero and the validity of (2.22),we consider equation (2.8) and apply the Implicit Function Theorem. Note thatequation (2.8) can be written in the form F ( λ, ε ) = 0 where F is a function of class C in the variables ( λ, ε ) ∈ ]0 , ∞ [ × [0 , F ( λ,
0) = − λ + 2 N ω N lM ,F ′ λ ( λ,
0) = − ,F ′ ε ( λ,
0) = λ (cid:18) M N ω N − ν l (1 + ν l ) (cid:19) + λ (cid:18) N − ν l + (2 − N ) N ω N ν l (1 + ν l ) M (cid:19) − N ω N lM (cid:18) N − − ω N M − ν l (cid:19) (2.23)By (1.2), λ l = N ω N l/M hence F ( λ l ,
0) = 0. Since F ′ λ ( λ l , = 0, the ImplicitFunction Theorem combined with the continuity of the functions λ l ( · ) allows toconclude that functions λ l ( · ) are of class C around zero.We now compute the derivative of λ l ( · ) at zero. Using the equality N ω N /M = λ l /l and recalling that ν l = l + N/ − F ′ ε ( λ l ,
0) = λ l (cid:18) l λ l − ν l (1 + ν l ) (cid:19) + λ l (cid:18) − l + λ l (2 − N )2 lν l (1 + ν l ) (cid:19) − λ l (cid:18) − l − λ l N l (cid:19) = λ l (cid:18) ν l (1 + ν l ) (cid:18) − N l − (cid:19) + 2 N l (cid:19) + 43 λ l l = 4 λ l N + 2 N l + 43 λ l l. Finally, formula λ ′ l (0) = − F ′ ε ( λ l , /F ′ λ ( λ l ,
0) yields (1.6) and the validity of (2.22). (cid:3)
Corollary 2.24.
For any l ∈ N \ { } there exists δ l such that the function λ l ( · ) isstrictly increasing in the interval [0 , δ l [ . In particular, λ l < λ l ( ε ) for all ε ∈ ]0 , δ l [ . Figure 1.
Solution branches of equation (2.6) with N = 2, M = π for ( ε, λ ) ∈ ]0 , × ]0 , l in(2.6): blue ( l = 0), red ( l = 1), green ( l = 2), purple ( l = 3),orange ( l = 4). Figure 2.
Solution branches of equation (2.6) with N = 2, M = π for ( ε, λ ) ∈ ]0 , × ]0 ,
50[ . The colors refer to the choice of l in (2.6):blue ( l = 0), red ( l = 1), green ( l = 2), purple ( l = 3), orange( l = 4), cyan ( l = 5), pink ( l = 6). TEKLOV EIGENVALUES 11
Estimates for the remainders.
This subsection is devoted to the proof ofa few technical estimates used in the proof of Lemma 2.7.
Lemma 2.25.
The function R defined by (2.26) J ν ( z ) J ′ ν ( z ) = zν + z ν (1 + ν ) + R ( z ) , is O ( z ) as z → .Proof. Recall the well-known following representation of the Bessel functions of thefirst species(2.27) J ν ( z ) = (cid:16) z (cid:17) ν + ∞ X j =0 ( − j j !Γ( j + ν + 1) (cid:16) z (cid:17) j . For clarity, we simply write(2.28) J ν ( z ) = z ν ( a + a z + a z + O ( z )) , hence(2.29) J ′ ν ( z ) = z ν − ( νa + ( ν + 2) a z + ( ν + 4) a z + O ( z ))where the coefficients a , a , a are defined by (2.27). By (2.28), (2.29) and standardcomputations it follows that J ν ( z ) J ′ ν ( z ) = zν − a ν a z + O ( z ) , which gives exactly (2.26). (cid:3) Lemma 2.30.
For any λ > the remainders R ( λ, ε ) and ˆ R ( λ, ε ) defined in theproof of Lemma 2.7 are O ( ε ) , O ( ε √ ε ) , respectively, as ε → . Moreover, thesame holds true for the corresponding partial derivatives ∂ λ R ( λ, ε ) , ∂ λ ˆ R ( λ, ε ) .Proof. First, we consider R ( a ) = R ( √ λε (1 − ε )) where R is defined in Lemma 2.25and we differentiate it with respect to λ . We obtain ∂R ( a ) ∂λ = aR ′ ( a )2 λ , hence by Lemma 2.25 we can conclude that R ( a ) and ∂R ( a ) ∂λ are O ( ε √ ε ) as ε → R ( b ) and R ( b ) defined in (2.12), (2.13). Since λ >
0, we havethat b > b and we can write √ λ ∂R ( b ) ∂λ = εb ( ε ) √ ε (1 − ε ) + ∞ X k =4 b k − ε k − ( k − − ε ) k − (cid:16) Y ′ ν ( b ) J ( k +1) ν ( b ) − J ′ ν ( b ) Y ( k +1) ν ( b ) (cid:17) + b ( ε ) √ ε + ∞ X k =4 ε k b k k !(1 − ε ) k (cid:16) Y ′ ν ( b ) J ( k +1) ν ( b ) − J ′ ν ( b ) Y ( k +1) ν ( b ) (cid:17) ′ . Here and in the sequel we write ν instead of ν l . Using the fact that b = p λ/εb ( ε )and Lemma 3.2 we conclude that all the cross products of the form Y ′ ν ( b ) J ( k +1) ν ( b ) − J ′ ν ( b ) Y ( k +1) ν ( b ) and their derivatives ( Y ′ ν ( b ) J ( k +1) ν ( b ) − J ′ ν ( b ) Y ( k +1) ν ( b )) ′ are O ( √ ε )and O ( ε ) respectively, as ε →
0. It follows that R ( b ) and ∂ λ R ( b ) are O ( ε √ ε ) as ε → Similarly, √ λ ∂R ( b ) ∂λ = εb ( ε ) √ ε (1 − ε ) + ∞ X k =3 b k − ε k − ( k − − ε ) k − (cid:16) J ν ( b ) Y ( k +1) ν ( b ) − Y ν ( b ) J ( k +1) ν ( b ) (cid:17) + b ( ε ) √ ε + ∞ X k =3 ε k b k k !(1 − ε ) k (cid:16) J ν ( b ) Y ( k +1) ν ( b ) − Y ν ( b ) J ( k +1) ν ( b ) (cid:17) ′ , hence R ( b ) and ∂ λ R ( b ) are O ( ε ) as ε → R ( λ, ε ) = R ( a ) (cid:20) επ (1 − ε ) (cid:18) ν b − (cid:19) + ε π (1 − ε ) (cid:18) − ν b (cid:19) + ε b π (1 − ε ) (cid:18) ν + 11 ν b − ν b + 1 (cid:19)(cid:21) + R ( b ) (cid:20) aν + a ν (1 + ν ) (cid:21) + R ( b ) ab + R ( a ) R ( b ) . We conclude that R ( λ, ε ) is O ( ε ) as ε →
0. Moreover, it easily follows that ∂R ( λ,ε ) ∂λ is also O ( ε ) as ε → R and its derivatives is similar and we omit it. (cid:3) Remark 2.31.
According to standard Landau’s notation, saying that a function f ( z ) is O ( g ( z )) as z → means that there exists C > such that | f ( z ) | ≤ C | g ( z ) | for any z sufficiently close to zero. Thus, using Landau’s notation in the statementsof Lemmas 2.7, 2.30 understands the existence of such constants C , which in prin-ciple may depend on λ > . However, a careful analysis of the proofs reveals thatgiven a bounded interval of the type [ A, B ] with < A < B then the appropriateconstants C in the estimates can be taken independent of λ ∈ [ A, B ] . The case N = 1 . We include here a description of the case N = 1 for thesake of completeness. Let Ω be the open interval ] − , ( u ′′ ( x ) = 0 , for x ∈ ] − , ,u ′ ( ±
1) = ± λ M u ( ± , in the unknowns λ and u . It is easy to see that the only eigenvalues are λ = 0 and λ = M and they are associated with the constant functions and the function x ,respectively. As in (1.3), we define a mass density ρ ε on the whole of ] − ,
1[ by ρ ε ( x ) = (cid:26) M ε − ε if x ∈ ] − , − ε [ ∪ ]1 − ε, ,ε if x ∈ ] − ε, − ε [ . Note that for any x ∈ ] − ,
1[ we have ρ ε ( x ) → ε →
0, and R − ρ ε dx = M forall ε >
0. Problem (1.4) for N = 1 reads(2.33) (cid:26) − u ′′ ( x ) = λρ ε ( x ) u ( x ) , for x ∈ ] − , ,u ′ ( −
1) = u ′ (1) = 0 . It is well-known from Sturm-Liouville theory that problem (2.33) has an increasingsequence of non-negative eigenvalues of multiplicity one. We denote the eigenvaluesof (2.33) by λ l ( ε ) with l ∈ N . For any ε ∈ ]0 , λ ( ε )and the corresponding eigenfunctions are the constant functions.We establish an implicit characterization of the eigenvalues of (2.33). TEKLOV EIGENVALUES 13
Proposition 2.34.
The nonzero eigenvalues λ of problem (2.33) are given implic-itly as zeros of the equation (2.35) 2 s ε (cid:18) M ε − ε (cid:19) cos (2 √ λε (1 − ε )) sin ε s λ (cid:18) M ε − ε (cid:19)! + " − M ε + 1 + (cid:18) M ε − ε (cid:19) cos ε s λ (cid:18) M ε − ε (cid:19)! sin (cid:16) √ λε (1 − ε ) (cid:17) = 0 . Proof.
Given an eigenvalue λ >
0, a solution of (2.33) is of the form u ( x ) = A cos ( √ λρ x ) + B sin ( √ λρ x ) , for x ∈ ] − , − ε [ ,C cos ( √ λρ x ) + D sin ( √ λρ x ) , for x ∈ ] − ε, − ε [ ,E cos ( √ λρ x ) + F sin ( √ λρ x ) , for x ∈ ]1 − ε, , where ρ = ε, ρ = M ε − ε and A, B, C, D, E, F are suitable real numbers. Weimpose the continuity of u and u ′ at the points x = − ε and x = 1 − ε andthe boundary conditions, obtaining a homogeneous system of six linear equationsin six unknowns of the form M v = 0, where v = ( A, B, C, D, E, F ) and M is thematrix associated with the system. We impose the condition det M = 0. Thisyields formula (2.35). (cid:3) Note that λ = 0 is a solution for all ε >
0, then we consider only the case ofnonzero eigenvalues. Using standard Taylor’s formulas, we easily prove the following
Lemma 2.36.
Equation (2.35) can be rewritten in the form (2.37) M − λM λM (cid:18) λ (cid:18) M (cid:19)(cid:19) ε + R ( λ, ε ) = 0 , where R ( λ, ε ) = O ( ε ) as ε → . Finally, we can prove the following theorem. Note that formula (2.39) is thesame as (2.22) with N = 1 , l = 1. Theorem 2.38.
The first eigenvalue of problem (2.33) has the following asymptoticbehavior (2.39) λ ( ε ) = λ + 23 ( λ + λ ) ε + o ( ε ) as ε → , where λ = 2 /M is the only nonzero eigenvalue of problem (2.32) . Moreover, for l > we have that λ l ( ε ) → + ∞ as ε → .Proof. The proof is similar to that of Theorem 2.21. It is possible to prove that theeigenvalues λ l ( ε ) of (2.33) depend with continuity on ε >
0. We consider equation(2.37) and apply the Implicit Function Theorem. Equation (2.37) can be written inthe form F ( λ, ε ) = 0, with F of class C in ]0 , + ∞ [ × [0 ,
1[ with F ( λ,
0) = M − λM , F ′ λ ( λ,
0) = − M and F ′ ε ( λ,
0) = λM (1 + λ (2 + M )).Since λ = M , F ( λ ,
0) = 0 and F ′ λ ( λ , = 0, the zeros of equation (2.39) ina neighborhood of ( λ,
0) are given by the graph of a C -function ε λ ( ε ) with λ (0) = λ . We note that λ ( ε ) = λ ( ε ) for all ε small enough. Indeed, assuming bycontradiction that λ ( ε ) = λ l ( ε ) with l ≥
2, we would obtain that, possibly passingto a subsequence, λ ( ε ) → ¯ λ as ε →
0, for some ¯ λ ∈ [0 , λ [. Then passing to the limit in (2.37) as ε → λ ( · ) is of class C in a neighborhood of zero and λ ′ (0) = − F ′ ε ( λ , /F ′ λ ( λ ,
0) which yields formula(2.39).The divergence as ε → λ l ( ε ) with l >
1, is clearlydeduced by the fact that the existence of a converging subsequence of the form λ l ( ε n ), n ∈ N would provide the existence of an eigenvalue for the limiting problem(2.32) different from λ and λ , which is not admissible. (cid:3) Appendix
We provide here explicit formulas for the cross products of Bessel functions usedin this paper.
Lemma 3.1.
The following identities hold Y ν ( z ) J ′ ν ( z ) − J ν ( z ) Y ′ ν ( z ) = − πz ,Y ν ( z ) J ′′ ν ( z ) − J ν ( z ) Y ′′ ν ( z ) = 2 πz ,Y ′ ν ( z ) J ′′ ν ( z ) − J ′ ν ( z ) Y ′′ ν ( z ) = 2 πz (cid:18) ν z − (cid:19) , Proof.
It is well-known (see [1, § J ν ( z ) Y ′ ν ( z ) − Y ν ( z ) J ′ ν ( z ) = J ν +1 ( z ) Y ν ( z ) − J ν ( z ) Y ν +1 ( z ) = 2 πz , which gives the first identity in the statement. The second identity holds since J ν ( z ) Y ′′ ν ( z ) − Y ν ( z ) J ′′ ν ( z ) = (cid:0) J ν ( z ) Y ′ ν ( z ) − Y ν ( z ) J ′ ν ( z ) (cid:1) ′ = (cid:18) πz (cid:19) ′ = − πz . The third identity holds since Y ′ ν ( z ) J ′′ ν ( z ) − J ′ ν ( z ) Y ′′ ν ( z ) = Y ′ ν ( z ) (cid:16) J ν − ( z ) − νz J ν ( z ) (cid:17) ′ − J ′ ν ( z ) (cid:16) Y ν − ( z ) − νz Y ν ( z ) (cid:17) ′ = Y ′ ν ( z ) J ′ ν − ( z ) − J ′ ν ( z ) Y ′ ν − ( z ) + νz (cid:0) Y ′ ν ( z ) J ν ( z ) − J ′ ν ( z ) Y ν ( z ) (cid:1) = (cid:18) Y ′ ν ( z ) 12 ( J ν − ( z ) − J ν ( z )) − J ′ ν ( z ) 12 ( Y ν − ( z ) − Y ν ( z )) (cid:19) + 2 νπz = 12 (cid:0) Y ′ ν ( z ) J ν − ( z ) − J ′ ν ( z ) Y ν − ( z ) (cid:1) − (cid:0) Y ′ ν ( z ) J ν ( z ) − J ′ ν ( z ) Y ν ( z ) (cid:1) + 2 νπz = 12 (cid:0) J ′ ν ( z ) Y ν ( z ) − Y ′ ν ( z ) J ν ( z ) (cid:1) + ν − z (cid:0) Y ′ ν ( z ) J ν − ( z ) − J ′ ν ( z ) Y ν − ( z ) (cid:1) − πz + 2 νπz = ν − z (cid:16) J ν − ( z ) (cid:16) Y ν − ( z ) − νz Y ν ( z ) (cid:17) − Y ν − ( z ) (cid:16) J ν − ( z ) − νz J ν ( z ) (cid:17)(cid:17) − πz + 2 νπz = − ν ( ν − z ( Y ν ( z ) J ν − ( z ) − J ν ( z ) Y ν − ( z )) − πz + 2 νπz = 2 πz (cid:18) − ν z (cid:19) , TEKLOV EIGENVALUES 15 where the first, second and fourth equalities follow respectively from the well-knownformulas C ′ ν ( z ) = C ν − ( z ) − νz C ν ( z ), 2 C ′ ν ( z ) = C ν − ( z ) − C ν +1 ( z ) and C ν − ( z ) + C ν ( z ) = ν − z C ν − ( z ), where C ν ( z ) stands both for J ν ( z ) and Y ν ( z ) (see [1, § (cid:3) Lemma 3.2.
The following identities hold Y ν ( z ) J ( k ) ν ( z ) − J ν ( z ) Y ( k ) ν ( z ) = 2 πz ( r k + R ν,k ( z )) , (3.3) Y ′ ν ( z ) J ( k ) ν ( z ) − J ′ ν ( z ) Y ( k ) ν ( z ) = 2 πz ( q k + Q ν,k ( z )) , (3.4) for all k > and ν ≥ , where r k , q k ∈ { , , − } , and Q ν,k ( z ) , R ν,k ( z ) are fi-nite sums of quotients of the form c ν,k z m , with m ≥ and c ν,k a suitable constant,depending on ν, k .Proof. We will prove (3.3) and (3.4) by induction. Identities (3.3) and (3.4) holdfor k = 1 and k = 2 by Lemma 3.1. Suppose now that Y ν ( z ) J ( k ) ν ( z ) − J ν ( z ) Y ( k ) ν ( z ) = 2 πz ( r k + R ν,k ( z )) ,Y ′ ν ( z ) J ( k ) ν ( z ) − J ′ ν ( z ) Y ( k ) ( z ) = 2 πz ( q k + Q ν,k ( z )) , hold for all ν ≥
0. First consider Y ′ ν ( z ) J ( k +1) ν ( z ) − J ′ ν ( z ) Y ( k +1) ν ( z ) . We use the recurrence relations C ν +1 ( z )+ C ν − ( z ) = νz C ν ( z ) and 2 C ′ ( z ) = C ν − ( z ) −C ν +1 ( z ), where C ν ( z ) stands both for J ν ( z ) and Y ν ( z ) (see [1, § (3.5) Y ′ ν ( z ) J ( k +1) ν ( z ) − J ′ ν ( z ) Y ( k +1) ν ( z ) = Y ′ ν ( z )( J ′ ν ) ( k ) ( z ) − J ′ ν ( z )( Y ′ ν ) ( k ) ( z )= 14 h ( Y ν − ( z ) − Y ν +1 ( z )) ( J ν − ( z ) − J ν +1 ( z )) ( k ) − ( J ν − ( z ) − J ν +1 ( z )) ( Y ν − ( z ) − Y ν +1 ( z )) ( k ) i = 14 h(cid:16) Y ν − ( z ) J ( k ) ν − ( z ) − J ν − ( z ) Y ( k ) ν − ( z ) (cid:17) + (cid:16) Y ν +1 ( z ) J ( k ) ν +1 ( z ) − J ν +1 ( z ) Y ( k ) ν +1 ( z ) (cid:17) + (cid:16) J ν +1 ( z ) Y ( k ) ν − ( z ) − Y ν − ( z ) J ( k ) ν +1 ( z ) (cid:17) + (cid:16) J ν − ( z ) Y ( k ) ν +1 ( z ) − Y ν +1 ( z ) J ( k ) ν − ( z ) (cid:17)i = 14 (cid:20) πz (cid:0) r k + R ν − ,k ( z ) + r k + R ν +1 ,k ( z ) (cid:1) + 2 νz (cid:16) J ν ( z ) Y ( k ) ν − − Y ν ( z ) J ( k ) ν − ( z ) + J ν ( z ) Y ( k ) ν +1 ( z ) − Y ν ( z ) J ( k ) ν +1 ( z ) (cid:17) − (cid:16) J ν − ( z ) Y ( k ) ν − ( z ) − Y ν − ( z ) J ( k ) ν − ( z ) + J ν +1 ( z ) Y ( k ) ν +1 ( z ) − Y ν +1 J ( k ) ν +1 ( z ) (cid:17)i = 14 (cid:20) πz (cid:0) r k + R ν − ,k ( z ) + R ν +1 ,k ( z ) (cid:1) + 2 νz (cid:16) J ν ( z ) ( Y ν − ( z ) + Y ν +1 ( z )) ( k ) − Y ν ( z ) ( J ν − ( z ) + J ν +1 ( z )) ( k ) (cid:17)(cid:21) = 1 πz (cid:0) r k + R ν − ,k ( z ) + R ν +1 ,k ( z ) (cid:1) + ν z J ν ( z ) (cid:18) z Y ν ( z ) (cid:19) ( k ) − Y ν ( z ) (cid:18) z J ν ( z ) (cid:19) ( k ) ! = 2 πz (cid:20) r k + 12 (cid:0) R ν − ,k ( z ) + R ν +1 ,k ( z ) (cid:1) − ν z k X j =0 k !( − k − j j ! z k − j +1 ( r j + R ν,j ( z )) . We prove now (3.4) (3.6) Y ν ( z ) J ( k +1) ν ( z ) − J ν ( z ) Y ( k +1) ν ( z ) = (cid:16) Y ν ( z ) J ( k ) ν ( z ) − J ν ( z ) Y ( k ) ν ( z ) (cid:17) ′ − (cid:16) Y ′ ν ( z ) J ( k ) ν ( z ) − J ′ ν ( z ) Y ( k ) ν ( z ) (cid:17) = 2 πz (cid:18) − q k − Q ν,k ( z ) − r k z − R ν,k ( z ) z + R ′ ν,k ( z ) (cid:19) . This concludes the proof. (cid:3)
Corollary 3.7.
The following formulas hold J ν ( z ) Y ′′′ ν ( z ) − Y ν ( z ) J ′′′ ν ( z ) = 2 πz (cid:18) ν z − (cid:19) ; Y ′ ν ( z ) J ′′′ ν ( z ) − J ′ ν ( z ) Y ′′′ ν ( z ) = 2 πz (cid:18) − ν z (cid:19) ; Y ′ ν ( z ) J ′′′′ ν ( z ) − J ′ ν ( z ) Y ′′′′ ν ( z ) = 2 πz (cid:18) − ν z + ν + 11 ν z (cid:19) . TEKLOV EIGENVALUES 17
Proof.
From Lemma 3.2 (see in particular (3.6)) it follows J ν ( z ) Y ′′′ ν ( z ) − Y ν ( z ) J ′′′ ν ( z ) = − πz (cid:20) − q − Q ν, ( z ) − r z − R ν, ( z ) z + R ′ ν, ( z ) (cid:21) = 2 πz (cid:18) ν z − (cid:19) . Next we compute Y ′ ν ( z ) J ′′′ ν ( z ) − J ′ ν ( z ) Y ′′′ ν ( z ) = 2 πz " r + R ν, ( z ) − ν z X j =0 − − j j ! z − j +1 ( r j + R ν,j ( z )) = 2 πz (cid:18) − ν z (cid:19) . Finally, by (3.5) with k = 3, we have Y ′ ν ( z ) J ′′′′ ν ( z ) − J ′ ν ( z ) Y ′′′′ ν ( z ) = 2 πz (cid:20) r + 12 ( R ν − , ( z ) + R ν +1 , ( z )) − ν z X j =0 − − j j ! z − j +1 ( r j + R ν,j ( z )) = 2 πz (cid:18) − ν z + ν + 11 ν z (cid:19) . (cid:3) Acknowledgments.
Large part of the computations in this paper have beenperformed by the second author in the frame of his PhD Thesis under the guidanceof the first author. The authors acknowledge financial support from the researchproject ‘Singular perturbation problems for differential operators’, Progetto di Ate-neo of the University of Padova and from the research project ‘INdAM GNAMPAProject 2015 - Un approccio funzionale analitico per problemi di perturbazione sin-golare e di omogeneizzazione’. The authors are members of the Gruppo Nazionaleper l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of theIstituto Nazionale di Alta Matematica (INdAM).
References [1] M. Abramowitz, I. Stegun,
Handbook of Mathematical Functions with Formulas, Graphs andMathematical Tables , eds(1972) New York: Dover Publications, ISBN 978-0-486-61272-0[2] J. M. Arrieta, A. Jimenez-Casas, A. Rodriguez-Bernal,
Flux terms and Robin boundary con-ditions as limit of reactions and potentials concentrating in the boundary.
Rev. Mat. Iberoam. (2008), no. 1, 183–211.[3] J.M. Arrieta, P.D. Lamberti, Spectral stability results for higher-order operators under per-turbations of the domain.
C. R. Math. Acad. Sci. Paris 351 (2013), no. 19–20, pp 725–730.[4] C. Bandle,
Isoperimetric inequalities and applications . Monographs and Studies in Mathe-matics, 7. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980.[5] D. Buoso, L. Provenzano,
A few shape optimization results for a biharmonic Steklov problem ,J. Differential Equations 259 (2015), no. 5, 1778-1818.[6] R. Courant R, D. Hilbert,
Methods of mathematical physics vol. I, Interscience, New York,1953.[7] G. Folland,
Introduction to partial differential equations . Second edition. Princeton UniversityPress, Princeton, NJ, 1995.[8] A. Girouard, I. Polterovich,
Spectral geometry of the Steklov problem , to appear in J. SpectralTheory. [9] D. G´omez, M. Lobo, S.A. Nazarov, E. P´erez,
Spectral stiff problems in domains surroundedby thin bands: asymptotic and uniform estimates for eigenvalues . J. Math. Pures Appl. (9)85 (2006), no. 4, 598-632.[10] D. G´omez, M. Lobo, S.A. Nazarov, E. P´erez,
Asymptotics for the spectrum of the Wentzellproblem with a small parameter and other related stiff problems.
J. Math. Pures Appl. (9)86 (2006), no. 5, 369-402.[11] P.D. Lamberti,
Steklov-type eigenvalues associated with best Sobolev trace constants: domainperturbation and overdetermined systems.
Complex Var. Elliptic Equ. 59 (2014), no. 3, 309-323.[12] P.D. Lamberti, M. Lanza de Cristoforis,
A real analyticity result for symmetric functionsof the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator , J.Nonlinear Convex Anal. 5 (2004), 19-42.[13] P.D. Lamberti, M. Perin
On the sharpness of a certain spectral stability estimate for theDirichlet Laplacian.
Eurasian Math. J. 1 (2010), Vol. 1, no. 1, 111-122.[14] P.D. Lamberti, L. Provenzano
Viewing the Steklov eigenvalues of the Laplace operator as crit-ical Neumann eigenvalues , in Current Trends in Analysis and Its Applications, Proceedingsof the 9th ISAAC Congress, Krak´ow 2013, Birkh¨auser Basel, 2015, 171-178.[15] P.D. Lamberti, L. Provenzano,
A maximum principle in spectral optimization problems forelliptic operators subject to mass density perturbations , Eurasian Mathematical Journaln(2013), Vol. 4, no. 3, 70-83.[16] V.A. Kozlov, V.G. Maz’ya, J. Rossmann,
Spectral problems associated with corner singulari-ties of solutions to elliptic equations . Mathematical Surveys and Monographs, , AmericanMathematical Society, Providence, RI, 2001.[17] W-M. Ni, X. Wang, On the first positive Neumann eigenvalue.
Discrete Contin. Dyn. Syst.17 (2007), no. 1, 1–19.[18] W. Stekloff,
Sur les probl´emes fondamentaux de la physique math´ematique (suite et fin).
Ann. Sci. ´Ecole Norm. Sup. (3), 19 (1902), 455-490.
Dipartimento di Matematica, Universit`a degli Studi di Padova, Via Trieste, 63, 35126Padova, Italy
E-mail address : [email protected] Dipartimento di Matematica, Universit`a degli Studi di Padova, Via Trieste, 63, 35126Padova, Italy
E-mail address ::