American Institute of Mathematical Sciences

The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the fractional Hardy inequality

\begin{document}$\Lambda_{N}\equiv \Lambda_{N}(\Omega): = \inf\limits_{\{\varphi\in \mathbb{E}^{s}(\Omega, D), \varphi \ne0\}}\dfrac{\frac{a_{d, s}}{2}\displaystyle\int_{\mathbb R^d}\int_{\mathbb R^d}\dfrac{|\varphi(x)-\phi(y)|^{2}}{|x-y|^{d+2s}}dx dy}{\displaystyle\int_{\Omega}\frac{\varphi^2}{|x|^{2s}}\, dx}, $ \end{document}

where \begin{document}$\Omega$\end{document} is a bounded domain of \begin{document}$\mathbb R^d$\end{document}, \begin{document}$0<s<1$\end{document}, \begin{document}$D\subset \mathbb R^d\setminus \Omega$\end{document} a nonempty open set, \begin{document}$N = (\mathbb R^d\setminus \Omega)\setminus\overline{D}$\end{document} and

\begin{document}$\mathbb{E}^{s}(\Omega, D) = \{ u \in H^s(\mathbb R^d):\, u = 0 \text{ in } D\}.$ \end{document}

The second aim of the paper is to study the mixed Dirichlet-Neumann boundary problem associated to the minimization problem and related properties; precisely, to study semilinear elliptic problem for the fractional Laplacian, that is,

\begin{document}${P_\lambda } \equiv \left\{ {\begin{array}{*{20}{l}}{{{\left( { - \Delta } \right)}^s}u\;\;\; = \;\;\;\lambda \frac{u}{{|x{|^{2s}}}} + {u^p}}&{{\rm{in}}\;\Omega ,}\\{\;\;\;\;\;\;\;\;\;u\;\;\; > \;\;\;0}&{{\rm{in}}\;\Omega ,}\\{\;\;\;\;\;\;{{\cal B}_s}u\;\;\;: = \;\;u{\chi _D} + {{\cal N}_s}u{\chi _N} = 0}&{{\rm{in}}\;{{\mathbb {R}}^d}\backslash \Omega ,}\end{array}} \right.$\end{document}

with \begin{document}$N$\end{document} and \begin{document}$D$\end{document} open sets in \begin{document}$\mathbb R^{d}\backslash\Omega$\end{document} such that \begin{document}$N \cap D = \emptyset$\end{document} and \begin{document}$\overline{N}\cup \overline{D} = \mathbb R^{d}\backslash\Omega$\end{document}, \begin{document}$d>2s$\end{document}, \begin{document}$\lambda> 0$\end{document} and \begin{document}$<p\le 2_s^*-1_s^* = \frac{2d}{d-2s}$\end{document}. We emphasize that the nonlinear term can be critical.

The operators \begin{document}$(-\Delta)^s $\end{document}, fractional Laplacian, and \begin{document}$\mathcal{N}_{s}$\end{document}, nonlocal Neumann condition, are defined below in (7) and (8) respectively.

We construct symplectomorphisms in dimension d ≥ 4 having a semi-local robustly transitive partially hyperbolic set containing C²-robust homoclinic tangencies of any codimension $c$ with 0 < c ≤ d/2.

We study the two membranes problem for two different fully nonlinear operators. We give a viscosity formulation for the problem and prove existence of solutions. Then we prove a general regularity result and the optimal \begin{document} $C^{1, 1}$ \end{document} regularity when the operators are the Pucci extremal operators. We also give an example that shows that no regularity for the free boundary is to be expected to hold in general.

We study the graph property for Lagrangian minimizing submanifolds of the geodesic flow of a Riemannian metric in the torus \begin{document}$ (T^{n},g) $\end{document}, \begin{document}$ n>2 $\end{document}. It is well known that the transitivity of the geodesic flow in a minimizing Lagrangian submanifold implies the graph property. We replace the transitivity by three kind of assumptions: (1) \begin{document}$ r $\end{document}-density of the set of recurrent orbits for some \begin{document}$ r>0 $\end{document} depending on \begin{document}$ g $\end{document}, (2) \begin{document}$ r $\end{document}-density of the limit set, (3) every point is nonwandering. Then we show that a Lagrangian, minimizing torus satisfying one of such assumptions is a graph.

We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete computation of several scattering maps. A complete description of the different kinds of scattering maps taking place as well as the existence of piecewise smooth global scattering maps is also provided.

We provide an integral estimate for a non-divergence (non-varia-tional) form second order elliptic equation \begin{document}$a_{ij}u_{ij} = u^p$\end{document}, \begin{document}$u≥ 0$, $p∈[0, 1)$\end{document}, with bounded discontinuous coefficients \begin{document}$a_{ij}$\end{document} having small BMO norm. We consider the simplest discontinuity of the form \begin{document}$x\otimes x|x|^{-2}$\end{document} at the origin. As an application we show that the free boundary corresponding to the obstacle problem (i.e. when \begin{document}$p = 0$\end{document}) cannot be smooth at the points of discontinuity of \begin{document}$a_{ij}(x)$\end{document}.

To implement our construction, an integral estimate and a scale invariance will provide the homogeneity of the blow-up sequences, which then can be classified using ODE arguments.

We study metastable motions in weakly damped Hamiltonian systems. These are believed to inhibit the transport of energy through Hamiltonian, or nearly Hamiltonian, systems with many degrees of freedom. We investigate this question in a very simple model in which the breather solutions that are thought to be responsible for the metastable states can be computed perturbatively to an arbitrary order. Then, using a modulation hypothesis, we derive estimates for the rate at which the system drifts along this manifold of periodic orbits and verify the optimality of our estimates numerically.

Hénon map is a well-studied classical example of area-contracting maps, modelling dissipative dynamics. The rich phenomena of coexistence of stable islands and their separatrices is typical of area-preserving maps, modelling conservative dynamics. In this paper we use the Hénon map to ascertain that coexistence of sinks is greater and greater approaching the conservative case, and that part of it can be organized following a renormalization argument. Using a numerical continuation that we devised, and called "dribbling method" [5], one can follow bifurcation paths from the area-preserving case into the dissipative one, organizing families of coexisting attractive periodic orbits with diverging period. When the dissipation parameter goes to zero, we will give numerical evidence of the increasing coexistence of such periodic orbits, in the coordinate and parameter space values. Vanishing dissipation and diverging period constitute a double limit that we study as such, giving evidence of a singularity in the limit. The families we study all appear as homoclinic bifurcation, and the fixed point causing the homoclinic onset also structures the renormalization scheme. One of the goals of this paper is to improve the results obtained by looking to higher periods, and to approach dissipation down to an area-contraction factor of \begin{document}$1- 10^{-8}$\end{document}. Using the same dribbling method, as further promising application, we also deal with the dissipative Standard map.

We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach relies on the method of 'correctly aligned windows'. We subdivide 'windows' into cubical complexes, and we assign to the vertices of the cubes labels determined by the dynamics. In this way, we encode the information on the dynamics into combinatorial structure. We use a version of Sperner's Lemma to infer that, if the labeling satisfies certain conditions, then there exist fixed points/periodic orbits/orbits with prescribed itineraries. The method developed here does not require the computation of algebraic topology-type invariants, as only combinatorial information is needed; our arguments are elementary.

We deal with a problem of target control synthesis for dynamical bilinear discrete-time systems under uncertainties (which describe disturbances, perturbations or unmodelled dynamics) and state constraints. Namely we consider systems with controls that appear not only additively in the right hand sides of the system equations but also in the coefficients of the system. We assume that there are uncertainties of a set-membership kind when we know only the bounding sets of the unknown terms. We presume that we have uncertain terms of two kinds, namely, a parallelotope-bounded additive uncertain term and interval-bounded uncertainties in the coefficients. Moreover the systems are considered under constraints on the state ("under viability constraints"). We continue to develop the method of control synthesis using polyhedral (parallelotope-valued) solvability tubes. The technique for calculation of the mentioned polyhedral tubes by the recurrent relations is presented. Control strategies, which can be constructed on the base of the polyhedral solvability tubes, are proposed. Illustrative examples are considered.

This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type Ⅱ, important for its application to random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type Ⅱ in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.

This article addresses the regularity issue for minimizing fractional harmonic maps of order s∈(0, 1/2) from an interval into a smooth manifold. Hölder continuity away from a locally finite set is established for a general target. If the target is the standard sphere, then Hölder continuity holds everywhere.

Dynamical systems appear in many models in all sciences and in technology. They can be either discrete or continuous, finite or infinite dimensional, deterministic or with random terms.

Many theoretical results, the related algorithms and implementations for careful simulations and a wide range of applications have been obtained up to now. But still many key questions remain open. They are mainly related either to global aspects of the dynamics or to the lack of a sufficiently good agreement between qualitative and quantitative results.

In these notes a sample of questions, for which the author is not aware of the existence of a good solution, are presented. Of course, it is easy to largely extend the list.

We prove the existence of quasi-periodic small-amplitude solutions for quasi-linear Hamiltonian perturbation of the fifth order KdV equation on the torus in presence of a quasi-periodic forcing.

We consider the nonlinear elliptic PDE driven by the fractional Laplacian with asymptotically linear term. Some results regarding existence and multiplicity of non-trivial solutions are obtained. More precisely, information about multiple non-trivial solutions is given under some hypotheses of asymptotically linear condition; non-local elliptic equations with combined nonlinearities are also studied, and some results of local existence and global existence are obtained. Finally, an \begin{document}$L^{∞}$\end{document} regularity result is also given in the appendix, using the De Giorgi-Stampacchia iteration method.

Existence of trajectory, global and pullback attractors for an incompressible non-Newtonian fluid (namely, for the mathematical model which describes a weak aqueous polymer solutions motion) in 2D and 3D bounded domains is studied in this paper. For this aim the approximating topological method is effectively combined with the theory of attractors of trajectory spaces.

In the present paper we establish the existence of weak solutions to one fractional Voigt type model of viscoelastic fluid. This model takes into account a memory along the motion trajectories. The investigation is based on the theory of regular Lagrangean flows, approximation of the problem under consideration by a sequence of regularized Navier-Stokes systems and the following passage to the limit.