Two positive solutions for a nonlinear four-point boundary value problem with a p-Laplacian operator

In this paper, we study the existence of positive solutions for a nonlinear four-point boundary value problem with a $p$-Laplacian operator. By using a three functionals fixed point theorem in a cone, the existence of double positive solutions for the nonlinear four-point boundary value problem with a $p$-Laplacian operator is obtained. This is different than previous results

Adler-Bardeen theorem and manifest anomaly cancellation to all orders in gauge theories

We reconsider the Adler-Bardeen theorem for the cancellation of gauge anomalies to all orders, when they vanish at one loop. Using the Batalin-Vilkovisky formalism and combining the dimensional-regularization technique with the higher-derivative gauge invariant regularization, we prove the theorem in the most general perturbatively unitary renormalizable gauge theories coupled to matter in four dimensions, and identify the subtraction scheme where anomaly cancellation to all orders is manifest, namely no subtractions of finite local counterterms are required from two loops onwards. Our approach is based on an order-by-order analysis of renormalization, and, differently from most derivations existing in the literature, does not make use of arguments based on the properties of the renormalization group. As a consequence, the proof we give also applies to conformal field theories and finite theories.Comment: 43 pages; EPJ

Kohn-Sham theory with paramagnetic currents: compatibility and functional differentiability

Recent work has established Moreau-Yosida regularization as a mathematical tool to achieve rigorous functional differentiability in density-functional theory. In this article, we extend this tool to paramagnetic current-density-functional theory, the most common density-functional framework for magnetic field effects. The extension includes a well-defined Kohn-Sham iteration scheme with a partial convergence result. To this end, we rely on a formulation of Moreau-Yosida regularization for reflexive and strictly convex function spaces. The optimal $L^p$-characterization of the paramagnetic current density $L^1\cap L^{3/2}$ is derived from the $N$-representability conditions. A crucial prerequisite for the convex formulation of paramagnetic current-density-functional theory, termed compatibility between function spaces for the particle density and the current density, is pointed out and analyzed. Several results about compatible function spaces are given, including their recursive construction. The regularized, exact functionals are calculated numerically for a Kohn-Sham iteration on a quantum ring, illustrating their performance for different regularization parameters