Advances in Differential Equations

Generic simplicity of the eigenvalues of the Stokes system in two space dimensions

Jaime H. Ortega and Enrique Zuazua

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We analyze the multiplicity of the eigenvalues for the Stokes operator in a bounded domain of $\mathbb R^2$ with Dirichlet boundary conditions. We prove that, generically with respect to the domain, all the eigenvalues are simple. In other words, given a bounded domain of $\mathbb R^2,$ we prove the existence of arbitrarily small deformations of its boundary such that the spectrum of the Stokes operator in the deformed domain is simple. We prove that this can be achieved by means of deformations which leave invariant an arbitrarily large subset of the boundary. The proof combines Baire's lemma and shape differentiation. However, in contrast with the situation one encounters when dealing with scalar elliptic eigenvalue problems, the problem is reduced to a unique continuation question that may not be solved by means of Holmgrem's uniqueness theorem. We show however that this unique continuation property holds generically with respect to the domain and that this fact suffices to prove the generic simplicity of the spectrum.

Article information

Adv. Differential Equations Volume 6, Number 8 (2001), 987-1023.

First available in Project Euclid: 2 January 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 35P99: None of the above, but in this section 49J20: Optimal control problems involving partial differential equations 93B60: Eigenvalue problems


Ortega, Jaime H.; Zuazua, Enrique. Generic simplicity of the eigenvalues of the Stokes system in two space dimensions. Adv. Differential Equations 6 (2001), no. 8, 987--1023.

Export citation