Communications in Contemporary Mathematics

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber–Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart.

We study the positive solutions of the Lane–Emden problem -Δ_pu = λ_p|u|^q-2u in Ω, u = 0 on ∂Ω, where Ω ⊂ ℝ^N is a bounded and smooth domain, N ≥ 2, λ_p is the first eigenvalue of the p-Laplacian operator Δ_p, p > 1, and q is close to p. We prove that any family of positive solutions of this problem converges in to the function θ_pe_p when q → p, where e_p is the positive and L^∞-normalized first eigenfunction of the p-Laplacian and . A consequence of this result is that the best constant of the immersion is differentiable at q = p. Previous results on the asymptotic behavior (as q → p) of the positive solutions of the nonresonant Lane–Emden problem (i.e. with λ_p replaced by a positive λ ≠ λ_p) are also generalized to the space and to arbitrary families of these solutions. Moreover, if u_λ,q denotes a solution of the nonresonant problem for an arbitrarily fixed λ > 0, we show how to obtain the first eigenpair of the p-Laplacian as the limit in , when q → p, of a suitable scaling of the pair (λ, u_λ,q). For computational purposes the advantage of this approach is that λ does not need to be close to λ_p. Finally, an explicit estimate involving L^∞- and L¹-norms of u_λ,q is also derived using set level techniques. It is applied to any ground state family {v_q} in order to produce an explicit upper bound for ‖v_q‖_∞ which is valid for q ∈ [1, p + ϵ] where .

This paper is devoted to the description of the lack of compactness of the Sobolev space in the Orlicz space . The approach that we adopt to establish this characterization is in the spirit of the one adopted in the case of into the Orlicz space in [H. Bahouri, M. Majdoub and N. Masmoudi, On the lack of compactness in the 2D critical Sobolev embedding, J. Funct. Anal. 260 (2011) 208–252].

If a non-ergodic, partially hyperbolic diffeomorphism on the 3-torus is homotopic to an Anosov diffeomorphism A, it is topologically conjugate to A.

The aim of this paper is to study the solvability of the following problem,

We study the following singular elliptic equation

In this paper, we give measure estimates of nodal sets of ℍ-harmonic functions on the Heisenberg group ℍⁿ. We also introduce a definition of horizontal singular sets and show the geometric structure of the horizontal singular sets of ℍ-harmonic functions.

Our results in this paper are threefold: First, we establish the modular properties of the graded dimensions of principal subspaces of level one standard modules for , and of principal subspaces of certain higher level standard modules for . Second, we establish the modular properties of families of q-series that appear in identities due to Warnaar and Zudilin, which generalize Macdonald's identities and the Rogers–Ramanujan identities. Third, we formulate a number of conjectures regarding the modularity of series of this type related to A_N-1 root systems.