Courant-sharp Robin eigenvalues for the square: the case with small Robin parameter

Abstract

We consider the cases where there is equality in Courant's nodal domain theorem for the Laplacian with a Robin boundary condition on the square. We treated the cases where the Robin parameter h > 0 is large, small in [5], [6] respectively. In this paper we investigate the case where h < 0 .MSC classification (2010): 35P99, 58J50, 58J37.We treat the particular example where Ω is the square S = (− π 2 , π 2 ) 2 ⊂ R 2 . Our key question is whether it is possible to determine the Courant-sharp eigenvalues of the Robin Laplacian on S with parameter h < 0.In previous work [5,6], we considered the case where h > 0. In [5], we showed that there are finitely many Courant-sharp Robin eigenvalues when h > 0.is an eigenvalue of the Robin Laplacian on S with parameter h and k ≥ 520, then it is not Courant-sharp.In addition, we proved that for h sufficiently large, the only Courant-sharp Robin eigenvalues are for k = 1, 2, 4.Theorem 1.2. There exists h 1 > 0 such that for h ≥ h 1 , the Courant-sharp cases for the Robin problem on S are the same as those for h = +∞ (i.e. the Dirichlet case).It was shown in [7] that the only Courant-sharp Neumann eigenvalues of the square are for k = 1, 2, 4, 5, 9. On the other hand, in [6], we proved the following theorem.Theorem 1.3. There exists h 0 > 0 such that for 0 < h ≤ h 0 , the Courant-sharp cases for the Robin problem on S are the same, except the fifth one, as those for h = 0 (i.e. the Neumann case) .

Courant-sharp Robin eigenvalues for the square and other planar domains

“…We note that if (i+j) is odd for any pair (i, j) such that λ n,h (S) = π −2 (α 2 i +α 2 j ), then we get by (2.10), u(−x, −y) = −u(x, y) and as a consequence u has an even number of nodal domains. As we shall see later, other symmetries related to the finite group generated by the identity and the symmetries (x, y) → (−x, y) and (x, y) → (x, −y) can be considered (see [20]).…”

“…In a second paper, [20], we hope to look at the situation where the Robin parameter h tends to 0 and to discuss the following conjecture. Conjecture 1.3.…”

Courant-sharp Robin eigenvalues for the square: The case of negative Robin parameter

“…In previous work [6,7], we considered the case where h > 0. In [6], we proved the following theorem which asserts that there are finitely many Courant-sharp Robin eigenvalues when h > 0.…”

“…For the case where h < 0, numerically we have that h * 9 ≈ −1.6293. In the case where |h| is sufficiently small, the arguments of [7] for h > 0 also apply for h < 0. In addition, the labelling of the Robin eigenvalues λ k,h (S) is the same as that for the Neumann eigenvalues λ k,0 (S) (see Remark 4.3).…”

“…In order to prove the results of [6,7], we made use of the fact that when h > 0, the Robin eigenvalues interpolate between the Neumann eigenvalues (h = 0) and the Dirichlet eigenvalues (h = +∞). In addition, in [6] we employed the analogue of the Faber-Krahn inequality for the Robin eigenvalues (due to Bossel-Daners) which asserts that among all bounded, Lipschitz domains Ω ⊂ R n of prescribed volume, the ball minimises λ 1,h (Ω).…”