OPTIMALITY CONDITIONS OF THE FIRST EIGENVALUE OF A FOURTH ORDER STEKLOV PROBLEM

. In this paper we compute the ﬁrst and second general domain variation of the ﬁrst eigenvalue of a fourth order Steklov problem. We study optimality conditions for the ball among domains of given measure and among domains of given perimeter. We show that in both cases the ball is a local minimizer among all domains of equal measure and perimeter.

Any minimizer u solves the fourth order Steklov eigenvalue problem Note that the eigenvalue appears in the boundary condition. Let us summarize some properties which are well known by now. We refer to [1,3,4,5] and [6] for more details.
• Among all convex domains in R n having the same measure or perimeter as the unit ball, there exists an optimal one, minimizing d 1 (see Theorem 4.6 in [5] and Theorem 5 in [1]). • Bucur, Ferrero and Gazzola showed that the eigenvalue is differentiable for the perturbed unit ball (see p. 19 in [4]). In complete analogy we can show that the first eigenvalue is differentiable with respect to smooth perturbations.
• Moreover, (3) -(5) admits infinitely many (countable) eigenvalues. The only eigenfunction of one sign is the one corresponding to the first eigenvalue d 1 (Ω). All eigenfunctions are smooth inside Ω, and up to the boundary, they are as smooth as the boundary permits (i.e. u ∈ C 4,α (Ω)) (see Theorem 1 in [3] and Theorem 1 in [6]). • In a recent paper Raulot and Savo studied the Steklov problem on compact manifolds with boundary (see [10]). They obtained estimates for the first Steklov eigenvalue. • The case of higher Steklov eigenvalues (λ k ) k is less investigated. However in a recent paper Liu proved, among other results, some asymptotic formulas for λ k as k → ∞ (see [9]). In particular the author investigated asymptotic formula for rectangular parallelepipeds.
We are interested in the domain dependence of d 1 (Ω). This problem has recently drawn some attention, since it is known that the ball is not the absolute minimizer of d 1 (Ω) among all domains of given measure. Kuttler [8] could show -in two dimensions -that a square has a first eigenvalue d 1 , which is strictly smaller than the one of a disc. Ferrero, Gazzola and Weth (see [6] (1.14) and (1.15)) could improve Kuttler's inequality. More recently Antunes and Gazzola [1] gave numerical evidence that the optimal planar shape is the regular pentagon. We are interested in clarifying the role of the ball. Bucur, Ferrero and Gazzola [4] could show that the ball is a critical point of d 1 among all measure preserving smooth perturbations. One of our main results is: The ball is a strict local minimizer among all domains of equal measure.
Antunes and Gazzola [1] suggest that among all convex planar domains with a fixed perimeter d 1 is minimized by the disk. We could also show that the ball is a local minimizer for perturbations which are perimeter preserving in n dimensions.
Theorem 1.2. The ball is a strict local minimizer among all domains of equal perimeter.
The paper is organized as follows. In Section 2 we introduce some notation needed throughout the paper. From Section 3 on we consider any domain Ω of class C 4,α at least. Domains of this class will be referred to as smooth. Such a domain will be imbedded into a family of domains (Ω t ) t , where Ω t is smooth as well and Ω := Ω 0 . The construction of Ω t is given in Section 3. Throughout the paper we will refer to Ω t as a "smoothly perturbed" domain. There will be two families under consideration. The first one has prescribed measure in the sense that |Ω t | = |Ω| + o(t 2 ) (see Lemma 3.1). The second family has prescribed perimeter in the sense that |∂Ω t | = |∂Ω| + o(t 2 ) (Lemma 3.2). Finally we compute the first domain variation of d 1 among domains of given measure and among domains of given perimeter (see Theorem 3.7). From Section 4 on the unperturbed domain Ω is the unit ball B. We compute also the first and second variation of d 1 (B) among all domains of equal measure and among all domains of equal perimeter. Using results of the previous sections we provide a formula for d 1 (B) • in the measure preserving case (Theorem 4.5) • in the perimeter preserving case (Theorem 5.2).
In particular the second variation in both cases is a quadratic form in u alone. Then we discuss the sign of this quadratic form and derive the strict positivity (Theorem 4.6 Theorem 5.3).
Let ϑ : Ω → R n be a smooth vectorfield, then Furthermore we will write Let (τ i ) i the basis of the tangent space in a point in ∂Ω. Then for any smooth vector field θ we have the decomposition on ∂Ω. We denote by ∇ Γ f = ∇f − ν∂ ν f (7) the tangential gradient of a smooth function f : Ω → R.
In particular we have H = 1 R if Ω is a ball of radius R. We will frequently use the Gauss theorem on surfaces. Let Ω and u ∈ C 2 (Ω) then where ∂ 2 ν u = ν · (D 2 u ν) and where ∆ Γ u denotes the Laplace-Beltrami operator on ∂Ω.
We also recall the representation of the Laplace operator in polar coordinates 3.1. Measure and perimeter preserving domains. Let Ω t be a family of perturbations of the domain Ω ⊂ R n of the form where o(t 2 ) collects all terms such that o(t 2 ) t 2 → 0 for t → 0. For t ∈ (−t 0 , t 0 ) and t 0 > 0 sufficiently small, Φ(t, ·) : Ω → Ω t is a diffeomorphism.
By Jacobi's formula we have for small t In the sequel we will consider perturbation which are measure resp. perimeter preserving. A perturbation will be measure preserving, iff where |Ω| is the n-dimensional Lebesgue measure of Ω. In particular we have In view of (13) this implies and (ii) By Lemma 2.1 in [2] we get the following equivalence.
We get the following result for perimeter preserving perturbations (see also [2] Section 2.2.1): (ii) For the unit ball a perimeter preserving perturbation satisfies 3.2. Properties of the variation of the outer normal vector. Let ν t be the outer normal on ∂Ω t , then we set .
For brevity, we write ν and ν rather than ν (0, x) and ν (0, x). Let x ∈ Ω, then we define δ as the distance function to the boundary For a sufficiently small tubular neighbourhood of ∂Ω we define ν := ∇δ(x) as a smooth extension of ν. The extension of ν implies also an extension of ν into this tubular neighbourhood. A direct consequence of this extension is the formula Indeed, this follows from the fact that |∇δ(x)| = 1 for all x in a sufficiently small tubular neigbourhood. This also implies div Γ ν = div ν.
Since ν t · ν t = 1 on ∂Ω t we have for all sufficiently small t. Thus where summation over repeated indices is understood. Here (τ i ) i denotes again an orthonormal basis of the tangent space in some point of ∂Ω.

Lemma 3.3.
Let ∂Ω be smooth and ν and ν as above. Then Proof. First we use (25) and (7) then we get by differentiating Then (20) gives the claim. Theorem 3.4. Let θ and Ω be as in (12).
For the first domain variation for boundary integrals (see [7] p. 191), we get the following result: With these general first domain variations we can recover Lemma 3.1 (i) resp. Lemma 3.2 (i) by setting g(t) = 1.

First domain variation.
We apply the previous formulas to our Steklov eigenvalue problem. For that we consider the Steklov eigenvalue } for a family (Ω t ) t of perturbed domains in the sense of Section 3.1. We set as the first resp. the second domain variation. Further, let u(t, y) be the corresponding eigenfunction of d 1 (Ω t ) with y ∈ Ω t . We will abbreviate and write u(t) := u(t, y). Then u(t) solves the Euler Lagrange equation in Ω t (26) Let us now introduce the shape derivative u as ∂u(t, Φ(t, x)) ∂t t=0 .

OPTIMALITY CONDITIONS OF A STEKLOV PROBLEM 1849
First we investigate the behavior of the shape derivative on Ω and on ∂Ω.
Lemma 3.6. Let u(t) be a solution of (26)-(28). The shape derivative u satisfies Proof. We give the proof of (31). (29) and (30) follows with the same technique. First we note that y = x + tθ(x) for x ∈ ∂Ω. Thus we may rewrite (28) as equation on ∂Ω for all |t| < t 0 : We differentiate this equation with respect to t and get In t = 0 this gives Since u = 0 on ∂Ω, ∇u = ∂ ν u on ∂Ω. So we get By (20) this expression and by (24) d 1 (Ω)∇u·ν is zero. This proves Lemma 3.6.

1.
Let Ω t be a family of measure preserving perturbations of Ω as described in (12). Then Ω is a critical point of the energy d 1 (Ω t ), i.e. d 1 (Ω) = 0, if and only if 2.
Let Ω t be a family of perimeter preserving perturbations of Ω as described in (12) and c ∈ R. Then Ω is a critical point of the energy d 1 (Ω t ), if and only if Proof. By Theorem 3.4 the numerator integral of the Rayleigh quotient in (1) be- For the last equality we used the biharmonicity of u and u = 0 on ∂Ω.
The integral in the denominator is treated similarly. We apply Theorem 3.5 and (24). The first variation of the denominator integral is Note that the eigenfunction is unique up to scaling, so we can normalize Next we use (31). Then we get We consider the first integral on the right hand side. Since θ τ = θ − (θ · ν)ν (see (6)) we get . From (28) we deduce that So far we did not use the fact that the vector fields are either measure or perimeter preserving. Assume we consider vector fields which are measure preserving in the sense of (15). Assume d 1 (Ω) = 0. Then this is equivalent to the fact that (33) holds. This is a consequence of the fundamental lemma of the calculus of variations.
Similarly assume the vector fields are perimeter preserving in the sense of (18). Then d 1 (Ω) = 0 for all such vector fields implies (34).
Remark 3. We observe that the first variation does not depend on η or on the tangential components of the vector field θ.

Domain variation for the unit ball.
4.1. First variation for the ball. From now on we consider a family B t of perturbations of the unit ball B of the form as in (12). Some particular properties hold for the unit ball. Notice that the mean curvature simplifies to H = 1. In [3] it was shown that the first eigenfunction u is We easily compute For the shape derivative we get ∆ 2 u = 0 in B. The boundary conditions (30) and (31) are with the help of (35) u = 2θ · ν and ∆u − n ∂ ν u + 2n θ · ν = −2d 1 (B) in ∂B.
We recall the mean value formula for harmonic functions on a ball.
Proof. Any harmonic function w ∈ C 0 (B 1 ) satisfies the mean value formula w dx for all 0 < R < 1.
We differentiate this expression with respect to R. This gives the claim for 0 < R < 1. By continuity of w this holds for 0 < R ≤ 1. Proof. We integrate the second equation in (37) and get We apply Lemma 4.1 for ∆u . Then the first variation d 1 (B) is The right hand side is zero if either the perturbations are measure preserving (see (15)) or if they are perimeter preserving (see (18)) since H = 1. Now (37) takes the following form u = 2θ · ν and ∆u = n ∂ ν u − n u in ∂B.

4.2.
Second variation for the ball. We compute the second domain variation to investigate local properties of the critical point B. Two strategies are possible. Starting with the Rayleigh quotient (1) we could derive formulas as in Theorem 3.4 and in Theorem 3.5 for the second derivation of the numerator and denominator in (1). Alternatively we can use (28) where d 1 (B t ) appears explicitly in the boundary condition. It turns out that the second choice is simpler. We first use (28) in order to have an explicit expression for d (B). For that we use (32) and differentiate it again with respect to t. In t = 0 we then get The explicitly known eigenfunction u (see (35)) simplifies this expression to 0 =∆u + 2θ · ∇∆u + 2d 1 (B) We compute some of the terms in this sum explicitly for the ball.
We will analyze the first term on the right hand side of this equality. Since it is a tangential derivative of ∆u and since by Finally we use (9) to replace ∂ 2 ν u . This gives 2θ τ · ∇ Γ ∆u =2nθ · (D 2 u ν) + 2nθ τ · ∇u − 2n(θ · ν)∆u Then we apply (39) to replace ∆u on the right hand side of (43). This gives the claim.
After integration of (45), the fourth term on the right hand side of (45) can be integrated by parts (see (8) This and (46) simplify (45) and we get The following integral identity was proved in Lemma 5 from [2].
Lemma 4.4. For measure preserving perturbations there holds We rewrite this formula for our case. Note that the first integrand on the right hand side can be written with the help of (35) and (39) as We insert this into (47) and use (39) again. So we get and can formulate the main result of this section.
Proof. Since u is biharmonic we get by partial integration Then (49) and (39) give the claim.
It is a remarkable property of the model that d 1 (B) only depends on u and does not depend on u or on η. Thus d 1 (B) is a quadratic functional of u . We set for any shape derivative u . Recall that u ∈ H 2,2 (B) is a shape derivative iff u is biharmonic and satifies (39) for any domain variation that is measure preserving.
In the next subsection we will show that J vol > 0 for all such shape derivatives.

4.3.
Discussion of the sign of the second variation of the unit ball. To determine the sign of d 1 (B) resp. J vol we expand u and θ · ν with respect to spherical harmonics. Let s ∈ N∪{0} and i = 1, ..., d s for d s = (2s+n−2) (s+n−3)! s!(n−2)! . The function Y s,i (ξ) denotes the i-th spherical harmonic eigenfunction of order s. The Y s,i (ξ) form an orthonormal basis of L 2 (∂B). In particular we have Then for x ∈ B, r = |x| and ξ = x |x| . We also write Remark 4. For s = 0 the spherical harmonic Y 0,0 is the constant function. It is excluded by (15) and (18), so the sum in (53) and (54) begins with s = 1.
We have to solve ∆ 2 u (x) = 0. We write it as a system ∆u (x) = v(x) and ∆v(x) = 0. First we set With (11) the equation ∆v(x) = 0 is equivalent to We use (52) and get A solution of (57) is given by Note that the solution r −n−s+2 is singular, thus it will be neglected and the solution of ∆v(x) = 0 is Now we can solve ∆u (x) = v(x). By (53), (11) and (59) we get Neglecting singular solutions again we obtain Hence (53) can be rewritten as and (54) as Recall that the shape derivative u also solves ∆u = n∂ ν u − n u on ∂B. By (61) and (11) we compute and (s a s,i r s−1 + (s + 2)b s,i r s+1 )Y s,i (ξ).
For r = 1 we then get (2n + 4s) b s,i = n s a s,i + n(s + 2) b s,i − n a s,i − n b s,i .
Note that the second variation only depends on the dimension n and s. Now we can prove our first main result (see also Theorem 1.1): Theorem 4.6. The ball is a strict local minimizer among all domains of equal measure.
For all n ≥ 2 and s ≥ 1 the second domain variation J vol (n, s) is strictly positive.
5. Perimeter constraint. In this section we change the constraint. Now we consider domains with the same surface area rather than the same measure. Thus we consider the class of perturbations of the ball for which θ and η satisfy (18) and (19). The computation for d 1 (B) is the same as in Section 4. Thus we recall (47).
Analogously to (49) we then get for the second variation under perimeter constraint This implies the following theorem. If we compare this quadratic form with J vol , we observe that J per (u ) = J vol (u ) + n 2(n − 1) Note that the large bracket on the right hand side is zero if and only if the domain deformation consits of translations, i.e. θ · ν = a · x for some vector a ∈ R n . Otherwise it is positive (see Lemma 2 -Lemma 3 [2]). This gives our second main result (see also Theorem 1.2). Theorem 5.3. The ball is a strict local minimizer among all domains of equal perimeter.