Observability of a Ring Shaped Membrane via Fourier Series

. We study the inverse Ingham type inequality for a wave equation in a ring. This leads to a conjecture on the zeros of Bessel cross product functions. We motivate the validity of the conjecture through numerical results. We do a complete analysis in the particular case of radial initial data, where an improved time of observability is available

This proposition is a special case of a general result, which can be proved by micro-local analysis [2] with the critical time T 0 = 2 √ b 2 − a 2 , according to the geometrical ray condition.
Our aim here is to study this problem using a Fourier series approach [6,5] Note that such an approach has been tackled for the whole disk (see [7]); the case of an annulus leads to new difficulties.In particular, we are not able to treat the general case, as it relies on very precise estimates of the location of the zeros of Bessel cross product functions, that we did not find in the literature.Instead, we state a conjecture on these zeros and give some numerical results to support the conjecture.In the case of radial functions or more generally functions with a limited number of modes in the angle direction, we are able to get the observability estimates with this method, even for smaller times T > 2(b − a), using an asymptotic result of MacMahon (see [1], p. 374).In Section 2, we give the expression of the solution.In Section 3, we formulate a theorem for the special case of radial functions.Section 4 is devoted to the statement of the conjecture and its numerical illustration.Finally, we prove in Section 5 the theorem of Section 3.

Expression of the solution
Let J ν (resp.Y ν ) be the Bessel functions of first (resp.second) kind of order ν.We recall the following proposition (see [7,8]) form a strictly increasing sequence: (ii) The eigenfunctions of the Laplacian corresponding to (1) are R k,m (r)e imθ for k ∈ N * and m ∈ N, where (iii) For a dense set of initial data the solution of (1) is given by the formula with complex coefficients c ± k,m , all but finitely of which vanish.

Observability estimates for radial solutions
We recall from [1] the following estimate of MacMahon (1894): Thanks to this estimation, we can obtain the following theorem.
For each positive integer M there exists a constant c M > 0 such that for all solutions of (1) of the form (4) -This theorem covers the case of radial initial data, corresponding to the case is the critical observability time for general initial data.
-The theorem remains true if the integral in ( 6) is taken only over the outer boundary (the circle of radius b): see the estimate (16)) below.-We may obtain similar results by changing the boundary conditions.For example, we may take homogeneous Neumann condition on the inner boundary, and observe the solution only on the outer boundary (see [4] for the corresponding asymptotic gap estimate that is needed).

Conjecture and numerical illustration
We then state the following conjecture.
-There exists a positive integer Using a Bessel Zeros Computer [3] we can evaluate the zeros γ ν,α,k for several parameters ν, α and k.We plot on Figure 1 (top and bottom left) the values k ν (α) versus ν for different values of α.We observe that k ν (α) increases with ν for a fixed α.The dependance seems to be almost linear, except for small values of α where the dependence seems to be quadratic (see Figure 1, bottom left).On Figure 1 (bottom right), we see the relative difference with the gap π √ 1−α 2 (corresponding to the optimal value of Proposition 1).We observe that γ ν,α,kν (α)+1 − γ ν,α,kν (α) decreases and approaches to this gap as ν increases.

Proof of Theorem 1
We first express the norm in terms of the Fourier coefficients.We have This leads to the equalities and Now, using the orthogonality of the eigenvectors of the Laplacian operator we obtain and a similar computation gives we have to compute also the the first integral on the right side.We have and using the orthogonality of the eigenfunctions, we get Using all these results we obtain the equalities and a by the formula (5), we may apply Ingham's theorem (more precisely its version due to Haraux, see, e.g., [7]) to deduce from the last equality the existence of C 1,M > 0 such that for all complex sequences (c ± k,m ) with c ± k,m = 0 whenever m ≥ M .Now it remains to prove that there exists another constant C 2,M > 0 such that for m ≥ 1, and a similar inequality for m = 0.
We adapt an argument used in [7], p. 107.Let y satisfy the Bessel equation Let c > 0. Multiplying the equation by 2y and integrating over (ca, cb) (instead of (0, c) as in [7]), we get Recall that which satisfies (9), and thus we have (10) with y = y k,m and c = γ m,α,k b .We have (11) From (3) we have the relation Integrating by parts and using the relations Using the relation (12) hence we conclude that Setting  5) and the inequalities γ k,α,m > 0, we conclude that for a suitable constant C 3,M > 0. This proves (8) for m ≥ 1.
and we conclude as before.