Extremal domains of big volume for the first eigenvalue of the Laplace–Beltrami operator in a compact manifold
Pieralberto Sicbaldi · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2013
We prove the existence of new extremal domains for the first eigenvalue of the Laplace–Beltrami operator in some compact Riemannian manifolds of dimension n⩾2 . The volume of such domains is close to the volume of the manifold. If the first eigenfunction \phi _{0} of the Laplace–Beltrami operator over the manifold is a nonconstant function, these domains are close to the complement of geodesic balls centered at a nondegenerate critical point of \phi _{0} . If \phi _{0} is a constant function and n⩾4 , these domains are close to the complement of geodesic balls centered at a nondegenerate critical point of the scalar curvature.