Solutions for a Class of Fractional Laplacian Problems

We discuss the existence of positive solutions to nonlinear fractional Laplacian problems with Dirichlet external condition. We use degree theory combined with a re-scaling technique to show the existence of positive weak solutions for a class of superlinear problem when the bifurcation parameter is small

Nonnegative solutions of nonlinear fractional Laplacian equations

The study of reaction-diffusion equations involving nonlocal diffusion operators has recently flourished. The fractional Laplacian is an example of a nonlocal diffusion operator which allows long-range interactions in space, and it is therefore important from the application point of view. The fractional Laplacian operator plays a similar role in the study of nonlocal diffusion operators as the Laplacian operator does in the local case. Therefore, the goal of this dissertation is a systematic treatment of steady state reaction-diffusion problems involving the fractional Laplacian as the diffusion operator on a bounded domain and to investigate existence (and nonexistence) results with respect to a bifurcation parameter. In particular, we establish existence results for positive solutions depending on the behavior of a nonlinear reaction term near the origin and at infinity. We use topological degree theory as well as the method of sub- and supersolutions to prove our existence results. In addition, using a moving plane argument, we establish that, for a class of steady state reaction-diffusion problems involving the fractional Laplacian, any nonnegative nontrivial solution in a ball must be positive, and hence radially symmetric and radially decreasing. Finally, we provide numerical bifurcation diagrams and the profiles of numerical positive solutions, corresponding to theoretical results, using finite element methods in one and two dimensions

How does risk flow in the credit default swap market?

We develop a framework to analyse the credit default swap (CDS) market as a network of risk transfers among counterparties. From a theoretical perspective, we introduce the notion of flow-of-risk and provide sufficient conditions for a bow-tie network architecture to endogenously emerge as a result of intermediation. This architecture shows three distinct sets of counterparties: (i) Ultimate Risk Sellers (URS), (ii) Dealers (indirectly connected to each other), (iii) Ultimate Risk Buyers (URB). We show that the probability of widespread distress due to counterparty risk is higher in a bow-tie architecture than in more fragmented network structures. Empirically, we analyse a unique global dataset of bilateral CDS exposures on major sovereign and financial reference entities in 2011–2014. We find the presence of a bow-tie network architecture consistently across both reference entities and time, and that the flow-of-risk originates from a large number of URSs (e.g. hedge funds) and ends up in a few leading URBs, most of which are non-banks (in particular asset managers). Finally, the analysis of the CDS portfolio composition of the URBs shows a high level of concentration: in particular, the top URBs often show large exposures to potentially correlated reference entities

Nonnegative Solutions of Nonlinear Fractional Laplacian Equations

The study of reaction-diffusion equations involving nonlocal diffusion operators has recently flourished. The fractional Laplacian is an example of a nonlocal diffusion operator which allows long-range interactions in space, and it is therefore important from the application point of view. The fractional Laplacian operator plays a similar role in the study of nonlocal diffusion operators as the Laplacian operator does in the local case. Therefore, the goal of this dissertation is a systematic treatment of steady state reaction-diffusion problems involving the fractional Laplacian as the diffusion operator on a bounded domain and to investigate existence (and nonexistence) results with respect to a bifurcation parameter. In particular, we establish existence results for positive solutions depending on the behavior of a nonlinear reaction term near the origin and at infinity. We use topological degree theory as well as the method of sub- and supersolutions to prove our existence results. In addition, using a moving plane argument, we establish that, for a class of steady state reaction-diffusion problems involving the fractional Laplacian, any nonnegative nontrivial solution in a ball must be positive, and hence radially symmetric and radially decreasing. Finally, we provide numerical bifurcation diagrams and the profiles of numerical positive solutions, corresponding to theoretical results, using finite element methods in one and two dimensions

Kontinuum kladných řešení superlineární úlohy s frakcionálním laplaciánem

Je studována existence kontinua kladných slabých řešení pro úlohu s frakcionálním laplaciánem zahrnující superlineární reakční člen. Využíváme teorii stupně zpbrazení v kombinaci s metodou přeškálování bifurkačního parametru.We discuss the existence of a continuum of positive weak solutions to a fractional Laplacian problem involving superlinear reaction term when a bifurcation parameter is small. We employ degree theory combined with a re-scaling argument, boundary analysis, and a continuation theorem to obtain our result

Kontinuum kladných řešení superlineární úlohy s frakcionálním laplaciánem

Je studována existence kontinua kladných slabých řešení pro úlohu s frakcionálním laplaciánem zahrnující superlineární reakční člen. Využíváme teorii stupně zpbrazení v kombinaci s metodou přeškálování bifurkačního parametru.We discuss the existence of a continuum of positive weak solutions to a fractional Laplacian problem involving superlinear reaction term when a bifurcation parameter is small. We employ degree theory combined with a re-scaling argument, boundary analysis, and a continuation theorem to obtain our result

Existence kladných řešení pro rovnice s frakcionálním laplaciánem: teorie a numerické experimenty

Uvažujeme nelokální okrajové úlohy pro frakcionální laplacián se sublineárními nelinearitami a také nelinearitami logistického typu. Pro tyto úlohy dokazujeme existenci kladného řešení pomocí metody dolních a horních řešení. V článku se též zabýváme numerickými experimenty pomocí metody konečných prvků.We consider a class of nonlinear fractional Laplacian problems satisfying the homogeneous Dirichlet condition on the exterior of a bounded domain. We prove the existence of positive weak solution for classes of sublinear nonlinearities including logistic type. A method of sub- and supersolution, without monotone iteration, is established to prove our existence results. We also provide numerical bifurcation diagrams and the profile of positive solutions, corresponding to the theoretical results using the finite element method in one dimension