Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls

The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in {S^{n}}. In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and {\operatorname{SO}(n)}-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for {\operatorname{SO}(n)}-invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834].

On the horseshoe conjecture for maximal distance minimizers

We study the properties of sets Σ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality maxy∈M dist (y,Σ) ≤ r for a given compact set M ⊂ ℝ2 and some given r > 0. Such sets play the role of shortest possible pipelines arriving at a distance at most r to every point of M, where M is the set of customers of the pipeline. We describe the set of minimizers for M a circumference of radius R > 0 for the case when r < R ∕ 4 .98, thus proving the conjecture of Miranda, Paolini and Stepanov for this particular case. Moreover we show that when M is the boundary of a smooth convex set with minimal radius of curvature R, then every minimizer Σ has similar structure for r < R ∕ 5. Additionaly, we prove a similar statement for local minimizers.

Screening Properties of the Subbase of all Closed Connected Subsets of a Connectedly Generated Space

In [1] de Groot has introduced the notation “connectedly generated” (or eg) for those spaces in which the closed connected sets form a subbase for the topology. He pointed out that these are the semi-locally connected spaces of Whyburn. See [5; 6].If X is cg, then, since X is closed, X is the union of a finite number of closed connected sets and, thus, has only a finite number of components. If p is any point in a eg space, and Nv is any neighbourhood of p, then the complement of Nv may be covered by a finite number of closed connected sets, none of which contain p.In [1] and in [2] the concept of “screening” is introduced and shown to be usefully related to local connectedness and construction of compactifications for completely regular spaces. We review this concept in § 2.