No abstract

A systematic and rigorous treatment of the massless scalar field in two dimensions is presented by carefully taking into account the maximal (Krein) state space associated to the Wightman functions. This allows a simple and rigorous answer to controversial statements appearing in the literature about the uniqueness of the translation-invariant state, the construction of translation-invariant operators (infrared operators), the scale and special conformal transformations of the fields, the construction of the dual field and the breaking of the Lorentz transformations, and, more generally, the status of symmetry breaking in the model.

Given ρ > 0, we study the elliptic problemwhere Ω ⊂ R N is a bounded domain and p > 1 is Sobolev-subcritical, searching for conditions (about ρ, N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is L 2 -subcritical, i.e. 1 < p ≤ 1 + 4/N , the problem admits solution for every ρ > 0. In the L 2 -critical and supercritical case, i.e. when 1 + 4/N ≤ p < 2 * − 1, we show that, for any k ∈ N, the problem admits solutions having Morse index bounded above by k only if ρ is sufficiently small. Next we provide existence results for certain ranges of ρ, which can be estimated in terms of the Dirichlet eigenvalues of −∆ in H 1 0 (Ω), extending to general domains and to changing sign solutions some results obtained in [21] for positive solutions in the ball.

A new general treatment of fermion bosonization in two-dimensional quantum field theory is presented. The main feature is that the bosonization is completely achieved in terms of a single massless scalar field in two dimensions without the introduction of ad hoc selection rules and/or spurion operators. This is made possible by the exploitation of the infrared operators associated to the algebra of the massless scalar field. Fermionic charges, chiral charges, fermionic symmetries, θ angles, etc. are shown to follow naturally from charges and symmetries of the scalar field. As an application, the massless Thirring model is discussed.

In this paper, we provide a representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction in a simplified monodomain model describing the electrical activity of the heart. We derive such a result in the case of a nonlinear problem. Our long-term goal is the solution of the inverse problem related to the detection of regions affected by heart ischemic disease, whose position and size are unknown. We model the presence of ischemic regions in the form of small inhomogeneities. This leads to the study of a boundary value problem for a semilinear elliptic equation. We first analyze the well-posedness of the problem establishing some key energy estimates. These allow us to derive rigorously an asymptotic formula of the boundary potential perturbation due to the presence of the inhomogeneities, following an approach similar to the one introduced by Capdeboscq and Vogelius in [A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, Math. Model. Numer. Anal. 37 (2003) 159–173] in the case of the linear conductivity equation. Finally, we propose some ideas of the reconstruction procedure that might be used to detect the inhomogeneities.