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Abstract: Abstract A result of Chernoff gives sufficient condition for an \(L^2\) -function on \({\mathbb { R}}^n\) to be quasi-analytic, in the sense that the function and all its derivatives cannot vanish at a point. This is a generalization of the classical Denjoy–Carleman theorem on \({\mathbb { R}}\) and of the subsequent works on \({\mathbb { R}}^n\) by Bochner and Taylor. In this note we endeavour to obtain an exact analogue of the result of Chernoff for \(L^p, p\in [1,2]\) functions on the Riemannian symmetric spaces of noncompact type. No restriction on the rank of the symmetric spaces and no condition on the symmetry of the functions is assumed. PubDate: 2024-06-12

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Abstract: Abstract In this paper we consider steady inviscid three-dimensional stratified water flows of finite depth with a free surface and an interface. The interface plays the role of an internal wave that separates two layers of constant and different density. We study two cases separately: when the free surface and the interface are functions of one variable and when the free surface and the interface are functions of two variables. In both cases, considering effects of surface tension, we prove that the bounded solutions to the three-dimensional equations are essentially two-dimensional. More specifically, assuming that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other we prove that their third coordinate vanishes in both layers. Also we prove that the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion. PubDate: 2024-06-11

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Abstract: Abstract In this paper, we obtain the Gehring–Hayman type theorem on smoothly bounded pseudoconvex domains of finite type in \(\mathbb {C}^2\) . As an application, we provide a quantitative comparison between global and local Kobayashi distances near a boundary point for these domains. PubDate: 2024-06-05

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Abstract: Abstract In the setting of the thin-shell approximation of the Euler equations in spherical coordinates for oceanic flows with variable density on the spinning Earth, we study a vorticity equation for a pseudo stream function \(\psi \) , whereby the assumption of incompressibility allows us to express the density as a function of \(\psi \) . Via an elliptic comparison argument, we show that, under certain assumptions, the (explicit) solution in the case of zero rate of rotation (i.e., on a fixed sphere) in a bounded region with smooth boundary contained either in the Northern or in the Southern Hemisphere is an approximation, in a suitable sense, of the corresponding solution of the equation with positive rate of rotation in the same region. This provides new insight into the dynamics of ocean gyres. PubDate: 2024-06-05

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Abstract: Abstract In this paper we show that every connected extremal Kähler submanifold of a complex projective space has a natural extension which is a complete Kähler manifold and admits a holomorphic isometric immersion into the same ambient space. We also give an application to study the scalar curvatures of extremal Hypersurfaces of complex projective spaces. PubDate: 2024-06-04

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Abstract: Abstract In this paper we study a class of variable coefficient third order partial differential operators on \({\mathbb {R}}^{n+1}\) , containing, as a subclass, some variable coefficient operators of KdV-type in any space dimension. For such a class, as well as for the adjoint class, we obtain a Carleman estimate and the local solvability at any point of \({\mathbb {R}}^{n+1}\) . A discussion of possible applications in the context of dispersive equations is provided. PubDate: 2024-06-04

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Abstract: Abstract In this paper we describe all invariant complex Dirac structures with constant real index on a maximal flag manifold in terms of the roots of the Lie algebra which defines the flag manifold. We also completely classify these structures under the action of B-transformations. PubDate: 2024-06-01

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Abstract: Abstract If a Lie group admits a left invariant Randers metric of scalar flag curvature, then it is called of scalar Randers type. In this paper we determine all simply connected three dimensional Lie groups of scalar Randers type. It turns out that such groups must also admit a left invariant Riemannian metric with constant sectional curvature. PubDate: 2024-06-01

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Abstract: Abstract Let \(n\ge 2\) and \(\Omega \subset \mathbb {R}^n\) be a bounded Lipschitz domain. Assume that \(\textbf{b}\in L^{n*}(\Omega ;\mathbb {R}^n)\) and \(\gamma \) is a non-negative function on \(\partial \Omega \) satisfying some mild assumptions, where \(n^*:=n\) when \(n\ge 3\) and \(n^*\in (2,\infty )\) when \(n=2\) . In this article, we establish the unique solvability of the Robin problems $$\begin{aligned} \left\{ \begin{aligned} -\Delta u+\textrm{div}(u\textbf{b})&=f{} & {} \text {in}\ \ \Omega , \\ \left( \nabla u-u\textbf{b}\right) \cdot \varvec{\nu }+\gamma u&=u_R{} & {} \text {on}\ \ \partial \Omega \end{aligned}\right. \end{aligned}$$ and $$\begin{aligned} \left\{ \begin{aligned} -\Delta v-\textbf{b}\cdot \nabla v&=g{} & {} \text {in}\ \ \Omega , \\ \nabla v\cdot \varvec{\nu }+\gamma v&=v_R{} & {} \text {on}\ \ \partial \Omega \end{aligned}\right. \end{aligned}$$ in the Bessel potential space \(L^p_\alpha (\Omega )\) , where \(\alpha \in (0,2)\) and \(p\in (1,\infty )\) satisfy some restraint conditions, and \(\varvec{\nu }\) denotes the outward unit normal to the boundary \(\partial \Omega \) . The results obtained in this article extend the corresponding results established by Kim and Kwon (Trans Am Math Soc 375:6537–6574, 2022) for the Dirichlet and the Neumann problems to the case of the Robin problem. PubDate: 2024-06-01

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Abstract: Abstract We give a very general splitting type theorem for biholomorphic maps close to identity in the context of smoothly bounded pseudoconvex domains (Theorem 1.4). As a particular case, in the context of worm domains, we essentially reprove the splitting type result (Theorem 1.3) from Bracci et al. (Math Z 292:879–893, 2019) (by a different method). We also discuss some properties of the Nebenhülle of worm domains. PubDate: 2024-06-01

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Abstract: Abstract An almost abelian Lie group is a solvable Lie group with a codimension one normal abelian subgroup. We characterize almost Hermitian structures on almost abelian Lie groups where the almost complex structure is harmonic with respect to the Hermitian metric. Also, we adapt the Gray–Hervella classification of almost Hermitian structures to the family of almost abelian Lie groups. We provide several examples of harmonic almost complex structures in different Gray–Hervella classes on some associated compact almost abelian solvmanifolds. PubDate: 2024-06-01

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Abstract: Abstract We study varieties \(X \subseteq {\mathbb {P}}^N\) of dimension n such that \(T_X(k)\) is an Ulrich vector bundle for some \(k \in {\mathbb {Z}}\) . First we give a sharp bound for k in the case of curves. Then we show that \(k \le n+1\) if \(2 \le n \le 12\) . We classify the pairs \((X,{\mathcal {O}}_X(1))\) for \(k=1\) and we show that, for \(n \ge 4\) , the case \(k=2\) does not occur. PubDate: 2024-06-01

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Abstract: Abstract We consider the following higher-order prescribed curvature problem on \( {\mathbb {S}}^N: \) $$\begin{aligned} D^m {\tilde{u}}=\widetilde{K}(y) {\tilde{u}}^{m^{*}-1} \quad \text{ on } \ {\mathbb {S}}^N, \qquad {\tilde{u}} >0 \quad {\quad \hbox {in } }{\mathbb {S}}^N. \end{aligned}$$ where \(\widetilde{K}(y)>0\) is a radial function, \(m^{*}=\frac{2N}{N-2m}\) , and \(D^m\) is the 2m-order differential operator given by $$\begin{aligned} D^m=\prod _{i=1}^m\left( -\Delta _g+\frac{1}{4}(N-2i)(N+2i-2)\right) , \end{aligned}$$ where \(g=g_{{\mathbb {S}}^N}\) is the Riemannian metric. We prove the existence of infinitely many double-tower type solutions, which are invariant under some non-trivial sub-groups of O(3), and their energy can be made arbitrarily large. PubDate: 2024-06-01

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Abstract: Abstract We determine the structure of solvable Lie groups endowed with invariant stretched non-positive Weyl connections and find classes of solvable Lie groups admitting and not admitting such connections. In dimension 4 we fully classify solvable Lie groups which admit invariant SNP Weyl connections. PubDate: 2024-06-01

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Abstract: Abstract We revisit the well-known Brezis-Nirenberg problem $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u= u^{\frac{N+2}{N-2}}+\varepsilon u, &{}{{\text {in}}~\Omega },\\ u>0, &{}{{\text {in}}~\Omega },\\ u=0, &{}{\text {on}~\partial \Omega }, \end{array}\right. } \end{aligned}$$ where \(\varepsilon >0\) and \(\Omega \subset \mathbb {R}^N\) are a smooth bounded domain with \(N\ge 3\) . The existence of multi-bump solutions to above problem for small parameter \(\varepsilon >0\) was obtained by Musso and Pistoia (Indiana Univ Math J 51:541–579, 2002). However, to our knowledge, whether the multi-bump solutions are non-degenerate that is open. Here, we give some straightforward answer on this question under some suitable assumptions for the Green’s function of \(-\Delta \) in \(\Omega \) , which enriches the qualitative analysis on the solutions of Brezis-Nirenberg problem and can be viewed as a generalization of Grossi (Nonlinear Differ Equ Appl 12:227–241, 2005) where the non-degeneracy of a single-bump solution has been proved. And the main idea is the blow-up analysis based on the local Pohozaev identities. PubDate: 2024-06-01

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Abstract: Abstract The generalized Zermelo navigation problem looks for the shortest time paths in an environment, modeled by a Finsler manifold (M, F), under the influence of wind or current, represented by a vector field W. The main objective of this paper is to investigate the relationship between the isoparametric functions on the manifold M with and without the presence of the vector field W. Our work generalizes results in (Dong and He in Differ Geom Appl 68:101581, 2020; He et al. in Acta Math Sinica Engl Ser 36:1049–1060, 2020; He et al. in Differ Geom Appl 84:101937, 2022; Ming et al. in Pub Math Debr 97:449–474, 2020; Xu et al. in Isoparametric hypersurfaces induced by navigation in Lorentz Finsler geometry, 2021). For the positive-definite cases, we also compare the mean curvatures in the manifold. Overall, we follow a coordinate-free approach. PubDate: 2024-06-01

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Abstract: Abstract In this paper we study the following class of linearly coupled systems in the plane: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + u = f_1(u) + \lambda v,\quad \text{ in }\quad \mathbb {R}^2, \\ -\Delta v + v = f_2(v) + \lambda u,\quad \text{ in }\quad \mathbb {R}^2, \\ \end{array}\right. } \end{aligned}$$ where \(f_{1}, f_{2}\) are continuous functions with critical exponential growth in the sense of Trudinger-Moser inequality and \(0<\lambda <1\) is a parameter. First, for any \(\lambda \in (0,1)\) , by using minimization arguments and minimax estimates we prove the existence of a positive ground state solution. Moreover, we study the asymptotic behavior of these solutions when \(\lambda \rightarrow 0^{+}\) . This class of systems can model phenomena in nonlinear optics and in plasma physics. PubDate: 2024-06-01

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Abstract: Abstract Let G be a finite group, \(\mathbb {F}\) be one of the fields \(\mathbb {Q},\mathbb {R}\) or \(\mathbb {C}\) , and N be a non-trivial normal subgroup of G. Let \({\textrm{acd}}^{*}_{{\mathbb {F}}}(G)\) and \({\textrm{acd}}_{{\mathbb {F}}, \textrm{even}}(G N)\) be the average degree of all non-linear \(\mathbb {F}\) -valued irreducible characters of G and of even degree \(\mathbb {F}\) -valued irreducible characters of G whose kernels do not contain N, respectively. We assume the average of an empty set is zero for more convenience. In this paper we prove that if \(\textrm{acd}^*_{\mathbb {Q}}(G)< 9/2\) or \(0<\textrm{acd}_{\mathbb {Q},\textrm{even}}(G N)<4\) , then G is solvable. Moreover, setting \(\mathbb {F} \in \{\mathbb {R},\mathbb {C}\}\) , we obtain the solvability of G by assuming \({\textrm{acd}}^{*}_{{\mathbb {F}}}(G)<29/8\) or \(0<{\textrm{acd}}_{{\mathbb {F}}, \textrm{even}}(G N)<7/2\) , and we conclude the solvability of N when \(0<{\textrm{acd}}_{{\mathbb {F}}, \textrm{even}}(G N)<18/5\) . Replacing N by G in \({\textrm{acd}}_{{\mathbb {F}}, \textrm{even}}(G N)\) gives us an extended form of a result by Moreto and Nguyen. Examples are given to show that all the bounds are sharp. PubDate: 2024-06-01

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Abstract: Abstract In our previous paper, it is proved that for any positive flow-spine P of a closed, oriented 3-manifold M, there exists a unique contact structure supported by P up to isotopy. In particular, this defines a map from the set of isotopy classes of positive flow-spines of M to the set of isotopy classes of contact structures on M. In this paper, we show that this map is surjective. As a corollary, we show that any flow-spine can be deformed to a positive flow-spine by applying first and second regular moves successively. PubDate: 2024-06-01

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Abstract: Abstract We introduce a capillary Chambolle-type scheme for mean curvature flow with prescribed contact angle. Our scheme includes a capillary functional instead of just the total variation. We show that the scheme is well-defined and has consistency with the energy minimizing scheme of Almgren–Taylor–Wang type. Moreover, for a planar motion in a strip, we give several examples of numerical computation of this scheme based on the split Bregman method instead of a duality method. PubDate: 2024-06-01