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In this article, we prove a Lichnerowicz estimate for a compact convex domain of a Kähler manifold whose Ricci curvature satisfies Ric for some constant . When equality is achieved, the boundary of the domain is totally geodesic and there exists a nontrivial holomorphic vector field.
We show that a ball of sufficiently large radius in complex projective space provides an example of a strongly pseudoconvex domain which is not convex, and for which the Lichnerowicz estimate fails.
We derive sharp estimates on the modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time. As an application, we give an easy proof of the optimal lower bound on the first eigenvalue of the Laplacian on such a manifold as a function of diameter.
We study the asymptotic behavior of the eigenvalues of the Laplace–Beltrami operator on a compact hypersurface in as it is flattened into a singular double-sided flat hypersurface. We show that the limit spectral problem corresponds to the Dirichlet and Neumann problems on one side of this flat (Euclidean) limit, and derive an explicit three-term asymptotic expansion for the eigenvalues where the remaining two terms are of orders and .
In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity due to Robbiano, Zuily, Tataru and Hörmander. Some other consequences are also given for the controllability and the existence of a compact attractor.
Dans cet article, on prouve la décroissance exponentielle de l’équation des ondes semilinéaires avec un amortissement actif dans une zone satisfaisant seulement la condition de contrôle géométrique. La nonlinéarité est supposée sous-critique, défocalisante et analytique. La principale nouveauté par rapport aux résultats précédents est la preuve d’un résultat de prolongement unique en grand temps pour une solution non amortie. L’idée est d’utiliser un effet régularisant asymptotique prouvé par Hale et Raugel dans le contexte des systèmes dynamiques. Ensuite, une fois l’analyticité en temps prouvée, on applique un théorème de prolongement unique avec analyticité partielle dû à Robbiano, Zuily, Tataru et Hörmander. Des applications à la contrôlabilité et à l’existence d’attracteur global compact pour l’équation des ondes sont aussi données.
The stability question of the Lane–Emden stationary gaseous star configurations is an interesting problem arising in astrophysics. We establish both linear and nonlinear dynamical instability results for the Lane–Emden solutions in the framework of the Navier–Stokes–Poisson system with adiabatic exponent .
We study the long-time behavior of the 3-dimensional repulsive nonlinear Hartree equation with an external attractive Coulomb potential , which is a nonlinear model for the quantum dynamics of an atom. We show that, after a sufficiently long time, the average number of electrons in any finite ball is always smaller than ( in the radial case). This is a time-dependent generalization of a celebrated result by E.H. Lieb on the maximum negative ionization of atoms in the stationary case. Our proof involves a novel positive commutator argument (based on the cubic weight ) and our findings are reminiscent of the RAGE theorem.
In addition, we prove a similar universal bound on the local kinetic energy. In particular, our main result means that, in a weak sense, any solution is attracted to a bounded set in the energy space, whatever the size of the initial datum. Moreover, we extend our main result to Hartree–Fock theory and to the linear many-body Schrödinger equation for atoms.
We establish a connection between the stability of an eigenvalue under a magnetic perturbation and the number of zeros of the corresponding eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a graph and count the number of edges where the eigenfunction changes sign (has a “zero”). It is known that the -th eigenfunction has such zeros, where the “nodal surplus” is an integer between 0 and the first Betti number of the graph.
We then perturb the Laplacian with a weak magnetic field and view the -th eigenvalue as a function of the perturbation. It is shown that this function has a critical point at the zero field and that the Morse index of the critical point is equal to the nodal surplus of the -th eigenfunction of the unperturbed graph.
We present a natural proof of a recent and surprising result of Gregory Berkolaiko interpreting the Courant nodal defect as a Morse index. This proof is inspired by a nice paper of Miroslav Fiedler published in 1975.