Speaker
Camilla Polvara, Sapienza Università di Roma
Abstract
We consider the critical Neumann problem in cones. It is known that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones, which is defined through the first Neumann eigenvalue of the Laplace-Beltrami operator on the domain D on the unit sphere, which spans the cone. This immediately implies a symmetry breaking result for the minimizers of the Sobolev inequality. We then construct a one-parameter family of domains on the sphere whose first eigenvalue crosses the threshold at which the bubble loses stability. Under the assumption that this eigenvalue is simple, we prove, via the Crandall Rabinowitz bifurcation theorem, the existence of a branch of nonradial solutions bifurcating from the standard bubble. Moreover, we show that the bifurcation is global. These results are contained in joint works with G. Ciraolo, F. Pacella and L. Provenzano.
Camilla Polvara, Sapienza Università di Roma
Abstract
We consider the critical Neumann problem in cones. It is known that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones, which is defined through the first Neumann eigenvalue of the Laplace-Beltrami operator on the domain D on the unit sphere, which spans the cone. This immediately implies a symmetry breaking result for the minimizers of the Sobolev inequality. We then construct a one-parameter family of domains on the sphere whose first eigenvalue crosses the threshold at which the bubble loses stability. Under the assumption that this eigenvalue is simple, we prove, via the Crandall Rabinowitz bifurcation theorem, the existence of a branch of nonradial solutions bifurcating from the standard bubble. Moreover, we show that the bifurcation is global. These results are contained in joint works with G. Ciraolo, F. Pacella and L. Provenzano.