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We prove that the initial value problem with small data for the asymptotically flat spherically symmetric Einstein-Vlasov-Maxwell system admits the global in time solution in the case of the non zero shift vector. This result extends the one already known for chargeless case.
In this paper, the first Chen inequality is proved for general warped product submanifolds in Riemannian space forms, this inequality involves intrinsic invariants (δ-invariant and sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer for Chen’s Problem 1 relating to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold. As a geometric application, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in Riemannian space forms, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2. For further research directions, we address a couple of open problems ; namely Problem 3 and Problem 4.
Let $$$1\leq q\leq\alpha < \infty. \left\{(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d):\alpha\leq p\leq\infty \right\}$$$ is a nondecreasing family of Banach spaces such that the Lebesgue space is $$$L^{\alpha}(\mathbb{R}^d)$$$ its minimal element and the classical Morrey space $$$\mathcal{M}_{q}^{\alpha}(\mathbb{R}^d)$$$ is its maximal element. In this note we investigate some closed linear subspaces of $$$(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$$$. We give a characterization of the closure in $$$(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$$$ of the set of all its compactly supported elements and study the action of some classical operators on it. We also describe the closure in $$$(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$$$ of the set $$$\mathcal{C}_{\rm{c}}^{\infty}(\mathbb{R}^d)$$$ of all infinitely differentiable and compactly supported functions on $$$\mathbb{R}^{d}$$$ as an intersection of other linear subspaces of $$$(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$$$ and obtain the weak density of $$$\mathcal{C}_{\rm{c}}^{\infty}(\mathbb{R}^d)$$$ in some of these subspaces. We establish a necessary condition on a function $$$f$$$ in order that its Riesz potential $$$I_{\gamma}(|f|) \;(0<\gamma<1)$$$ be in a given Lebesgue space.
2024
Volume 24- 15
Numéro 1 (Janvier 2024)2023
Volume 23- 14
Numéro 1 (Janvier 2023)2022
Volume 22- 13
Numéro 1 (Janvier 2022)2021
Volume 21- 12
Numéro spécial : AUS-ICMS 20202020
Volume 20- 11
Numéro 1 (Mai 2020)