Programa – XIV ENAMA

The problem of prescribing the Gaussian curvature on compact surfaces is a classic one, and dates back to the works of Berger, Moser, Kazdan and Warner, etc.
The case of the sphere receives the name of Nirenberg problem and has deserved a lot of attention in the literature. In the first part of the talk we will review the known results about compactness and existence of solutions to that problem.

If the domain has a boundary, the most natural question is to prescribe also the geodesic curvature h(x) of the boundary. This problem reduces to solve a semilinear elliptic PDE under a nonlinear Neumann boundary condition. In this talk we will focus in the case of the standard disk.

First we perform a blow-up analysis for the solutions of this equation. We will show that, if a sequence of solutions blow-up, it tends to concentrate around a unique point in the boundary of the disk. We are able to give conditions on such point that, quite interestingly, depend on h(x) in a nonlocal way. This is joint work with A. Jevnikar, R. López-Soriano and M. Medina.

Secondly, we will give existence results. We will show how the blow-up analysis developed before can be used to compute the Leray-Schauder degree associated to the problem.

Utilizando a teoria da bifurcação, estudamos um problema não-local envolvendo operadores de dispersão que surgem em modelos logístico-populacionais.

In 1961 Bishop and Phelps proved that for any Banach space the set of continuous and linear functionals attaining their norms is norm dense in the topological dual. In 1970 Bollobás obtained a quantitative version of this result showing that a pair (x, x*) given by an element x in the unit sphere of a Banach space and a functional x in the unit sphere of the dual space such that x*(x) is close to 1 can be approximated in norm by another pair (y, y*) satisfying the same conditions and also that y* attains its norm at y. Bishop and Phelps posed the problem of extending their result to operators. Let X and Y Banach spaces. We say that the pair (X, Y ) has the Bishop-Phelps property if the set of norm attaining operators from X to Y is dense in the L(X,Y), the space of all bounded linear operators between Banach spaces X and Y, endowed with the usual operator norm. In 1963 Lindenstrauss exhibited a counterexample showing that in general the set of norm attaining operators is not dense in L(X,Y). He also provided that if X is a reflexive Banach space then the set of norm attaining operators from X to Y is dense in L(X, Y ), for any Banach space Y. Afterwards several authors proved interesting results and there are also many open problems in the subject. In the paper :[1] The Bishop-Phelps-Bollobás theorem for operators, J. Funct. Anal. 254 (2008), 2780–2799, Acosta, Aron, García and Maestre introduced a version of Bollobás result called Bishop-Phelps-Bollobás property for operators (BPBp in short). As we already mentioned, there are Banach spaces X and Y such that the pair (X, Y ) does not have the Bishop-Phelps property. In such cases the pair (X, Y) does not have the BPBp. In general these two properties are very different. After the paper [1] many results have been obtained in this topic. Let us say only that there are some geometric properties both on the domain and on the range implying the BPBp. For instance, a certain geometrical property on Y, called property \beta of Lindenstrauss, implies that the pair (X, Y ) has the Bishop-Phelps-Bollobás property for operators for any Banach space X. If a Banach space X is uniformly convex, then the pair (X, Y ) has the Bishop-Phelps-Bollob\’as property for operators for any Banach space Y. For example the pair (l_1, Y) for any Y Banach space has Bishop-Phelps property, however, in general the pair (l_1, Y ) does not have the BPBp. In the paper [1] the authors obtained a characterization of the Banach spaces Y such that (l_1, Y ) satisfies the BPBp. The property appearing in this characterization is quite technical and it is called the approximate hyperplane property. In this talk we are going to present some results obtained in the last 12 years in the context of Bishop-Phelps-Bollobás property for operators.