Positive ground state of coupled systems of Schrödinger equations in involving critical exponential growth

In this paper, we study the existence of a positive ground state solution to the following coupled system of nonlinear Schrödinger equations: where the nonlinearities f₁(x,s) and f₂(x,s) are superlinear at infinity and have exponential critical growth of the Trudinger‐Moser type. The potentials V₁(x) and V₂(x) are nonnegative and satisfy a condition involving the coupling term λ(x), namely, λ(x)²<δ²V₁(x)V₂(x) for some 0<δ<1. For this purpose, we use the minimization technique over the Nehari manifold and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap argument and L^q‐estimates, we get regularity and asymptotic behavior.     

Optimal <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Riemannian Gagliardo–Nirenberg inequalities

Let (M, g) be a compact Riemannian manifold of dimension n ≥  2 and 1 <  p ≤  2. In this work we prove the validity of the optimal Gagliardo–Nirenberg inequality

L 2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature

Let $x:M^{m}\to\bar{M}$ , m≥3, be an isometric immersion of a complete noncompact manifold M in a complete simply connected manifold $\bar{M}$ with sectional curvature satisfying $-k^{2}\leq K_{\bar{M}}\leq0$ , for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L ^m -norm. If $K_{\bar{M}}\not\equiv0$ , assume further that the first eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L ² harmonic 1-forms on M has finite dimension. Moreover, there exists a constant Λ>0, explicitly computed, such that if the total curvature is bounded from above by Λ then there are no nontrivial L ²-harmonic 1-forms on M.     

Building the Additive Utility Functions for CAD-UFRJ Evaluation Staff Criteria

This paper presents an application of the UTA method for building utility functions for the evaluation criteria defined by the Staff Evaluation Commission (CAD) of the Rio de Janeiro Federal University (UFRJ). Every year, the CAD-UFRJ gives the staff evaluation results for each Postgraduate Engineering Programme. However, the method used to generate the staff evaluation is assumed unknown. Trying to find the CAD-UFRJ preference structure, the evaluation results supplied by CAD-UFRJ are used to apply the UTA method. Some additional information obtained from the CAD-UFRJ data is incorporated in the optimal solutions analysis.     

Affine Properties of Convex Equal-Area Polygons

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygons admit natural definitions of the usual affine differential geometry concepts, like affine normal and affine curvature. These definitions lead to discrete analogous to the six-vertex theorem and an affine isoperimetric inequality. One can also define discrete counterparts of the affine evolute, parallels and the affine distance symmetry set preserving many of the properties valid for smooth curves.     

Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones

We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily subanalytic) between two subanalytic sets, then their tangent cones are bi-Lipschitz homeomorphic. As a consequence of this result, we show that any Lipschitz regular complex analytic set, i.e., any complex analytic set which is locally bi-Lipschitz homeomorphic to an Euclidean ball must be smooth. Finally, we give an alternative proof of S. Koike and L. Paunescu’s result about the bi-Lipschitz invariance of directional dimensions of subanalytic sets.     

Existence and concentration of solutions for a class of biharmonic equations

Some superlinear fourth order elliptic equations are considered. A family of solutions is proved to exist and to concentrate at a point in the limit. The proof relies on variational methods and makes use of a weak version of the Ambrosetti–Rabinowitz condition. The existence and concentration of solutions are related to a suitable truncated equation.     

Bessel Potentials in Ahlfors Regular Metric Spaces

In this paper we introduce Bessel potentials and the Sobolev potential spaces resulting from them in the context of Ahlfors regular metric spaces. The Bessel kernel is defined using a Coifman type approximation of the identity, and we show integration against it improves the regularity of Lipschitz, Besov and Sobolev-type functions. For potential spaces, we prove density of Lipschitz functions, and several embedding results, including Sobolev-type embedding theorems. Finally, using singular integrals techniques such as the T1 theorem, we find that for small orders of regularity Bessel potentials are inversible, its inverse in terms of the fractional derivative, and show a way to characterize potential spaces, concluding that a function belongs to the Sobolev potential space if and only if itself and its fractional derivative are in L^p. Moreover, this characterization allows us to prove these spaces in fact coincide with the classical potential Sobolev spaces in the Euclidean case.     

Applications of stratifying systems to the finitistic dimension

Given an Ext-injective stratifying system of Λ-modules satisfying that the projective dimension of Y is finite, we prove that the finitistic dimension of the algebra Λ is equal to the finitistic dimension of the category . Moreover, using the theory of stratifying systems we obtain bounds for the finitistic dimension of Λ. In particular, we get the optimal bound 2n-2 for the finitistic dimension of a standardly stratified algebra with n simples.