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Convex in x domain, eigenfunction, level sets, local rearrangement Erratum to the paper "Convexity in x of the level sets of the first Dirichlet eigenfunction": (Published in Math. Nachr. Keywords: convex in x domain; eigenfunction; level sets; local rearrangement EN convex in x domain eigenfunction level sets local rearrangement 1025 1025 1 03/21/23 20230301 NES 230301 Lemma 3.2 is incorrect. [Extracted from the article]

In this note we point out and correct a mistake in our paper "GlobalLpestimates for degenerate Ornstein–Uhlenbeck operators with variable coefficients", published in Math. Nachr. 286 (2013), no. 11–12, 1087–1101. [ABSTRACT FROM AUTHOR]

There exists a mistake in the proof of Theorem 4.2. We present a new proof of this theorem, which shows that the main results of the paper are still true. [ABSTRACT FROM AUTHOR]

We give a corrected proof of the last theorem (Theorem 4.11) of Regularized Riesz energies of submanifolds, published in Math. Nachr. 291 (2018), no. 8–9, 1356–1373. Namely, we prove that the Riesz energy EΩ(z) of a compact body Ω is Möbius invariant if and only if the dimension of Ω is even and z=−2dimΩ. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the well‐posedness, blow‐up phenomena, and continuity of the data‐to‐solution map of the Cauchy problem for a two‐component higher order Camassa–Holm (CH) system. The local well‐posedness is established in Besov spaces Bp,11p×Bp,12+1p$B_{p,1}^{\frac{1}{p}} \times B_{p,1}^{2+\frac{1}{p}}$ with 1≤p<∞$1 \le p < \infty$, which improves the local well‐posedness result proved before in Tang and Liu [Z. Angew. Math. Phys. 66 (2015), 1559–1580], Ye and Yin [arXiv preprint arXiv:2109.00948 (2021)], Zhang and Li [Nonlinear Anal. Real World Appl. 35 (2017), 414–440], and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. Next, we consider the continuity of the solution‐to‐data map, that is, the ill‐posedness is derived in Besov space Bp,∞s−2×Bp,∞s$B_{p,\infty }^{s - 2} \times B_{p,\infty }^s$ with 1≤p≤∞$1 \le p \le \infty$ and s>max{2+1p,52}$s>\max \lbrace 2+\frac{1}{p},\frac{5}{2}\rbrace$. Then, the nonuniform continuous and Hölder continuous dependence on initial data for this system are also presented in Besov spaces Bp,rs−2×Bp,rs$B_{p,r}^{s - 2} \times B_{p,r}^s$ with 1≤p,r<∞$1 \le p,r < \infty$ and s>max{2+1p,52}$s > \max \lbrace {2+\frac{1}{p},\frac{5}{2}}\rbrace$. Finally, the precise blow‐up criteria for the strong solutions of the two‐component higher order CH system is determined in the lowest Sobolev space Hs−2×Hs$H^{s-2}\times H^s$ with s>52$s>\frac{5}{2}$, which improves the blow‐up criteria result established before in He and Yin [Discrete Contin. Dyn. Syst. 37 (2016), no. 3, 1509–1537] and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. [ABSTRACT FROM AUTHOR]

Linear operators with off‐diagonal decay appear in many areas of mathematics including harmonic and numerical analysis, and their stability is one of the basic assumptions. In this paper, we consider a family of localized integral operators in the Beurling algebra with kernels having mild singularity near the diagonal and certain Hölder continuity property, and prove that their weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other on a space of homogeneous type with Ahlfors regular measure. [ABSTRACT FROM AUTHOR]

This paper studies the nonlocal fractional analog of the famous paper of Brezis and Nirenberg [Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477]. Namely, we focus on the following model: P−Δsu−λu=α|u|p−2u+β|u|2s∗−2uinΩ,u=0inRN∖Ω,$$\begin{align*}\hskip5pc {\left(\mathcal{P}\right)} {\left\{ \def\eqcellsep{&}\begin{array}{l} {\left(-\Delta \right)}^s u-\lambda u = \alpha |u|^{p-2}u + \beta |u|^{2^*_s-2}u \quad \mbox{in}\quad \Omega ,\\ u=0\quad \mbox{in}\quad \mathbb {R}^N\setminus \Omega , \end{array} \right.}\hskip-5pc \end{align*}$$where (−Δ)s$(-\Delta)^s$ is the fractional Laplace operator, s∈(0,1)$s \in (0,1)$, with N>2s$N > 2s$, 20,λ,α∈R$\beta >0,\, \lambda , \alpha \in \mathbb {R}$, and establish the existence of nontrivial solutions and sign‐changing solutions for the problem (P)$(\mathcal{P})$. [ABSTRACT FROM AUTHOR]

The product manifold S3×S3$\mathbb {S}^3\times \mathbb {S}^3$ is one of the only four homogeneous six‐dimensional nearly Kähler manifolds. It also admits a canonical almost product structure P, which is compatible with the almost complex structure (see Bolton et al., Tôhoku Math. J. 67 (2015), 1–17, and Moruz and Vrancken, Publ. Inst. Math. 103 (2018), no. 117, 147–158). In this paper, we investigate and describe the two‐dimensional surfaces of S3×S3$\mathbb {S}^3\times \mathbb {S}^3$ which are P‐invariant. [ABSTRACT FROM AUTHOR]

In this paper, we study second order linear Sturm-Liouville problems involving one or two homogeneous linear multi-point boundary conditions in the most general form. We obtain conditions for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Furthermore, we reveal the interlacing relations between the eigenvalues of such Sturm-Liouville problems and those of Sturm-Liouville problems with certain two-point separated boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new Lyapunov-type inequalities for a class of even-order linear differential equations, the results are new and generalize and improve some early results in this field. [ABSTRACT FROM AUTHOR]

A table of contents for the 2016 issue of the journal "Mathematische Nachrichten" is presented.

In this paper, we prove some new dynamic inequalities on time scales using Hölder's inequality and Keller's chain rule on time scales. These inequalities, as special cases when the time scale [ABSTRACT FROM AUTHOR]

In this paper we study local existence, uniqueness, and continuous dependence of an abstract integrodifferential equation. We also present a result on unique continuation and a blow-up alternative for mild solutions of the integrodifferential equation. Finally, we apply our results to an interesting strongly damped plate equation with memory. [ABSTRACT FROM AUTHOR]

We give a sharp estimate for the codimension of the poly-Bergman space in the poly-Bergman space over the punctured domain. It is established the behaviour at the infinity point of polyanalytic Bergman functions on the complement of closed disks. In the main result of the paper, we prove that for [ABSTRACT FROM AUTHOR]

A table of contents of the issue Mathematische Nachrichten is presented.

Continuing the study of bounded geometry for Riemannian foliations, begun by Sanguiao, we introduce a chart-free definition of this concept. Our main theorem states that it is equivalent to a condition involving certain normal foliation charts. For this type of charts, it is also shown that the derivatives of the changes of coordinates are uniformly bounded, and there are nice partitions of unity. Applications to a trace formula for foliated flows will be given in a forthcoming paper. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of decomposability in the space of Henstock-Kurzweil-Pettis integrable (for short HKP-integrable) functions. We show representations theorems for decomposable sets of HKP-integrable or Henstock integrable functions, in terms of the family of selections of suitable multifunctions. [ABSTRACT FROM AUTHOR]

We obtain a fundamental gap estimate for classes of bounded domains with quantitative control on the boundary in a complete manifold with integral bounds on the negative part of the Ricci curvature. This extends the result of Oden, Sung, and Wang [Trans. Amer. Math. Soc. 351 (1999), no. 9, 3533–3548] to Lp$L^p$‐Ricci curvature assumptions, p>n/2$p>n/2$. To achieve our result, it is shown that the domains under consideration are John domains, what enables us to obtain an estimate on the first nonzero Neumann eigenvalue, which is of independent interest. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of the Navier–Stokes equations in Rn$\mathbb {R}^n$ (n≥3)$n\ge 3)$ and establish several new extension criteria involving the pressure or its gradient. In particular, we improve the previous results by means of the homogeneous Besov space with negative differential orders in the case n=3$n=3$. Our method is based on the interpolation inequality and the trilinear estimate due to Gérard–Meyer–Oru (Séminaire É. D. P. (1996–1997), Exp. No. IV, 1–8) and Guo–Kučera–Skalák (J. Math. Anal. Appl. 458 (2018) 755–766), respectively. [ABSTRACT FROM AUTHOR]

We prove the Lp$L^p$ bound for the Hilbert transform along variable non‐flat curves (t,u(x)[t]α+v(x)[t]β)$(t,u(x)[t]^\alpha +v(x)[t]^\beta)$, where α and β satisfy α≠β,α≠1,β≠1$\alpha \ne \beta ,\ \alpha \ne 1,\ \beta \ne 1$. Compared with the associated theorem in the work (Guo et al. Proc. Lond. Math. Soc. 2017) investigating the case α=β≠1$\alpha =\beta \ne 1$, our result is more general while the proof is more involved. To achieve our goal, we divide the frequency of the objective function into three cases and take different strategies to control these cases. Furthermore, we need to introduce a "short" shift maximal function M[n]$\mathbf {M}^{[n]}$ to establish some pointwise estimates. [ABSTRACT FROM AUTHOR]

In this study, we consider the Oseen structure of the linearization of a compressible fluid–structure interaction (FSI) system for which the interaction interface is under the effect of material derivative term. The flow linearization is taken with respect to an arbitrary, variable ambient vector field. This process produces extra "convective derivative" and "material derivative" terms, which render the coupled system highly nondissipative. We show first a new well‐posedness result for the full incorporation of both Oseen terms, which provides a uniformly bounded semigroup via dissipativity and perturbation arguments. In addition, we analyze the long time dynamics in the sense of asymptotic (strong) stability in an invariant subspace (one‐dimensional less) of the entire state space, where the continuous semigroup is uniformly bounded. For this, we appeal to the pointwise resolvent condition introduced in Chill and Tomilov [Stability of operator semigroups: ideas and results, perspectives in operator theory Banach center publications, 75 (2007), Institute of Mathematics Polish Academy of Sciences, Warszawa, 71–109], which avoids an immensely technical and challenging spectral analysis and provides a short and relatively easy‐to‐follow proof. [ABSTRACT FROM AUTHOR]

The table of contents for the November 1, 2016 issue of "Mathematische Nachrichten" is presented.

The table of contents for this issue is presented.

The table of contents for the 2014 issue of the journal "Mathematische Nachrichten" is presented.