59 research outputs found
Construction of a solution for the two-component radial Gross-Pitaevskii system with a large coupling parameter
We consider strongly coupled competitive elliptic systems that arise in the
study of two-component Bose-Einstein condensates. As the coupling parameter
tends to infinity, solutions that remain uniformly bounded are known to
converge to a segregated limiting profile, with the difference of its
components satisfying a limit scalar PDE. In the case of radial symmetry, under
natural non-degeneracy assumptions on a solution of the limit problem, we
establish by a perturbation argument its persistence as a solution to the
elliptic system
On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space
In this paper, we study nonlinear Helmholtz equations (NLH)
in , where
denotes the Laplace-Beltrami operator in the hyperbolic space
and is chosen suitably. Using fixed point
and variational techniques, we find nontrivial solutions to (NLH) for all
and . The oscillatory behaviour and decay rates of radial
solutions is analyzed, with possible extensions to Cartan-Hadamard manifolds
and Damek-Ricci spaces. Our results rely on a new Limiting Absorption Principle
for the Helmholtz operator in . As a byproduct, we obtain simple
counterexamples to certain Strichartz estimates
Converse problem for the two-component radial Gross-Pitaevskii system with a large coupling parameter
summary:We consider strongly coupled competitive elliptic systems that arise in the study of two-component Bose-Einstein condensates. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. In the case of radial symmetry, under natural non-degeneracy assumptions on a solution of the limit problem, we establish by a perturbation argument its persistence as a solution to the elliptic system
Hidden dissipation and convexity for Kimura equations
In this paper we establish a rigorous gradient flow structure for
one-dimensional Kimura equations with respect to some Wasserstein-Shahshahani
optimal transport geometry. This is achieved by first conditioning the
underlying stochastic process to non-fixation in order to get rid of
singularities on the boundaries, and then studying the conditioned -process
from a more traditional and variational point of view. In doing so we complete
the work initiated in [Chalub et Al., Gradient flow formulations of discrete
and continuous evolutionary models: a unifying perspective. Acta App Math.,
171(1), 1-50], where the gradient flow was identified only formally. The
approach is based on the Energy Dissipation Inequality and Evolution
Variational Inequality notions of metric gradient flows. Building up on some
convexity of the driving entropy functional, we obtain new contraction
estimates and quantitative long-time convergence towards the stationary
distribution
Trivariate distribution of sticky Brownian motion
In this short note we derive a closed form for the trivariate distribution
(position, local time at the origin, and positive occupation time) of the
one-dimensional sticky Brownian motion, thereby filling some gaps and fixing
some mistakes in the literature
Singular radial solutions for the Lin-Ni-Takagi equation
We study singular radially symmetric solution to the Lin-Ni-Takagi equation for a supercritical power non-linearity in dimension N >= 3. It is shown that for any ball and any k >= 0, there is a singular solution that satisfies Neumann boundary condition and oscillates at leastktimes around the constant equilibrium. Moreover, we show that the Morse index of the singular solution is finite or infinite if the exponent is respectively larger or smaller than the Joseph-Lundgren exponent.Peer reviewe
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