Let . Let be a bounded open domain of of class . Let denote the outward unit normal to . We assume that the Steklov problem in , on has a multiple eigenvalue of multiplicity r. Then we consider an annular domain obtained by removing from a small cavity of class and size , and we show that under appropriate assumptions each elementary symmetric function of r eigenvalues of the Steklov problem in , on which converge to as ϵ tend to zero, equals real a analytic function defined in an open neighborhood of in and computed at the point for small enough. Here denotes the outward unit normal to , and and if . Such a result is an extension to multiple eigenvalues of a previous result obtained for simple eigenvalues in collaboration with S. Gryshchuk.
In this paper we consider a Steklov eigenvalue problem in a domain perforated by a hole. First we introduce the problem with no hole, and then we consider the case with the hole.
We consider an open bounded connected subset of of of class , for some , such that and such that the complement in of the closure is also connected. Then we consider the Steklov eigenvalue problem
Here denotes the outward unit normal to . By definition, a Steklov eigenvalue for Δ in is a real number λ for which problem (1.1) has a nontrivial solution in . As is well known, problem (1.1) has an increasing sequence of eigenvalues , and we write each eigenvalue as many times as its multiplicity (cf. e.g., Henrot [13, p. 113]). Here denotes the set of natural numbers including 0. In this paper, we assume that there exists such that the eigenvalue has multiplicity and that
Then problem (1.1) has an eigenspace of solutions of dimension r.
Next we make a hole in the domain . Namely, we consider another open bounded connected subset of of class such that and such that the complement in of is also connected, and we take such that for , and we consider the annular (or perforated) domain . Obviously, . For each , we consider the Steklov eigenvalue problem
Here denotes the outward unit normal to . By definition, a Steklov eigenvalue for Δ in is a real number λ for which problem (1.2) has a nontrivial solution. As is well known, problem (1.2) has an increasing sequence of Steklov eigenvalues , and we write each Steklov eigenvalue as many times as its multiplicity. We are interested in the behaviour of the eigenvalues of (1.2) as ϵ tends to 0. This type of problem has been considered for a long time. We mention the explicit computations of Dittmar [6] in a circular annulus. Then we mention Nazarov [29], who has proved that the eigenvalues of tend to those of as ϵ tends to zero and who has computed corresponding complete asymptotic expansions.
We also mention the asymptotic expansions for singularly perturbed domains with a peak of Nazarov [27,28], and the paper of Nazarov and Taskinen [30] on the spectrum in a domain with a peak. We also mention the work of Chiado Piat, Nazarov, Piatnitski [3], Douanla [8], Mel’nik [24], Pastukhova [31], Vanninathan [36], which concern the case of periodic perforations and who aim at understanding the limiting behaviour of the eigenvalues and the existence of asymptotic expansions.
In this paper instead, we generalize the work of [12] with Gryshchuk in the case of simple eigenvalues and by following a pattern for problems with multiple eigenvalues of Lamberti and the author in [18], of Lamberti [17], and of Buoso and Lamberti [1] for regular perturbations, we show that there exist , and real analytic functions from to such that
for (cf. Theorem 11.1). Here if , if .
By (1.3), if , then (1.3) implies that the following convergent series expansion holds
for sufficiently small and for all . Instead, if , then possibly shrinking , the s-th elementary symmetric functions of the eigenvalues for can be continued real analytically in , and the following convergent series expansion holds
for sufficiently small and for all . Here denotes a partial differentiation with respect to the first variable of . In case , we also prove that each can be continued analytically to (small) negative values of ϵ.
In our analysis we reduce our problem to a system of integral equations, and we mention that the reduction of the Steklov problem to integral equations has also been exploited by Kuznetsov and Motygin [16], and by Shamma [35]. In this paper, we have considered the case of a single hole. By the same ideas one could consider the case with a finite number of holes, at the price of having to consider systems of integral equations with more equations. One could also consider our problem under weaker regularity assumptions on the domain as long as the spectrum is discrete as in the case of Lipschitz sets. The author believes that the ideas of the present paper could be applied as long as the corresponding integral equations still correspond to Fredholm operators.
This paper is organized as follows. Section 2 is a section of preliminaries. In Section 3, we introduce some basic notation and results in potential theory. In Section 4, we formulate the Steklov eigenvalue problem on a domain as an eigenvalue problem for a compact selfadjoint operator in a Hilbert function space on the boundary. In Section 5, we formulate the Steklov eigenvalue problem in as an eigenvalue problem for a compact selfadjoint operator in the space with a specific scalar product . In Section 6, we show that the family of scalar products can be continued analytically for negative values of ϵ. In Section 7, we prove that the operator can be defined implicitly. In Section 8, we prove the real analytic representation formula for the family of Theorem 8.1. In Section 9, we prove Theorem 9.5 on the representation of the elementary symmetric functions of the eigenvalues of the operators of the family which split from a given multiple eigenvalue. In Section 10, we present the above mentioned continuity result for the Steklov eigenvalues of Nazarov [27, Thm. 2.1, p. 288]. In Section 11, we prove our main result Theorem 11.1 on the representation of the elementary symmetric functions of the Steklov eigenvalues.
Preliminaries and notation
We denote the norm on a normed space X by . Let X and Y be normed spaces. We endow the product space with the norm defined by , while we use the Euclidean norm for . We denote by the normed space of linear and continuous maps from X to Y, equipped with its usual norm of the uniform convergence on the unit sphere of X (and we set ). If T is a linear map from X to Y, then we set . For standard definitions of Calculus in normed spaces, we refer to Cartan [2] and to Prodi and Ambrosetti [32]. The symbol denotes the set of natural numbers including 0. Throughout the paper,
A dot ‘·’ denotes the inner product in . Let . Then denotes the closure of and denotes the boundary of . For all , , denotes the j-th coordinate of x, denotes the Euclidean modulus of x in , and denotes the ball . Let Ω be an open subset of . The space of m times continuously differentiable real-valued functions on Ω is denoted by , or more simply by . Let , . The s-th component of f is denoted and the Jacobian matrix of f is denoted . Let , . Then denotes . The subspace of of those functions f such that f and its derivatives of order can be extended with continuity to is denoted . The subspace of whose functions have m-th order derivatives that are Hölder continuous with exponent is denoted (cf. e.g. Gilbarg and Trudinger [11]). Now let Ω be a bounded open subset of . Then endowed with the norm is a Banach space. If , then its Hölder constant is defined as . The space , equipped with its usual norm , is well-known to be a Banach space. We say that a bounded open subset of is of class or of class , if its closure is a manifold with boundary imbedded in of class or , respectively (cf. e.g., Gilbarg and Trudinger [11, §6.2]). For standard properties of the functions of class both on a domain of or on a manifold imbedded in we refer to Gilbarg and Trudinger [11] (see also [20, §2, Lem. 3.1, 4.26, Thm. 4.28], [23, §2]). We retain the standard notation of spaces and of corresponding norms. Also, we find convenient to set
where is the -dmensional measure of and Y is a vector subspace of . We note that throughout the paper ‘analytic’ means ‘real analytic’. For the definition and properties of analytic operators, we refer to Deimling [5, p. 150] and to Prodi Ambrosetti [32, p. 89].
Some basic facts in potential theory
We denote by the function of to defined by
where denotes the dimensional measure of for . is well-known to be the fundamental solution of the Laplace operator. Now let Ω be an open bounded connected subset of of class for some . Let denote the outward unit normal to . Let
be the exterior of Ω. Then we set
for all , and we introduce a notaton for the corresponding boundary operators. Namely,
for all and . As is well known (cf. e.g., Folland [10, Prop. 3.25, p. 129]), if , then , and we set
Also, if , then admits a unique continuous extension to , which we denote by , and admits a unique continuous extension to , which we denote by . Then we have the following result of classical potential theory (cf. Miranda [25,26], see also [23, Thm. 3.1]).
Let. Let Ω be a bounded open subset ofof class. Letbe such that. Then the following statements hold.
The map fromtowhich takes μ tois linear and continuous. The map fromtowhich takes μ tois linear and continuous. Furthermore,on, andonfor all, andon, andon.
The map fromtowhich takes μ tois linear and continuous. The map fromtowhich takes μ tois linear and continuous. Furthermore,on, andon.
The map fromtowhich takes μ tois linear and continuous.
Then we have the following two known elementary lemmas.
Let. Let Ω be a bounded open connected subset ofof class. Then
The map Υ fromto the subspace ofof those functions which are harmonic in Ω and which takestois a linear homeomorphism.
The mapfromtowhich takestois a linear homeomorphism.
For a proof we refer for example to [12, Lem. 3.6], and to the mongraph [4, Thms. 6.41, 6.42] with Dalla Riva and Musolino. For a proof of the following lemma, we refer for example to paper [12, Lem. 3.10 (i)] with Gryshchuk together with the Open Mapping Theorem.
Let. Let Ω be a bounded open connected subset ofof classsuch that the exterioris connected. Then the operatoris an isomorphism ofonto itself.
Formulation of the Steklov eigenvalue problem as an eigenvalue problem for a compact selfadjoint operator in a Hilbert function space on the boundary
We plan to write a formulation of the Steklov problem in an open subset Ω of by means of integral equations on . To do so, one can resort to the representation Lemma 3.2(i) for harmonic functions in Ω and obtain and integral equation (cf. paper [12, Cor. 3.7] with Gryshchuk). However, such an equation is not an eigenvalue equation for a self adjoint operator in . In order to obtain an eigenvalue equation for a self adjoint operator in , we need the following preliminary statement.
Let. Let Ω be a bounded open connected subset ofof class. Then the mapfromtowhich takes τ tois a linear homeomorphism. In particular, problemhas a unique solutionin. Moreover,andgenerates.
Since the kernel of the integral operator has a weak singularity, is compact in . Since , is linear and continuous from to . By Folland [10, Prop. 3.34, 3.36], has dimension one and thus it has a generator . If , then is orthogonal to the space , which is generated by the characteristic function (cf. e.g., Folland [10, Prop. 3.34, 3.36]). Then equality implies that , a contradiction (cf. e.g., Folland [10, Prop. 3.38]). Hence, . By classical regularity theory, (cf. e.g., [21, Thm. 5.1 (i)]). By equality , we can apply Fredholm Alternative to the first component of and show that is surjective. If and , then there exists such that and condition implies that and is injective. Hence is a bijection and the Open Mapping Theorem implies that is a homeomorphism. □
By Lemma 4.1, restricts to an isomorphism from onto itself. Then we can prove the following.
Let. Let Ω be a bounded open connected subset ofof class. The setis contained inand the map fromto the set ofwhich satisfy the Steklov eigenvalue value problemwhich takestois a bijection. In particular, ifand, then.
We first assume that . Then satisfies equation
which we rewrite as
Since the kernel in brackets has a weak singularity, a classical regularization argument implies that (cf. e.g., Folland [10, Prop. 3.13]). Then (cf. e.g., Miranda [26], and paper [7, Thm. 7.2] with Dondi). Since μ satisfies equation (4.4), a classical regularity result implies that (cf. e.g., [21, Thm. 5.1 (i)]). Then the jump relations for the single layer of Theorem 3.1 imply that belongs to and satisfies the Steklov eigenvalue value problem (4.3). In particular, if belongs to , then its image by the map of the statement solves the Steklov eigenvalue problem. If , belong to and if , , then Lemma 3.2(i) implies that and that . Hence, the map of the statement is injective.
We now assume that satisfies the Steklov eigenvalue value problem (4.3). Then Lemma 3.2(i) and the jump relations for the single layer potential ensure that there exists such that and . Then we can integrate on and deduce that . Since and , we deduce that , and that accordingly on . Since the integral of the right hand side on equals 0, the right hand side belongs to and thus Lemma 4.1 implies that the right hand side belongs to the image of the operator of and thus we can write , and belongs to . In particular, the map of the statement is surjective. □
Proposition 4.2 provides a formulation of our problem in . To obtain a formulation in , we exploit the following decomposition of , where is the constant function equal to 1 on .
Let. Let Ω be a bounded open connected subset ofof class. Then the map π fromontowhich takes μ tois a projection ontoalong the subspace ofgenerated by the function. In particular,, where the direct sum is both topological and orthogonal in.
We now consider the inclusion of into , and we show that the map defined by
is selfadjoint in endowed by a scalar product, which we introduce in the following proposition.
Let. Let Ω be a bounded open connected subset ofof class. Then the bilinear formfromtodefined byis continuous and the following statements hold.
is a real scalar product on.
. In particular the normwhich is canonically associated to the scalar productis equivalent to the usual norm of.
is obviously bilinear and symmetric. The continuity of in follows by the continuity of π, by the continuity of , by the continuity of the ordinary scalar product in and of the integral in . By Lemma 4.1, is a linear homeomorphism in , and thus we have . Hence,
for all . Since the sum is topological, the map from to which takes μ to is a continuous isomorphism. Then the Open Mapping Theorem implies the existence of such that
and thus is positive definite and accordingly a scalar product. By the same inequality statement (ii) holds true. □
We are now ready to prove the following.
Let. Let Ω be a bounded open connected subset ofof class. Then the operatordefined in (
4.5
) is compact and selfadjoint in the real Hilbert space. Moreover, the map from the setto the setdefined in (
4.2
) which takestois a bijection. In particular, ifbelongs toand, then.
Since has a weakly singular kernel, it defines a compact operator in . Since is the composition of linear and continuous operators and of a compact operator, it is compact.
We now show that is selfadjoint. Let , . Since the image of is contained in , the π-projection equals the identity map on the image of and the integral of an element of the image of equals zero. Then the second Green identity implies that
Now by interchanging the roles of and in the first two equalities, the right hand side equals , and thus is selfadjoint. The last part of the statement follows because the membership of in implies that , and accordingly that . □
We conclude this section by showing that if , then is uniquely determined by an implicit relation. Namely, we show the following.
Let. Let Ω be a bounded open connected subset ofof class. Let. Then the problemhas a unique solutionand.
A boundary Hilbert space formulation of the Steklov eigenvalue problem in the perforated domain
We shall consider the following assumptions for some .
Now let , be as in (5.1). Then there exists
A simple topological argument shows that is connected, and that has exactly the two connected components and , and that for all . For brevity, we set
Obviously, , and for all , where if , if . In order to shorten our notation, we set
and we emphasize that has a natural real Hilbert space structure (although not the only one that we will consider). We now convert the Steklov eigenvalue problem (1.2) in the perforated domain for small enough into an eigenvalue problem for a selfadjoint operator in by exploiting the results of the previous section. By Proposition 4.5, the nonzero Steklov eigenvalues of problem (1.2) are precisely the reciprocals of those of the compact selfadjoint operator in the Hilbert space . So we now introduce an isomorphism from onto and exploit it to define an ϵ-dependent map in which corresponds to and an ϵ-dependent scalar product on which corresponds to . To do so, we define to be the isomorphism from onto defined by
for each (cf. (5.2)). Then the operator
belongs to and can be considered to be the operator in which corresponds to and the bilinear map
is a scalar product in which can be considered to be the scalar product in which corresponds to . Next we note that the equalities
and the equivalence of the norm
and of the usual norm of and Proposition 4.4(ii) applied to imply that the norm is equivalent to the usual norm
of for all . Then we have the following consequence of Propositions 4.2, 4.5
Let. Let,be as in (
5.1
). If, then the linear operatoris compact and selfadjoint in the real Hilbert space. Moreover, the map from the setto the set of pairswhich satisfy the Steklov eigenvalue problem (
1.2
) in, which takesto, is a bijection. In particular,is an eigenvalue of multiplicityofif and only ifis an eigenvalue of multiplicity r of the Steklov eigenvalue value problem (
1.2
) and if so, we must necessarily have.
We now devote the next three sections to the behavior of and as ϵ is close to 0.
Real analytic continuation for the family of scalar products
We are now ready to show the existence of an analytic continuation of the family to negative values of ϵ. To do so, we write out explicitly by exploiting formula (5.3).
Let. Let,be as in (
5.1
),. Letfor all. If, then the bilinear and symmetric mapdefined byfor all,is continuous. Moreover, the following statements hold.
for all.
for all,. In particular, the continuous symmetric bilinear formis positive semidefinite.
The map fromtowhich takesto, is analytic, andis a real analytic family in the spaceof real valued symmetric bilinear and continuous maps on.
Since the integral is a linear and continuous functional on and , the operator is linear and continuous from to and the operator is linear and continuous from to . Since the pointwise product is continuous from to and from to and the terms in braces are compositions of linear and continuous operators in and in , we conclude that is bilinear and continuous in .
We now verify the formula of statement (i). To do so, we compute the pull back of the bilinear form (cf. (4.6), (5.3)). Clearly,
for all . Let . By the rule of change of variables in integrals, we have
for all , and
for all and for all , and
for all . By combining formulas (4.6), (5.3) and (6.3)–(6.9), and by the rule of change of variables in integrals over , we deduce the validity of the formula of statement (i).
(ii) By setting , we obtain
and
and the right hand side equals (cf. (4.6)).
(iii) Next we show that is analytic in . Since the integral is a linear and continuous functional on and on , the last addendum in the right hand side of the definition (6.2) of is analytic in . Since the pointwise product is continuous from to and from to , it suffices to show that the terms in braces in the right hand side of (6.2) define real analytic maps from to in case of the first addendum and to in case of the second addendum. We consider the first term in braces. Since the integral is a linear and continuous functional, the maps and are real analytic in . Since is linear and continuous in , we conclude that is analytic from to . By an analyticity result for integral operators with real analytic kernel, the map from to which takes to the function of the variable is analytic (cf. [22, Prop. 4.1 (ii)]). Then the map from to which takes to the function of the variable is analytic for . Hence, the first term in braces in the right hand side of (6.2) defines a real analytic map from to . The proof for the second term in braces is similar. Hence the proof of the analyticity of in the variable is complete.
Next we prove that the map from to , which takes ϵ to the bilinear map is analytic. By the formula for the second order differential of the monomial associated to the symmetric bilinear and continuous map , we have for all , for all (cf. e.g., Prodi and Ambrosetti [32, (10.1)]). By the proof above, the map in the variable is analytic, and accordingly its second order partial differential is analytic from to the space . Hence, the map from to , which takes to is analytic. Since such a map is constant with respect to , the map from to the space , which takes ϵ to is analytic and the proof is complete. □
An implicit definition of the operators of the family
The proof of a real analytic representation formula for the family is based upon an implicit relation satisfied by the operators and which we derive by pulling back on the characterization of of Proposition 4.6. We do so in the following statement.
Let. Let,be as in (
5.1
). Letbe as in Theorem
6.1
. Let.
If, then the pairis the only solution inof the system of the following three equations
By Proposition 4.6, , is characterized to be the only solution of the following problem
We have already computed the terms in the left hand side in terms of , (see the computations of (6.4), (6.8), (6.9) where we can replace by and by ). By formulas (6.6), (6.7), we have
Hence,
Then we can compute for and for by exploiting the rule of change of variables in integrals, and we can invoke formula (6.4) in which we replace by and rewrite system (7.4) in the form of the system of the three equations (7.1)–(7.3). □
Next we note that by letting ϵ tend to 0 in (7.1)–(7.3), we obtain the following ‘limiting system’
Now assume that , satisfy the limiting system. Then Proposition 4.6 with and the last two equations of the limiting system imply that
(cf. (4.5)). Since the exterior is connected, Lemma 3.3 implies that is an isomorphism in and accordingly, the first equation of the limiting system implies that
on . Then we introduce the following.
Let . Let , be as in (5.1). We define the limiting operator to be the operator from to itself which maps of to the pair of defined by the right hand sides of (7.6), (7.7).
We note that is independent of its first functional variable and that the following holds.
Let. Let,be as in (
5.1
). The positive semidefinite bilinear and symmetric formis positive definite on the subspaceof. In particular,is positive definite on all the eigenspaces of.
By Theorem 6.1(ii), is positive semidefinite and if belongs to , then . Thus if , then we have . Since , there exists such that . Since , Proposition 4.6 implies that and must satisfy the second and the third equation of the limiting system (7.5) and accordingly
In particular, is constant on and thus the Maximum Principle implies that is constant on the whole of . In particular,
Since satisfies (7.7), we have and thus and is positive definite on . □
Since is entirely determined by its first component , which is a selfadjoint and compact operator in whose nonzero eigenvalues coincide precisely with the reciprocals of those of the Steklov problem (1.1) in , we can prove the following, which corresponds to Proposition 5.1 in case .
Let. Let,be as in (
5.1
). Then the linear operatoris compact and the following equality holdsfor all,. Moreover, the mapfrom the setdefined in (
4.7
) withto the setwhich takesto, whereon, is a bijection. In particular, ifand, then. Finally, if, then the mapis an isomorphism from the spaceonto the space.
Since the integral operator associated to a single layer potential has a weakly singular kernel, it is compact. Then the operator delivered by the right hand side of (7.7) equals the composition of the bounded operator and of a compact operator and is accordingly compact. Then we already know that is compact and we can conclude that is compact. By Theorem 6.1(ii) and by the Definition 7.2 of , we have . Since is selfadjoint, we have , and by switching the roles of and , we conclude that . By Definition 7.2 of , we have if and only if , i.e., , and
Hence, the second part of the statement holds. Then the last part follows by the linearity of . □
A real analytic representation formula for the family
We now turn to show a real analytic representation formula for the family in a sense which we clarify below. The starting point is that is implicitly defined by equations (7.1)–(7.3). Then we observe that if we replace the term in equations (7.1)–(7.3) by a new variable ι, we obtain a system of equations whose terms are analytic in all functional variables and in ϵ, ι and that can be analyzed by an appropriate application of the Implicit Function Theorem in Banach spaces.
Let. Let,be as in (
5.1
). Letbe as in Theorem
6.1
. Then the following statements hold.
There exists,and a real analytic familyinsuch thatandfor all.
By Proposition 7.1, is implicitly defined by system (7.1)–(7.3) for all in . Thus we now recast system (7.1)–(7.3) into an abstract functional equation in Banach space, which we analyze by means of the Implicit Function Theorem.
To do so, we introduce a map from to as follows.
We define to be equal to the difference between the left hand side and the right hand side of equation (7.1), once we have replaced the term which appears in the right hand side by ι, for all .
We define to be equal to the difference between the left hand side and the right hand side of equation (7.2), once we have replaced the term which appears in the right hand side by ι, for all .
We define to be equal to the left hand side of equation (7.3), for all .
By definition of , if , then we can rewrite the system of equations (7.1)–(7.3) as
and is the unique solution of such equation whenever . So equation (8.2) characterizes for each . We now wish to introduce an operator equation which characterizes the operator . Thus we introduce the operator valued map from the set to which takes with to the operator defined by the following formula
Since the integral is a linear and continuous functional on and on , we conclude that , depend analytically on (cf. (6.1)). Similarly, and depend analytically on . Then we know that the linear operators , and , are linear and continuous in and in , respectively. Moreover, an analyticity result for integral operators with real analytic kernel implies that the maps from to which take to the function and to the function , and the maps from to which take to the function , and to the function , are analytic (cf. [22, Prop. 4.1(ii)]). As a consequence, the corresponding Fréchet derivatives with respect to the second variable are analytic and thus the maps from to which take ϵ to , are analytic, and the maps from to which take ϵ to , are analytic.
Hence, the map is real analytic from to . Next we note that equation is just another way of writing the limiting system (7.5), which implies that formulas (7.6), (7.7) hold and that accordingly (cf. Definition 7.2). Then the definition of implies that . We now wish to compute the Fréchet differential of at the point with respect to its last (operator) argument. Since is linear with respect to its last argument, we have for all . We now show that the linear map from to which takes to is a bijection.
Let . We must show that there exists a unique such that
By the definition of , we rewrite such an equality in the form
In order to shorten our notation, we set , . Then we can rewrite equation (8.4) in the following form.
By Lemma 4.1, the map from onto which takes to the pair is an isomorphism. Then the last two equations can be rewritten as
and the right hand side is linear and continous in . Since is connected, Lemma 3.3 implies that is an isomorphism in , and thus the first equation of (8.5) can be rewritten as
and the right hand side is linear and continous in . In particular, equation (8.3) has one and only one solution for each B as above and is a linear isomorphism. Since is continuous, the Open Mapping Theorem implies that it is a homeomorphism. Then the Implicit Function Theorem in Banach space implies the existence of and of , and of an open neighborhood U of in and of a real analytic family as in (8.1) in U such that
In particular, and thus (ii) holds true. Possibly shrinking , we can assume that belongs to for all . By the definition of , we know that for all ϵ in , and we know that is the only solution in of such an equation. Hence, for all . □
Symmetric functions of multiple eigenvalues of the operators of the family
Propositions 4.2, 4.5, 7.4 imply that if we fix , then λ is an eigenvalue of multiplicity for the Steklov problem (1.1) in if and only if is an eigenvalue of multplicity r for the operator , if and only if is an eigenvalue of multplicity r for .
Similarly, Propositions 4.2, 4.5, 5.1 imply that if , then a number is an eigenvalue of multiplicity for the Steklov problem (1.2) in if and only if is an eigenvalue of multplicity r for .
Since the operators and are compact and self adjoint with respect to suitable scalar products, their nonzero eigenvalues are real isolated and positive.
We now assume that is an eigenvalue of mutiplicity of the Steklov problem (1.1) in . Then is a nonzero isolated eigenvalue of multiplicity r for .
In order to study the behavior of the eigenvalues of as ϵ approaches zero, we plan to complexify the operators , in . The complexification of coincides with the space . Since , we have . If Q is a real bilinear form on , then its complexification coincides with the sesquilinear form on defined by
If Q is symmetric, then is conjugate symmetric, i.e., for all . If Q is a scalar product on the real space X, then is a scalar product on the complex space . So for example, the complexification of the usual scalar product
in , coincides with the usual scalar product
in the complexified space . Then if T is a linear operator from to , the complexified operator of T is defined by for all . If T is selfadjoint in the real Hilbert space , then is easily verified to be selfadjoint in the complex Hilbert space .
We also note that if λ is a real eigenvalue of finite geometric multiplicity r of the operator T in , i.e., the real dimension of the eigenspace corresponding to λ equals r, then λ is also an eigenvalue for the operator in of geometric multiplicity r, i.e., the complex dimension of the eigenspace corresponding to λ equals r. By the Spectral Stability Theorem, we immediately deduce the validity of the following statement.
Let,. Let,be as in (
5.1
). Letbe an eigenvalue of multiplicity r of. Thenis an eigenvalue of geometric multiplicity r of the compact complexified operatorand is an isolated point of the spectrum of. Let,be as in Theorem
8.1
.
Letsuch thatdoes not contain 0 and does not contain any point of the spectrum of. Then there existsuch thatand such that the spectrum of the complexified operatordoes not contain elements of the setfor all.
Since is a real eigenvalue of multiplicity r for , then is also an eigenvalue of geometric multiplicity r for its complexified operator . Since is compact and , is an isolated eigenvalue of and there exists δ as in the statement. Then is an open set which contains the spectrum of . Since is a real analytic family in , then the family of complexified operators is real analytic in . In particular, such a family is continuous at and . Then the Spectral Stability Theorem ensures that there exist such that the spectrum of is contained in U for all (cf. e.g., Rudin [34, Thm. 10.20, p. 257]). Possibly shrinking , we can ensure the validity of (9.1). □
By the exploiting Kato Projection, we can prove the following.
Let,. Let,be as in (
5.1
). Letbe an eigenvalue of multiplicity r of. Let δ,,be as in Theorem
9.1
. Let ς be the curve defined byfor all. Then the following statements hold.
If, then the operatoris a projection of the complexified spaceonto an-invariant subspace of.
The map fromtowhich takestois real analytic.
coincides with the projection onto the eigenspace ofcorresponding to the eigenvalue, which has complex dimension equal to r.
There existsuch thatfor alland such that if, then the image ofhas complex dimension equal to r.
If, thenis selfadjoint inand the restriction ofto the-invariant subspacehas precisely r real eigenvalues counted with their multiplicity, in the interval. Moreover,contains no other eigenvalue of.
For a proof of statement (i), we refer to Kato [15, p. 178]. We now turn to the proof of (ii). The space of bounded and continuous functions from the compact set to with the -norm is a complex Banach algebra with unity. Let denote the set of invertible elements of . Then is the set of invertible elements of and is open in the space of bounded and continuous functions from the compact set to , and the map from to itself which takes an element to its inverse is real analytic (cf. e.g., Hille and Phillips [14, Thms. 4.3.2 and 4.3.4]).
Since the map which takes to the map of the variable of is analytic, and the line integral is a linear and continuous from to , we conclude that is real analytic from to .
For a proof of statement (iii), we refer to Kato [15, p. 178]. Indeed, is the only element of the spectrum of which belongs to .
We now consider statement (iv). Since is a projection onto the eigenspace of corresponding to , which has finite dimension equal to r a known result of functional analysis implies that there exists a neighborhood of in such that all projection operators which belong to such a neighborhood have an image of dimension r (cf. Dunford and Schwartz [9, Ch. VII, Lem. 7]). Since in , there exists such that for all . By Theorem 8.1, . Then Proposition 5.1 implies that is selfadjoint in and accordingly that is selfadjoint in . Then the restriction of to the r-dimensional invariant subspace has r real eigenvalues counted with their multiplicity.
By Theorem 9.1, the intersection of and the spectrum of the operator is a spectral set and the image of is the corresponding projection (cf. (9.2)). Then the intersection of the spectrum of and of equals the spectrum of the restriction of the operator to , i.e., the set of the eigenvalues , …, (cf. Dunford and Schwartz [9, Ch. VII, Thm. 20]). □
Next we turn to show that possibly shrinking , , we can choose r vectors for which generate the space and which satisfy the orthonormality conditions
for all and which depend real analytically on when . Namely, we prove the following by a variant of an argument of the proof of a corresponding result of paper [18, Prop. 2.20] with Lamberti.
Let,. Let,be as in (
5.1
). Letbe an eigenvalue of multiplicity r of. Let δ,,be as in Theorem
9.2
.
Then there existsuch thatfor alland r real analytic functionsfor, fromtowhich satisfy the orthonormality conditions (
9.3
) and such thatgenerates the spacefor all.
Since is an eigenvalue of multiplicity r of , Proposition 7.4 ensures that is an eigenvalue of multiplicity r of the compact self adjoint operator in . Then the eigenspace of corresponding to has an orthonormal basis , and we have for all . Now let be the function defined by the right hand side of formula (7.8) when , . By Proposition 7.4, , is a basis of the eigenspace of corresponding to and Theorem 6.1(ii) ensures the validity of the equalities for all . Then is a basis of the (complex) eigenspace of the complexification of corresponding to the real eigenvalue . By the definition of the complexified Hermitian form , we have for all . By Lemma A.1 of the Appendix, and by the continuity of in at , there exist such that the restriction is injective when and accordingly, is a system of r linearly independent vectors which generates . If , we know that is a sesquilinear conjugate symmetric form. If , then we also know that is a scalar product in and that accordingly is a scalar product on . By Proposition 7.3, is positive definite on the subspace of and accordingly on the eigenspace of corresponding to . Hence, is a scalar product on the eigenspace of corresponding to the real eigenvalue . Now the equalities
for all , and the continuity of at and of at ensure that possibly shrinking , for all , for all . Then by following the Gram–Schmidt procedure, we set for all pairs in . Next we argue by induction and we assume that possibly shrinking , , the real analytic functions have been defined for with and that generates the space generated by for all and we define . To do so, we consider the vector . For , we have
Indeed, is positive semidefinite, and it is positive definite on the eigenspace of corresponding to , and if the number in (9.4) were to be equal to zero, then we would have the equality , and would belong to the space generated by the set , which by inductive assumption equals the space generated by the set , a contradiction. By inequality (9.4) and possibly shrinking , , we can assume that
is strictly positive for all . Then we can set
for all . Then conditions (9.3) hold true for all and generates the space generated by for all .
So by finite induction, the same is true for and satisfies conditions (9.3) and generates the space for all . Possibly shrinking , we can assume that for all . □
Next we introduce an -dependent family of (complex) matrices, which represents the matrix of the restriction of to the invariant subspace . We do so by means of the following.
Let. Let,be as in (
5.1
). Letbe an eigenvalue of multiplicity r of. Let δ,,,be as in Proposition
9.3
. Letbe the map from the setto the spaceofmatrices with complex entries defined byfor all. Then the following statements hold.
is real analytic.
If, thenis the Hermitianmatrix associated to the restriction ofto the invariant spaceand has precisely r real eigenvalues counted with their multiplicity, in the interval. Moreover,contains no other eigenvalue of.
is the Hermitian matrix of the restriction ofto the eigenspaceofcorresponding to, which has the real eigenvaluewith multiplicity r, and we set.
Since depends real analytically on and , depend real analytically on , then depends real analytically on for all h, .
(ii) By Proposition 5.1 and Theorem 8.1(i), is selfadjoint in . Then is selfadjoint in . By Proposition 9.3, is an orthonormal basis of the invariant space of , which has dimension r (cf. Theorem 9.2(iv)). Hence, is the matrix of the restriction of to of . Hence, is Hermitian and has r real eigenvalues counted with their multiplicity. Also, Theorem 9.2 implies that can contain no other eigenvalue of .
(iii). Since is analytic, then it is continuous at , and accordingly must be Hermitian. By Proposition 9.3, is an orthonormal basis of the -invariant space , which equals the eigenspace of corresponding to the eigenvalue (see Theorem 9.2(iii)). Then the Hermitian matrix is the matrix of the restriction of to , which has only as eigenvalue and accordingly has r real eigenvalues counted with their multplicity, all of them equal to . □
We are now ready to analyze the behavior of the eigenvalues , …, of Proposition 9.4(ii), which ‘split’ from the multiple eigenvalue of (or of ). We do so by means of the following.
Let,. Let,be as in (
5.1
). Letbe an eigenvalue of multiplicity r of. Let δ be as in Theorem
9.1
. Then there existandand r real analytic functionsfromtosuch thatfor all, and such thathas precisely r real eigenvaluescounted with their multiplicity in the intervalandfor all. Moreover,andIf we further assume that, then there existssuch that for eachthere exists an analytic functionfromtosuch thatfor all, and.
Let , be as in Proposition 9.3. Then we take , . If , Proposition 9.4(ii) ensures that the operator has precisely r real eigenvalues counted with their multiplicity in the interval . Since is the complexification of , the above real eigenvalues are also eigenvalues of and have the same multiplicity. Then we note that if I is the identity matrix in , then we must have
for all . Indeed, is the Hermitian matrix of the restriction of to (see Proposition 9.4). We now define to be the coefficient of multiplied by of the polynomial for all and . Since the are sums of products of the entries of the matrix , Proposition 9.4 ensures that the functions are analytic in . Moreover, equality (9.8) ensures the validity of the equalities in (9.5) and that equality (9.6) holds true.
Since and the spectrum of equals , the Spectral Stability Theorem implies that equality (9.7) holds true (cf. e.g., Rudin [34, Thm. 10.20, p. 257]).
If we further assume that , then by applying the Rellich Theorem to the analytic family of Hermitian matrixes, we deduce the existence of and of the analytic functions (cf. e.g., Rellich [33, Thm. 1, p. 57]). □
Continuity of the eigenvalues of the operators of the family
If , then the operator is self adjoint in the Hilbert space and thus all of its eigenvalues ar real and positive, and we can write the nonzero ones as a decreasing sequence , which has 0 as limiting point and we know that for all (cf. Proposition 5.1).
Then we know that the nonzero eigenvalues of coincide with those of the compact self adjoint operator , which in turn equal the reciprocals of the nonzero Steklov eigenvalues in . Thus the eigenvalues of are real and positive and we can write the nonzero ones as a decreasing sequence , which has 0 as limiting point. Moreover, we know that for all (cf. Proposition 7.4).
By Theorems 6.1 and 8.1, the family of operators of and the family of bilinear and continuous maps on are continuous in ϵ and a continuity result for parameter dependent self adjoint and compact operators in a Hilbert space with a parameter dependent scalar product implies that for each , the eigenvalue depends continuously on (cf. e.g., paper [19, Thm. 5.5] with Lamberti). Then by the continuity result for at of Nazarov [27, Thm. 2.1, p. 288], we deduce the validity of the following.
Let,. Let,be as in (
5.1
). Then the map fromtowhich takes ϵ tois continuous for alland the map fromtowhich takes ϵ tois continuous andfor all.
One could also prove Theorem 10.1 by exploiting Theorem 9.5, but for brevity we omit such a proof.
Symmetric functions of multiple Steklov eigenvalues
Let. Let,be as in (
5.1
). Letbe such thatis an eigenvalue of multiplicityof the Steklov problem (
1.1
) inand thatThen there existandand r real analytic functionsfrom the settosuch thatfor all, and such that the equalities in (
1.3
) hold true for all. Moreover,If we further assume that, then there existssuch that for eachthere exists an analytic functionfromtosuch thatfor alland.
Since is a non zero eigenvalue of multiplicity r of the Steklov eigenvalue problem (1.1) in , then Proposition 4.2 implies that is an eigenvalue of multiplicity r of . Then Proposition 7.4 implies that is an eigenvalue of multiplicity r of . Then Theorem 9.1 implies that there exists such that does not contain 0 and does not contain any point of the spectrum of and Theorem 9.5 implies that there exist and and r real analytic functions from to such that for all , and such that has precisely r real eigenvalues counted with their multiplicity in the interval and that equalities (9.5) and (9.6) hold true.
By the continuity Theorem 10.1, for all . Then possibly shrinking we can assume that and thus we must necessarily have for all , and thus for all (see Proposition 5.1). Then we set for all , and for all and for all , then the equalities in (1.3) and (11.1) hold true. Here ℜ denotes the real part.
Then the last part of the statement follows by the equalities for all and for all and by the last part of the statement of Theorem 9.5. □
Footnotes
Acknowledgements
The author acknowledges the support of ‘Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni’, of ‘INdAM’ and of the project ‘BIRD168373/16: Singular perturbation problems for the heat equation in a perforated domain’ of the University of Padova.
An elementary lemma of operator theory
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