Weak observability estimates for 1-D wave equations with rough coefficients

In this paper we prove observability estimates for 1-dimensional wave equations with non-Lipschitz coefficients. For coefficients in the Zygmund class we prove a "classical" observability estimate, which extends the well-known observability results in the energy space for $BV$ regularity. When the coefficients are instead log-Lipschitz or log-Zygmund, we prove observability estimates "with loss of derivatives": in order to estimate the total energy of the solutions, we need measurements on some higher order Sobolev norms at the boundary. This last result represents the intermediate step between the Lipschitz (or Zygmund) case, when observability estimates hold in the energy space, and the H\"older one, when they fail at any finite order (as proved in \cite{Castro-Z}) due to an infinite loss of derivatives. We also establish a sharp relation between the modulus of continuity of the coefficients and the loss of derivatives in the observability estimates. In particular, we will show that under any condition which is weaker than the log-Lipschitz one (not only H\"older, for instance), observability estimates fail in general, while in the intermediate instance between the Lipschitz and the log-Lipschitz ones they can hold only admitting a loss of a finite number of derivatives. This classification has an exact counterpart when considering also the second variation of the coefficients.Comment: submitte

Dispersive estimates with loss of derivatives via the heat semigroup and the wave operator

This paper aims to give a general (possibly compact or noncompact) analog of Strichartz inequalities with loss of derivatives, obtained by Burq, G\'erard, and Tzvetkov [19] and Staffilani and Tataru [51]. Moreover we present a new approach, relying only on the heat semigroup in order to understand the analytic connexion between the heat semigroup and the unitary Schr\"odinger group (both related to a same self-adjoint operator). One of the novelty is to forget the endpoint $L^1-L^\infty$ dispersive estimates and to look for a weaker $H^1-BMO$ estimates (Hardy and BMO spaces both adapted to the heat semigroup). This new point of view allows us to give a general framework (infinite metric spaces, Riemannian manifolds with rough metric, manifolds with boundary,...) where Strichartz inequalities with loss of derivatives can be reduced to microlocalized $L^2-L^2$ dispersive properties. We also use the link between the wave propagator and the unitary Schr\"odinger group to prove how short time dispersion for waves implies dispersion for the Schr\"odinger group.Comment: 48 page

On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment

We study the following modification of the Landau–Kolmogorov problem: Let k; r ∈ ℕ, 1 ≤ k ≤ r − 1, and p, q, s ∈ [1,∞]. Also let MM^m, m ∈ ℕ; be the class of nonnegative functions defined on the segment [0, 1] whose derivatives of orders 1, 2,…,m are nonnegative almost everywhere on [0, 1]. For every δ > 0, find the exact value of the quantity We determine the quantity in the case where s = ∞ and m ∈ {r, r − 1, r − 2}. In addition, we consider certain generalizations of the above-stated modification of the Landau–Kolmogorov problem.Дослiджується наступна модифiкацiя задачi Ландау – Колмогорова. Нехай k,r∈N,1≤k≤r−1, p,q,s∈[1,∞] i MM^m,m∈N, — клас невiд’ємних функцiй, що заданi на вiдрiзку [0,1] та мають майже скрiзь на [0,1] невiд’ємнi похiднi порядкiв 0,1,...,m. Для кожного δ>0 необхiдно знайти величину У данiй роботi величину знайдено у випадку s=∞ таm∈{r,r—1,r—2}. Також розглянуто деякi узагальнення вказаної модифiкацiї задачi Ландау – Колмогорова

Non-extendability of holomorphic functions with bounded or continuously extendable derivatives

We consider the spaces $H_{F}^{\infty}(\Omega)$ and $\mathcal{A}_{F}(\Omega)$ containing all holomorphic functions $f$ on an open set $\Omega \subseteq \mathbb{C}$, such that all derivatives $f^{(l)}$, $l\in F \subseteq \mathbb{N}_0=\{ 0,1,...\}$, are bounded on $\Omega$, or continuously extendable on $\overline{\Omega}$, respectively. We endow these spaces with their natural topologies and they become Fr\'echet spaces. We prove that the set $S$ of non-extendable functions in each of these spaces is either void, or dense and $G_\delta$. We give examples where $S=\varnothing$ or not. Furthermore, we examine cases where $F$ can be replaced by $\widetilde{F}=\{ l\in \mathbb{N}_0:\min F \leqslant l \leqslant \sup F\}$, or $\widetilde{F}_0= \{ l\in \mathbb{N}_0:0\leqslant l \leqslant \sup F\}$ and the corresponding spaces stay unchanged

The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities

We study the long term evolution of the distance between two Keplerian confocal trajectories in the framework of the averaged restricted 3-body problem. The bodies may represent the Sun, a solar system planet and an asteroid. The secular evolution of the orbital elements of the asteroid is computed by averaging the equations of motion over the mean anomalies of the asteroid and the planet. When an orbit crossing with the planet occurs the averaged equations become singular. However, it is possible to define piecewise differentiable solutions by extending the averaged vector field beyond the singularity from both sides of the orbit crossing set. In this paper we improve the previous results, concerning in particular the singularity extraction technique, and show that the extended vector fields are Lipschitz-continuous. Moreover, we consider the distance between the Keplerian trajectories of the small body and of the planet. Apart from exceptional cases, we can select a sign for this distance so that it becomes an analytic map of the orbital elements near to crossing configurations. We prove that the evolution of the 'signed' distance along the averaged vector field is more regular than that of the elements in a neighborhood of crossing times. A comparison between averaged and non-averaged evolutions and an application of these results are shown using orbits of near-Earth asteroids.Comment: 29 pages, 8 figure