Communications in Contemporary Mathematics

In this paper, we study the existence of sign-changing solutions for the p-Laplacian equation

In this paper we will use variational methods and limiting approaches to give a complete solution to the maximization problems of the Dirichlet/Neumann eigenvalues of the one-dimensional p-Laplacian with integrable potentials of fixed L¹-norm.

In this paper we are dealing with a Schrödinger–Maxwell system in a bounded domain of R³; the unknowns are the charged standing waves ψ = e^-iωtu(x) in equilibrium with a purely electrostatic potential ϕ. The system is not autonomous, in the sense that the coupling depends on a function q = q(x). The non-homogeneous Neumann boundary condition on ϕ prescribes the flux of the electric field 𝔉 and gives rise to a necessary condition. On the other hand we consider the usual normalizing condition in L² for u. Under mild assumptions involving 𝔉 and the function q = q(x), we prove that this problem has a variational framework: its solutions can be characterized as constrained critical points. Then, by means of the Ljusternick–Schnirelmann theory, we get the existence of infinitely many solutions.

We use knot Floer surgery exact sequences and torsion invariants to compute the knot Floer homology of certain fibered knots in the double cover of S³ branched along the closure of an alternating braid.

When the vacuum Einstein equations are cast in the form of Hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold of Riemannian metrics on a Cauchy hypersurface Σ. As in every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection. In an appendix, we develop some aspects of diffeology, the basic framework for our treatment of function spaces.

We define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed symplectic hypersurface V in a symplectic 4-manifold (X, ω) at prescribed points with prescribed contact orders (in addition to insertions on X\V). We obtain invariants of the deformation class of (X, V, ω). Two large issues must be tackled to define such invariants: (1) curves lying in the hypersurface V and (2) genericity results for almost complex structures constrained to make V pseudo-holomorphic (or almost complex). Moreover, these invariants are refined to take into account rim-tori decompositions. In the latter part of the paper, we extend the definition to disconnected submanifolds and construct relative Gromov–Taubes invariants.