In this work we are concerned with the multiplicity of the eigenvalues of the Neumann Laplacian in regions of R n which are invariant under the natural action of a compact subgroup G of O(n). We give a partial positive answer (in the Neumann case) to a conjecture of V. Arnold [1] on the transversality of the transformation given by the Dirichlet integral to the stratification in the space of quadratic forms according to the multiplicities of the eigenvalues. We show, for some classes of subgroups of O(N ) that, generically in the set of G−invariant, C 2 -regions, the action is irreducible in each eigenspace Ker(∆ + λ). These classes include finite subgroups with irreducible representations of dimension not greater than 2 and, in the case n = 2, any compact subgroup of O(2). We also obtain some partial results for general compact subgroups of O(n).

In this note, we prove triviality and nonexistence results for gradient Ricci soliton warped metrics. The proofs stem from the construction of gradient Ricci solitons that are realized as warped products, from which we know that the base spaces of these products are Ricci-Hessian type manifolds. We study this latter class of manifolds as the most appropriate setting to prove our results.

In this paper, we establish a kind of splitting theorem for the eigenvalues of a specific family of operators on the base of a warped product. As a consequence, we prove a density theorem for a set of warping functions that makes the spectrum of the Laplacian a warped-simple spectrum. This is then used to study the generic situation of the eigenvalues of the Laplacian on a class of compact G-manifolds. In particular, we give a partial answer to a question posed in 1990 by Steven Zelditch about the generic situation of multiplicity of the eigenvalues of the Laplacian on principal bundles.2010 Mathematics Subject Classification. Primary 47A75; Secondary 35P05, 47A55.

In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces. Firstly, we prove that if a Banach space E admits a fundamental biorthogonal system, then there exists a continuous vector field f : E → E such that the autonomous differential equation u ′ = f (u) has no solutions at any time. The proof relies on a key result asserting that every infinite-dimensional Fréchet space with a fundamental biorthogonal system possesses a nontrivial separable quotient. The later, is the byproduct of a mixture of known results on barrelledness and two fundamental results of Banach space theory (namely, a result of Pe lczyński on Banach spaces containing L 1 (µ) and the ℓ 1 -theorem of Rosenthal). Next, we introduce a natural notion of weak-approximate solutions for the non-autonomous Cauchy-Peano problem in Banach spaces, and prove that a necessary and sufficient condition for the existence of such an approximation is the absence of ℓ 1 -isomorphs inside the underline space. We also study a kind of algebraic genericity for the Cauchy-Peano problem in spaces E having complemented subspaces with unconditional Schauder basis. It is proved that if K(E) denotes the family of all continuous vector fields f : E → E for which u ′ = f (u) has no solutions at any time, then K(E) {0} is spaceable in sense that it contains a closed infinite dimensional subspace of C(E), the locally convex space of all continuous vector fields on E with the linear topology of uniform convergence on bounded sets.MSC 2010: 34G20, 34A34, 46B20, 47H10, 34K07.

In this paper we address the problem of determining whether the eigenspaces of a class of weighted Laplacians on Cayley graphs are generically irreducible or not. This work is divided into two parts. In the first part, we express the weighted Laplacian on Cayley graphs as the divergence of a gradient in an analogous way to the approach adopted in Riemannian geometry. In the second part, we analyze its spectrum on left-invariant Cayley graphs endowed with an invariant metric in both directed and undirected cases. We give some criteria for a given eigenspace being generically irreducible. Finally, we introduce an additional operator which is comparable to the Laplacian, and we verify that the same criteria hold.