Necessary Conditions for Interpolation by Multivariate Polynomials

Let $$\Omega $$ Ω be a smooth, bounded, convex domain in $${\mathbb {R}}^n$$ R n and let $$\Lambda _k$$ Λ k be a finite subset of $$\Omega $$ Ω . We find necessary geometric conditions for $$\Lambda _k$$ Λ k to be interpolating for the space of multivariate polynomials of degree at most k. Our results are asymptotic in k. The density conditions obtained match precisely the necessary geometric conditions that sampling sets are known to satisfy and are expressed in terms of the equilibrium potential of the convex set. Moreover we prove that in the particular case of the unit ball, for k large enough, there are no bases of orthogonal reproducing kernels in the space of polynomials of degree at most k.

The Keller–Segel system on bounded convex domains in critical spaces

Consider the classical Keller–Segel system on a bounded convex domain $$\varOmega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 . In contrast to previous works it is not assumed that the boundary of $$\varOmega $$ Ω is smooth. It is shown that this system admits a local, strong solution for initial data in critical spaces which extends to a global one provided the data are small enough in this critical norm. Furthermore, it is shown that this system admits for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution.

A hypersurface containing the support of a Radon transform must be an ellipsoid. II: The general case

If the Radon transform of a compactly supported distribution f ≠ 0 {f\neq 0} in ℝ n {\mathbb{R}^{n}} is supported on the set of tangent planes to the boundary ∂ ⁡ D {\partial D} of a bounded convex domain D, then ∂ ⁡ D {\partial D} must be an ellipsoid. The special case of this result when the domain D is symmetric was treated in [J. Boman, A hypersurface containing the support of a Radon transform must be an ellipsoid. I: The symmetric case, J. Geom. Anal. 2020, 10.1007/s12220-020-00372-8]. Here we treat the general case.

Global well-posedness in a chemotaxis system with oxygen consumption

<p style='text-indent:20px;'>Motivated by the studies of the hydrodynamics of the tethered bacteria <i>Thiovulum majus</i> in a liquid environment, we consider the following chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{split} &amp; n_t = \Delta n-\nabla\cdot\left(n\chi(c)\nabla{c}\right)+nc, &amp;x\in \Omega, t&gt;0, \ &amp; c_t = \Delta c-{\bf u}\cdot\nabla c-nc, &amp;x\in \Omega, t&gt;0\ \end{split} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded convex domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^d(d\in\{2, 3\}) $\end{document}</tex-math></inline-formula> with smooth boundary. For any given fluid <inline-formula><tex-math id="M2">\begin{document}$ {\bf u} $\end{document}</tex-math></inline-formula>, it is proved that if <inline-formula><tex-math id="M3">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula>, the corresponding initial-boundary value problem admits a unique global classical solution which is uniformly bounded, while if <inline-formula><tex-math id="M4">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>, such solution still exists under the additional condition that <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\chi\leq \frac{1}{16\|c(\cdot, 0)\|_{L^\infty(\Omega)}} $\end{document}</tex-math></inline-formula>.</p>

Оn a Space of Holomorphic Functions on a Bounded Convex Domain of ${\mathbb C}^n$ and Smooth Up to the Boundary and its Dual Space

В работе рассматривается локально выпуклое пространство функций, голоморфных в ограниченной выпуклой области многомерного комплексного пространства и гладких вплоть до границы, с топологией, определяемой счетным семейством норм, образованных при помощи семейства ${\mathfrak M}$ логарифмически выпуклых последовательностей положительных чисел специального вида. Благодаря условиям на указанные последовательности данное пространство является пространством Фреше - Шварца. Изучается задача описания сильного сопряженного для этого пространства в терминах преобразования Лапласа функционалов. Интерес к ней связан с исследованиями Б. А. Державца классических проблем теории линейных дифференциальных операторов с постоянными коэффициентами, А. В. Абанина, С. В. Петрова и К. П. Исаева современных проблем теории абсолютно представляющих систем в различных пространствах функций, голоморфных в выпуклых областях комплексного пространства, с заданной граничной гладкостью, при решении которых важную роль сыграли полученные ими теоремы типа Пейли - Винера - Шварца. Основной результат работы, полученный в теореме 1, утверждает, что преобразование Лапласа линейных непрерывных функционалов устанавливает изоморфизм между сильным сопряженным к рассматриваемому функциональному пространству и некоторым пространством целых функций экспоненциального типа в ${\mathbb C}^n $, представляющим собой внутренний индуктивный предел весовых банаховых пространств целых функций. Отметим, что в рассматриваемом случае удалось получить аналитическую реализацию сопряженного пространства при меньших ограничениях на семейство ${\mathfrak M}$ по сравнению с работой автора 2002 г. Основу доказательства теоремы 1 в настоящей работе составляют схема, предложенная М. Наймарком и Б. А. Тейлором, и ряд предыдущих результатов автора.

A Sharp Multidimensional Hermite–Hadamard Inequality

Let $\Omega \subset {\mathbb{R}}^d $, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to{\mathbb{R}}$ be a non-negative subharmonic function. In this paper, we prove the inequality $$\begin{equation*} \frac{1}{|\Omega|}\int_{\Omega} f(x)\, \textrm{d}x \leq \frac{d}{|\partial\Omega|}\int_{\partial\Omega} f(x)\, \textrm{d}\sigma(x)\,. \end{equation*}$$Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $\Omega \subset{\mathbb{R}}^d$ is a bounded convex domain and $u$ is the solution of $-\Delta u =1$ with homogeneous Dirichlet boundary conditions, then $$\begin{equation*} \|\nabla u\|_{L^\infty(\Omega)} &lt; d\frac{|\Omega|}{|\partial\Omega|}\,. \end{equation*}$$Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al. [2].

On the monotonicity of the principal frequency of the p-Laplacian

For any fixed integer {D>1} we show that there exists {M\in[e^{-1},1]} such that for any open, bounded, convex domain {\Omega\subset{\mathbb{R}}^{D}} with smooth boundary for which the maximum of the distance function to the boundary of Ω is less than or equal to M, the principal frequency of the p-Laplacian on Ω is an increasing function of p on {(1,\infty)} . Moreover, for any real number {s>M} there exists an open, bounded, convex domain {\Omega\subset{\mathbb{R}}^{D}} with smooth boundary which has the maximum of the distance function to the boundary of Ω equal to s such that the principal frequency of the p-Laplacian is not a monotone function of {p\in(1,\infty)} .

Minimization problems for inhomogeneous Rayleigh quotients

In this paper, the minimization problem [Formula: see text] where [Formula: see text] is studied when [Formula: see text] ([Formula: see text]) is an open, bounded, convex domain with smooth boundary and [Formula: see text]. We show that [Formula: see text] is either zero, when the maximum of the distance function to the boundary of [Formula: see text] is greater than [Formula: see text], or it is a positive real number, when the maximum of the distance function to the boundary of [Formula: see text] belongs to the interval [Formula: see text]. In the latter case, we provide estimates for [Formula: see text] and show that for [Formula: see text] sufficiently large [Formula: see text] coincides with the principal frequency of the [Formula: see text]-Laplacian in [Formula: see text]. Some particular cases and related problems are also discussed.

Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$

Let $\Omega \subset \mathbb{C}^2$ be a bounded convex domain with $C^1$-smooth boundary and $\varphi \in C^1(\overline{\Omega})$ such that $\varphi $ is harmonic on the non-trivial disks in the boundary. We estimate the essential norm of the Hankel operator $H_{\varphi }$ in terms of the $\overline{\partial}$ derivatives of $\varphi$ “along” the non-trivial disks in the boundary.

Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system

This work considers the Keller–Segel-type parabolic system [Formula: see text] in a smoothly bounded convex domain [Formula: see text], [Formula: see text], under no-flux boundary conditions, which has recently been proposed as a model for processes of stripe pattern formation via so-called “self-trapping” mechanisms. In the two-dimensional case, in stark contrast to the classical Keller–Segel model in which large-data solutions may blow up in finite time, for all suitably regular initial data the associated initial value problem is seen to possess a globally-defined bounded classical solution, provided that the motility function [Formula: see text] is uniformly positive. In the corresponding higher-dimensional setting, it is shown that certain weak solutions exist globally, where in the particular three-dimensional case this solution actually is bounded and classical if the initial data are suitably small in the norm of [Formula: see text]. Finally, if still [Formula: see text] but merely the physically interpretable quantity [Formula: see text] is appropriately small, then the above-weak solutions are proved to become eventually smooth and bounded.