Search results for "hyperbolic"

On the critical curve for systems of hyperbolic inequalities in an exterior domain of the half-space

2023

We establish blow-up results for a system of semilinear hyperbolic inequalities in an exterior domain of the half-space. The considered system is investigated under an inhomogeneous Dirichlet-type boundary condition depending on both time and space variables. In certain cases, an optimal criterium of Fujita-type is derived. Our results yield naturally sharp nonexistence criteria for the corresponding stationary wave system and equation.

Mean-field games and two-point boundary value problems

2014

A large population of agents seeking to regulate their state to values characterized by a low density is considered. The problem is posed as a mean-field game, for which solutions depend on two partial differential equations, namely the Hamilton-Jacobi-Bellman equation and the Fokker-Plank-Kolmogorov equation. The case in which the distribution of agents is a sum of polynomials and the value function is quadratic is considered. It is shown that a set of ordinary differential equations, with two-point boundary value conditions, can be solved in place of the more complicated partial differential equations associated with the problem. The theory is illustrated by a numerical example.

Global integrability of the gradients of solutions to partial differential equations

1994

Annihilation Operators for Exponential Spaces in Subdivision

2022

We investigate properties of differential and difference operators annihilating certain finite-dimensional subspaces of exponential functions in two variables that are connected to the representation of real-valued trigonometric and hyperbolic functions. Although exponential functions appear in a variety of contexts, the motivation behind this work comes from considering subdivision schemes with the capability of preserving those exponential functions required for an exact description of surfaces parametrized in terms of trigonometric and hyperbolic functions.

Appendix: Diophantine Approximation on Hyperbolic Surfaces

2002

In this (independent) appendix, we study the Diophantine approximation properties for the particular case of the cusped hyperbolic surfaces, in the spirit of Sect. 2 (or [11]), and the many still open questions that arise for them. We refer to [9], [10]for fundamental results and further developments. We study in particular the distance to a cusp of closed geodesics on a hyperbolic surface.

On bounds for total absolute curvature of surfaces in hyperbolic 3-space

2003

Abstract We construct examples of surfaces in hyperbolic space which do not satisfy the Chern–Lashof inequality (which holds for immersed surfaces in Euclidean space). To cite this article: R. Langevin, G. Solanes, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Approximation von extremalflächenstücken (hyperbolischen typs) durch charakteristische räumliche vierecke

1982

We consider solutions z of the Cauchy-problem for hyperbolic Euler-Lagrange equations derived from a general Lagrangian f(x, y, z; zx, zy) in two independent variables x, y. z is supposed to be an extremal of the corresponding variational problem. Visualizing z as a surface in R3 we give a geometric interpretation of Lewy's well-known characteristic approximation scheme for the numerical solution of second order hyperbolic equations by approximating z via a polyhedral construction built up from subunits which consist of two characteristic triangles having one side in common but lying on different planes in R3. Utilizing ideas from Cartan-geometry one can (in an appropriate sense) introduce …

Seifert manifolds admitting partially hyperbolic diffeomorphisms

2017

We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow.

Non-wandering sets with non-empty interiors

2003

We study diffeomorphisms of a closed connected manifold whose non-wandering set has a non-empty interior and conjecture that C1-generic diffeomorphisms whose non-wandering set has a non-empty interior are transitive. We prove this conjecture in three cases: hyperbolic diffeomorphisms, partially hyperbolic diffeomorphisms with two hyperbolic bundles, and tame diffeomorphisms (in the first case, the conjecture is folklore; in the second one, it follows by adapting the proof in Brin (1975 Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature Funct. Anal. Appl. 9 9–19)).We study this conjecture without global assumptions and pro…

Robust existence of nonhyperbolic ergodic measures with positive entropy and full support

2021

We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic measures which are ergodic, fully supported and have positive entropy. To do so, we formulate abstract conditions sufficient for the construction of an ergodic, fully supported measure $\mu$ which has positive entropy and is such that for a continuous function $\phi\colon X\to\mathbb{R}$ the integral $\int\phi\,d\mu$ vanishes. The criterion is an extended version of the control at any scale with a long and sparse tail technique coming from the previous w…