Recent zbMATH articles in MSC 31https://www.zbmath.org/atom/cc/312022-05-16T20:40:13.078697ZWerkzeugGreedy energy minimization can count in binary: point charges and the van der Corput sequencehttps://www.zbmath.org/1483.111552022-05-16T20:40:13.078697Z"Pausinger, Florian"https://www.zbmath.org/authors/?q=ai:pausinger.florianFor a sequence \(X = (x_n)_{n \ge 0}\) of points in \([0,1)\), its \textit{discrepancy} associates to any \(N \ge 1\) the quantity \[ D_N(X) = \sup_{0 \le \alpha < \beta < 1}\left|\frac{1}{N}\sharp\{1 \le n \le N \colon \alpha \le x_n < \beta\}-(\beta - \alpha)\right|. \] \textit{W. M. Schmidt} [Acta Arith. 21, 45--50 (1972; Zbl 0244.10035)] showed that there is no sequence \(X\) so that one can have \(D_N(X) = o(N^{-1}\log N)\). \textit{J. G. van der Corput} [Proc. Akad. Wet. Amsterdam 38, 1058--1066 (1935; Zbl 0013.05703)] had previously constructed a sequence, \(S_2\) satisfying \(D_N(S_2) \le c N^{-1} \log N\) for some \(c > 0\), where \(x_k\) is explicitly given in terms of the binary expansion of \(k\).\\
The general theme of the paper is the inductive construction of sequences in \([0, 1)\) satisfying \[x_N = \operatorname{arg min}_{x \in (0,1)} \sum_{n=0}^{N-1} f(|x-x_n)|), \tag{1}\] where \( f \colon [0, 1] \rightarrow \mathbb{R}\) satisfies \[ f(t) = f(1-t) \; , \; f \text{ is twice differentiable on } (0, 1) \; \text{ and } f'' \text{ is positive on } (0,1). \]
In Theorem 5.5, the author proves that for any \(N\), if \(\{x_0, x_1, \ldots, x_{N-1}\}\) consists of the \(N\) first elements of the van der Corput sequence \(S_2\), then the element of \(S_2\) with index \(N+1\) satisfies (1).\\
In his main result, Theorem 2.1, the author proves that if a sequence \(X=(x_n)_{n \ge 0}\) with \(x_0=0\) satisfies the inductive property (1), then
\begin{itemize}
\item[1.] all its terms belong to \textit{permuted} van der Corput sequences,
\item[2.] for all \(N\), one has \(D_N(X) = D_N(S_2)\).
\end{itemize}
In particular, this leads to the proof a special case of a conjecture by
\textit{S. Steinerberger} [Monatsh. Math. 191, No. 3, 639--655 (2020; Zbl 1471.11222)].\\
A nice paper at the border of dynamical systems, combinatorics, number theory and mathematical physics with interesting results and questions.
Reviewer: Jean-Marc Deshouillers (Bordeaux)Rate of growth of distributionally chaotic functionshttps://www.zbmath.org/1483.300592022-05-16T20:40:13.078697Z"Gilmore, Clifford"https://www.zbmath.org/authors/?q=ai:gilmore.clifford"Martínez-Giménez, Félix"https://www.zbmath.org/authors/?q=ai:martinez-gimenez.felix"Peris, Alfred"https://www.zbmath.org/authors/?q=ai:peris.alfredoSummary: We investigate the permissible growth rates of functions that are distributionally chaotic with respect to differentiation operators. We improve on the known growth estimates for \(D\)-distributionally chaotic entire functions, where growth is in terms of average \(L^p\)-norms on spheres of radius \(r>0\) as \(r\rightarrow\infty\), for \(1\leq p\leq\infty\). We compute growth estimates of \(\partial/\partial x_k\)-distributionally chaotic harmonic functions in terms of the average \(L^2\)-norm on spheres of radius \(r>0\) as \(r\rightarrow\infty\). We also calculate sup-norm growth estimates of distributionally chaotic harmonic functions in the case of the partial differentiation operators \(D^\alpha\).Global boundedness of functions of finite order that are bounded outside small setshttps://www.zbmath.org/1483.300602022-05-16T20:40:13.078697Z"Khabibullin, Bulat N."https://www.zbmath.org/authors/?q=ai:khabibullin.b-nPolyanalytic boundary value problems for planar domains with harmonic Green functionhttps://www.zbmath.org/1483.300722022-05-16T20:40:13.078697Z"Begehr, Heinrich"https://www.zbmath.org/authors/?q=ai:begehr.heinrich"Shupeyeva, Bibinur"https://www.zbmath.org/authors/?q=ai:shupeyeva.bibinurThe authors characterize the solvability of three boundary value problems for the inhomogeneous polyanalytic equation in planar domains (having a harmonic Green function), namely the well-posed iterated Schwarz problem, and two over-determined iterated problems of Dirichlet and Neumann type. Solutions formulas are also obtained and, in particular, it is concluded that the polyanalytic Cauchy-Pompeiu representation formula provides the solution to the Dirichlet problem (for any degree \(n\), and in the cases for which the solution exists).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)On existence of quasi-Strebel structures for meromorphic \(k\)-differentialshttps://www.zbmath.org/1483.300822022-05-16T20:40:13.078697Z"Shapiro, Boris"https://www.zbmath.org/authors/?q=ai:shapiro.boris-zalmanovich"Tahar, Guillaume"https://www.zbmath.org/authors/?q=ai:tahar.guillaumeA meromorphic differential \(\Psi\) of order \(k\geq 2\) on a compact orientable Riemann surface \(Y\) without boundary is a meromorphic section of the \(k\)-th tensor power \((T^{*}_{\mathbb{C}}Y)^{\otimes k}\) of the holomorphic cotangent bundle of \(Y\). Zeros and poles of \(\Psi\) constitute the set of critical points.
For a differential of order \(k\) given locally by \(f(z)dz^{k}\) in a neighborhood of a non-critical point, there are \(k\) locally distinct directions, called \textit{horizontal}, which are given by the conditions that \(f(z)dz^{k}\) is real and positive.
Recall that a quadratic differential is called \textit{Strebel} if almost all horizontal trajectories are closed. Such a phenomenon can never happen for a \(k\)-differential of order \(k\geq 3\) unless it is a power of a \(1\)-form or a quadratic differential. The authors introduce thus the notion of quasi-Strebel structure and give sufficient conditions for a meromorphic \(k\)-differential to be quasi-Strebel. The main result is the following:
\textbf{Theorem 1.} Let \(\Psi\) be a meromorphic \(k\)-differential such that
\begin{itemize}
\item no poles have of order smaller than \(-k\);
\item at a pole of order \(-k\), the residue belongs to \(i^{k}\mathbb{R}\).
\end{itemize}
Then,
\begin{itemize}
\item[1.] if \(k>2\) is even, then \(\Psi\) has a quasi-Strebel structure;
\item[2.] if \(k>2\) is odd, and, up to a common factor, the period of \(\sqrt[k]{\Psi}\) along every path connecting any two singularities of \(\Psi\) belongs to \(\mathbb{Q}[e^{\frac{2i\pi}{k}}]\), then \(\Psi\) has a quasi-Strebel structure.
\end{itemize}
Reviewer: Andrea Tamburelli (Houston)On the coefficients of certain subclasses of harmonic univalent mappings with nonzero polehttps://www.zbmath.org/1483.310012022-05-16T20:40:13.078697Z"Bhowmik, Bappaditya"https://www.zbmath.org/authors/?q=ai:bhowmik.bappaditya"Majee, Santana"https://www.zbmath.org/authors/?q=ai:majee.santanaSummary: Let \(Co(p), p\in (0,1]\) be the class of all meromorphic univalent functions \(\varphi\) defined in the open unit disc \(\mathbb{D}\) with normalizations \(\varphi (0)=0=\varphi^{\prime} (0)-1\) and having simple pole at \(z=p\in (0,1]\) such that the complement of \(\varphi (\mathbb{D})\) is a convex domain. The class \(Co(p)\) is called the class of concave univalent functions. Let \(S_H^0 (p)\) be the class of all sense preserving univalent harmonic mappings \(f\) defined on \(\mathbb{D}\) having simple pole at \(z=p\in (0,1)\) with the normalizations \(f(0)=f_z (0)-1=0\) and \(f_{\bar{z}}(0)=0\). We first derive the exact regions of variability for the second Taylor coefficients of \(h\) where \(f=h+\overline{g}\in S_H^0 (p)\) with \(h-g\in Co(p)\). Next we consider the class \(S_H^0 (1)\) of all sense preserving univalent harmonic mappings \(f\) in \(\mathbb{D}\) having simple pole at \(z=1\) with the same normalizations as above. We derive exact regions of variability for the coefficients of \(h\) where \(f=h+\overline{g}\in S_H^0 (1)\) satisfying \(h-e^{2i\theta}g\in Co(1)\) with dilatation \(g^{\prime} (z)/h^{\prime} (z)=e^{-2i\theta}z\), for some \(\theta, 0\leq \theta <\pi\).Extreme points and support points of families of harmonic Bloch mappingshttps://www.zbmath.org/1483.310022022-05-16T20:40:13.078697Z"Deng, Hua"https://www.zbmath.org/authors/?q=ai:deng.hua"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathan"Qiao, Jinjing"https://www.zbmath.org/authors/?q=ai:qiao.jinjingSummary: In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the unit ball of the normalized harmonic Bloch spaces (resp. the normalized little harmonic Bloch spaces) in the unit disk \(\mathbb{D}\). Then we show that a harmonic Bloch mapping \(f\) is a support point of the unit ball of the normalized harmonic Bloch spaces in \(\mathbb{D}\) if and only if the Bloch unit-valued set of \(f\) is not empty. We also give a characterization for the support points of the unit ball of the harmonic Bloch spaces in \(\mathbb{D}\).Harmonic measure of the outer boundary of colander setshttps://www.zbmath.org/1483.310032022-05-16T20:40:13.078697Z"Glücksam, Adi"https://www.zbmath.org/authors/?q=ai:glucksam.adiSummary: We present two companion results: Phragmén-Lindelöf type tight bounds on the minimal possible growth of subharmonic functions with a recurrent zero set, and tight bounds on the maximal possible decay of the harmonic measure of the outer boundary of colander sets.The existence of minimizers of energy for diffeomorphisms between two-dimensional annuli in \(\mathbb{R}^2\) and \(\mathbb{R}^3\)https://www.zbmath.org/1483.310042022-05-16T20:40:13.078697Z"Kalaj, David"https://www.zbmath.org/authors/?q=ai:kalaj.david"Zhu, Jian-Feng"https://www.zbmath.org/authors/?q=ai:zhu.jianfengSummary: In this paper, we consider the minimization for the Dirichlet energy of Sobolev homeomorphisms between two-dimensional annuli in \(\mathbb{R}^2\) and \(\mathbb{R}^3\), respectively. It should be noticed that in this case a Nitsche phenomenon occurs. The main result of this paper partly extends the corresponding result in [\textit{K. Astala} et al., Arch. Ration. Mech. Anal. 195, No. 3, 899--921 (2010; Zbl 1219.30011)].Completely monotone sequences and harmonic mappingshttps://www.zbmath.org/1483.310052022-05-16T20:40:13.078697Z"Long, Bo-Yong"https://www.zbmath.org/authors/?q=ai:long.boyong"Sugawa, Toshiyuki"https://www.zbmath.org/authors/?q=ai:sugawa.toshiyuki"Wang, Qi-Han"https://www.zbmath.org/authors/?q=ai:wang.qihanSummary: In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.A new property of a diskhttps://www.zbmath.org/1483.310062022-05-16T20:40:13.078697Z"Volchkov, V. V."https://www.zbmath.org/authors/?q=ai:volchkov.valerii-vladimirovich"Volchkov, Vit. V."https://www.zbmath.org/authors/?q=ai:volchkov.vitalii-vladimirovichSummary: It is shown that some overdetermined boundary Neumann problem has a solution only for a disk.Some properties of certain close-to-convex harmonic mappingshttps://www.zbmath.org/1483.310072022-05-16T20:40:13.078697Z"Wang, Xiao-Yuan"https://www.zbmath.org/authors/?q=ai:wang.xiaoyuan"Wang, Zhi-Gang"https://www.zbmath.org/authors/?q=ai:wang.zhigang|wang.zhigang.1"Fan, Jin-Hua"https://www.zbmath.org/authors/?q=ai:fan.jinhua"Hu, Zhen-Yong"https://www.zbmath.org/authors/?q=ai:hu.zhenyongSummary: In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr's inequalities for a subclass of close-to-convex harmonic mappings, whose analytic parts are Ma-Minda convex functions.Constant generalized Riesz potential functions and polarization optimality problemshttps://www.zbmath.org/1483.310082022-05-16T20:40:13.078697Z"Bosuwan, Nattapong"https://www.zbmath.org/authors/?q=ai:bosuwan.nattapongSummary: An extension of a conjecture of \textit{N. Nikolov} and \textit{R. Rafailov} [Geom. Dedicata 167, 69--89 (2013; Zbl 1287.52007)] by considering the following potential function defined on \(\mathbb{R}^2\):
\[
f_s(x)=\sum_{j=1}^N \left(|x-x_j|^2+h\right)^{-s/2},\quad \quad h\geq0,
\]
for \(s=2-2N\) is given. We obtain a characterization of sets of \(N\) distinct points \(\{x_1,x_2,\ldots,x_N\}\) such that \(f_{2-2N}\) is constant on some circle in \(\mathbb{R}^2.\) Using this characterization, we prove some special cases of this new conjecture. The other problems considered in this paper are polarization optimality problems. We find all maximal and minimal polarization constants and configurations of two concentric circles in \(\mathbb{R}^2\) using the above potential function for certain values of \(s\).Pólya-Schur inequality and the Green energy of a discrete chargehttps://www.zbmath.org/1483.310092022-05-16T20:40:13.078697Z"Dubinin, V. N."https://www.zbmath.org/authors/?q=ai:dubinin.vladimir-nSummary: The classical Pólya-Schur inequality for the logarithmic energy of a point charge distributed on a circle is generalized to the Green energy with respect to the concentric circular ring.On the Mayer series of two-dimensional Yukawa gas at inverse temperature in the interval of collapsehttps://www.zbmath.org/1483.310102022-05-16T20:40:13.078697Z"Kroschinsky, Wilhelm"https://www.zbmath.org/authors/?q=ai:kroschinsky.wilhelm"Marchetti, Domingos H. U."https://www.zbmath.org/authors/?q=ai:marchetti.domingos-h-uSummary: We prove a theorem on the minimal specific energy for a \(\pm 1\) charged particles system, interacting through a class of pair potential \(v\), that may be stated as follows: suppose \(v\) may be represented by a scale mixtures of \(d\)-dimensional Euclid's hat. If the number of particles \(n\) is even, then their interacting energy \(U_n\) divided by \(n\) is minimized by a constant \(B\) at the configurations with total charge zero and all particles collapsed to a point; if \(n\) is odd, then the ratio \(U_n/(n-1)\) is minimized by a constant \(\bar{B}=B\) at the configurations with total charge \(\pm 1\) and all particles collapsed to a point. The theorem is then used to investigate the convergence of the Mayer series for a gas of \(\pm 1\) charged particles interacting through the two-dimensional Yukawa pair potential \(v\) for inverse temperatures in the collapse interval \([4\pi ,8\pi )\). The convergence is proved in the present paper up to the second threshold \(6\pi\) using the decomposition of the Yukawa potential into scales of modified Bessel functions (standard) and into scale mixtures of Euclid's hat. Moreover, assuming that \textbf{(i)} neutral subclusters of size smaller than an odd number \(k>1\) do not collapse inside a cluster of size larger than \(k\) for \(\beta\) in the threshold interval \([8\pi (k-2)/(k-1),8\pi k/(k+1))\) and \textbf{(ii)} they satisfy a technical condition, then the Mayer series, discarding the first even coefficients of order smaller than \(k,\) converges.How to keep a spot cool?https://www.zbmath.org/1483.310112022-05-16T20:40:13.078697Z"Solynin, Alexander Yu."https://www.zbmath.org/authors/?q=ai:solynin.alexander-yuSummary: Let \(D\) be a planar domain, let \(a\) be a \textit{reference} point fixed in \(D\), and let \(b_k, k=1,\dots,n\), be \(n\) \textit{controlling} points fixed in \(D\setminus\{a\}\). Suppose further that each \(b_k\) is connected to the boundary \(\partial D\) by an arc \(l_k\). In this paper, we propose the problem of finding a shape of arcs \(l_k\), \(k=1,\dots,n\), which provides the minimum to the harmonic measure \(\omega(a,\bigcup_{k=1}^n l_k,D\setminus \bigcup_{k=1}^n l_k)\). This problem can also be interpreted as a problem on the minimal temperature at \(a\), in the steady-state regime, when the arcs \(l_k\) are kept at constant temperature \(T_1\) while the boundary \(\partial D\) is kept at constant temperature \(T_0< T_1\).
In this paper, we mainly discuss the first non-trivial case of this problem when \(D\) is the unit disk \(\mathbf{D}=\{z:|z|< 1\}\) with the reference point \(a=0\) and two controlling points \(b_1=ir\), \(b_2=-ir\), \(0< r< 1\). It appears, that even in this case our minimization problem is highly nontrivial and the arcs \(l_1\) and \(l_2\) providing minimum for the harmonic measure are not the straight line segments as it could be expected from symmetry properties of the configuration of points under consideration.Stability and the inverse gravimetry problem with minimal datahttps://www.zbmath.org/1483.310122022-05-16T20:40:13.078697Z"Isakov, Victor"https://www.zbmath.org/authors/?q=ai:isakov.victor"Titi, Aseel"https://www.zbmath.org/authors/?q=ai:titi.aseelSummary: The inverse problem in gravimetry is to find a domain \(D\) inside the reference domain \(\Omega\) from boundary measurements of gravitational force outside \(\Omega\). We found that about five parameters of the unknown \(D\) can be stably determined given data noise in practical situations. An ellipse is uniquely determined by five parameters. We prove uniqueness and stability of recovering an ellipse for the inverse problem from minimal amount of data which are the gravitational force at three boundary points. In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for natural parameters of an ellipse. To illustrate the technique, we use these equations in numerical examples with various location of measurements points on \(\partial\Omega\). Similarly, a rectangular \(D\) is considered. We consider the problem in the plane as a model for the three-dimensional problem due to simplicity.Quantitative stratification of \(F\)-subharmonic functionshttps://www.zbmath.org/1483.310132022-05-16T20:40:13.078697Z"Chu, Jianchun"https://www.zbmath.org/authors/?q=ai:chu.jianchunSummary: In this paper, we study the singular sets of $F$-subharmonic functions $u: B_{2}(0^{n})\rightarrow\mathbf{R}$, where $F$ is a subequation. The singular set $\mathcal{S}(u)\subset B_{2}(0^{n})$ has a stratification $\mathcal{S}^{0}(u)\subset\mathcal{S}^{1}(u)\subset\cdots\subset\mathcal{S}^{k}(u)\subset\cdots\subset\mathcal{S}(u)$, where $x\in\mathcal{S}^{k}(u)$ if no tangent function to $u$ at $x$ is $(k+1)$-homogeneous. We define the quantitative stratifications $\mathcal{S}_{\eta}^{k}(u)$ and $\mathcal{S}_{\eta,r}^{k}(u)$ satisfying $\mathcal{S}^{k}(u)=\cup_{\eta}\mathcal{S}_{\eta}^{k}(u)=\cup_{\eta}\cap_{r}\mathcal{S}_{\eta,r}^{k}(u)$.
When homogeneity of tangents holds for $F$, we prove that $\dim_{H}\mathcal{S}^{k}(u)\leq k$ and $\mathcal{S}(u)=\mathcal{S}^{n-p}(u)$, where $p$ is the Riesz characteristic of $F$. And for the top quantitative stratification $\mathcal{S}_{\eta}^{n-p}(u)$, we have the Minkowski estimate $\text{Vol}(B_{r}(\mathcal{S}_{\eta}^{n-p}(u)\cap B_{1}(0^{n})))\leq C\eta^{-1}(\int_{B_{1+r}(0^{n})}\Delta u)r^{p}$.
When uniqueness of tangents holds for $F$, we show that $S_{\eta}^{k}(u)$ is $k$-rectifiable, which implies $\mathcal{S}^{k}(u)$ is $k$-rectifiable.
When strong uniqueness of tangents holds for $F$, we introduce the monotonicity condition and the notion of $F$-energy. By using refined covering argument, we obtain a definite upper bound on the number of $\{\Theta(u,x)\geq c\}$ for $c> 0$, where $\Theta(u,x)$ is the density of $F$-subharmonic function $u$ at $x$.
Geometrically determined subequations $F(\mathbb{G})$ are a very important type of subequation (when $p=2$, homogeneity of tangents holds for $F(\mathbb{G})$; when $p> 2$, uniqueness of tangents holds for $F(\mathbb{G})$). By introducing the notion of $\mathbb{G}$-energy and using quantitative differentation argument, we obtain the Minkowski estimate of quantitative stratification $\text{Vol}(B_{r}(\mathcal{S}_{\eta,r}^{k}(u))\cap B_{1}(0^{n}))\leq Cr^{n-k-\eta}$.On some topological characteristics of harmonic polynomialshttps://www.zbmath.org/1483.310142022-05-16T20:40:13.078697Z"Darinskii, B. M."https://www.zbmath.org/authors/?q=ai:darinskii.boris-m"Loboda, A. V."https://www.zbmath.org/authors/?q=ai:loboda.alexander-vasilevich"Saiko, D. S."https://www.zbmath.org/authors/?q=ai:saiko.d-sSummary: In this paper, we study geometric and topological properties of harmonic homogeneous polynomials. Based on the study of zero-level lines of such polynomials on the unit sphere, we introduce the notion of their topological type. We describe topological types of harmonic polynomials up to the third degree inclusive. In the case of complex-valued harmonic polynomials, we consider distributions of their critical points in those regions on the sphere, where their real and imaginary parts have constant signs. We demonstrate that when passing from real to complex polynomials, the number of such regions increases and the maximal values of the square of the modulus of the harmonic polynomial decrease. Using the Euler formula, we make certain conclusions about the number of critical points of functions under consideration.Boundary growth rates and exceptional sets for superharmonic functions on the real hyperbolic ballhttps://www.zbmath.org/1483.310152022-05-16T20:40:13.078697Z"Hirata, Kentaro"https://www.zbmath.org/authors/?q=ai:hirata.kentaroSummary: We present a sharp upper bound of the Hausdorff dimension of an exceptional set in the unit sphere where a positive superharmonic function with respect to the hyperbolic Laplacian on the unit ball blows up faster than a prescribed order.Harmonic functions in slabs and half-spaceshttps://www.zbmath.org/1483.310162022-05-16T20:40:13.078697Z"Madych, W. R."https://www.zbmath.org/authors/?q=ai:madych.wolodymyr-r|madych.wally-rSummary: The usual solution to the Dirichlet problem for the Laplace equation \(\Delta u = 0\) in the slab \({\mathbb R}^n \times (a,b)\), where \(- \infty < a < b < \infty \), and the half-space \({\mathbb R}^n \times (0,\infty )\) involves convolution of the data with a Poisson kernel. Interestingly, the class of distributions which is convolvable with the natural Poisson kernel \(Q\) for the slab is considerably wider than that which is convolvable with the classical Poisson kernel \(P\) for the half-space. We investigate this curious phenomenon and observe that arbitrary tempered distributions can be convolved with \(Q\), resulting in functions harmonic in the slab with no greater than polynomial growth in the interior and distributionally bounded on hyperplanes parallel to the boundary. Conversely, we show that all harmonic functions in the slab which enjoy no greater than polynomial growth in the interior and are distributionally bounded on hyperplanes parallel to the boundary can be characterized as Poisson integrals of tempered distributions. In the case of the half-space we observe that the classical Poisson kernel can be modified so that the result is applicable to all tempered distributions and gives rise to harmonic functions in the half-space with the prescribed boundary values. In both cases if the boundary data is given by polynomials then so is the resulting harmonic function. In the appendix we record some additional properties of the kernel \(Q\) and offer several pertinent comments and observations.
For the entire collection see [Zbl 1470.42003].On the exact asymptotics of exit time from a cone of an isotropic \(\alpha\)-self-similar Markov process with a skew-product structurehttps://www.zbmath.org/1483.310172022-05-16T20:40:13.078697Z"Palmowski, Zbigniew"https://www.zbmath.org/authors/?q=ai:palmowski.zbigniew"Wang, Longmin"https://www.zbmath.org/authors/?q=ai:wang.longminSummary: In this paper we identify the asymptotic tail of the distribution of the exit time \(\tau_C\) from a cone \(C\) of an isotropic \(\alpha\)-self-similar Markov process \(X_t\) with a skew-product structure, that is, \(X_t\) is a product of its radial process and an independent time changed angular component \(\Theta_t\). Under some additional regularity assumptions, the angular process \(\Theta_t\) killed on exiting the cone \(C\) has a transition density that can be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asymptotic properties of the exponential functional of a killed Lévy process related to the Lamperti representation of the radial process, we prove that
\[
P_x (\tau_C >t)\sim h(x)t^{-\kappa_1}
\]
as \(t\rightarrow\infty\) for \(h\) and \(\kappa_1\) identified explicitly. The result extends the work of \textit{R. D. De Blassie} [``Remark on exit times from cones in \(\mathbb R^n\) of Brownian motion'', Probab. Theory Related Fields 79, 95--97 (1988)] and \textit{R. Bañuelos} and \textit{R. G. Smits} [Probab. Theory Relat. Fields 108, No. 3, 299--319 (1997; Zbl 0884.60037)] concerning the Brownian motion.Two inequalities for convex equipotential surfaceshttps://www.zbmath.org/1483.310182022-05-16T20:40:13.078697Z"Zhou, Yajun"https://www.zbmath.org/authors/?q=ai:zhou.yajunSummary: We establish two geometric inequalities, respectively, for harmonic functions in exterior Dirichlet problems, and for Green's functions in interior Dirichlet problems, where the boundary surfaces are smooth and convex. Both inequalities involve integrals over the mean curvature and the Gaussian curvature on an equipotential surface, and the normal derivative of the harmonic potential thereupon. These inequalities generalize a geometric conservation law for equipotential curves in dimension two, and offer solutions to two free boundary problems in three-dimensional electrostatics.Infinitely many embedded eigenvalues for the Neumann-Poincaré operator in 3Dhttps://www.zbmath.org/1483.310192022-05-16T20:40:13.078697Z"Li, Wei"https://www.zbmath.org/authors/?q=ai:li.wei.8|li.wei.10|li.wei.7|li.wei-wayne|li.wei.11|li.wei.9|li.wei.2|li.wei.4|li.wei|li.wei.3"Perfekt, Karl-Mikael"https://www.zbmath.org/authors/?q=ai:perfekt.karl-mikael"Shipman, Stephen P."https://www.zbmath.org/authors/?q=ai:shipman.stephen-pFirst-kind boundary integral equations for the Dirac operator in 3-dimensional Lipschitz domainshttps://www.zbmath.org/1483.310202022-05-16T20:40:13.078697Z"Schulz, Erick"https://www.zbmath.org/authors/?q=ai:schulz.erick"Hiptmair, Ralf"https://www.zbmath.org/authors/?q=ai:hiptmair.ralfA family of Riesz distributions for differential forms on Euclidian spacehttps://www.zbmath.org/1483.310212022-05-16T20:40:13.078697Z"Fischmann, Matthias"https://www.zbmath.org/authors/?q=ai:fischmann.matthias"Ørsted, Bent"https://www.zbmath.org/authors/?q=ai:orsted.bentSummary: In this paper, we introduce a new family of operator-valued distributions on Euclidian space acting by convolution on differential forms. It provides a natural generalization of the important Riesz distributions acting on functions, where the corresponding operators are \((-\Delta )^{-\alpha/2}\), and we develop basic analogous properties with respect to meromorphic continuation, residues, Fourier transforms, and relations to conformal geometry and representations of the conformal group.Dynamics of particles on a curve with pairwise hyper-singular repulsionhttps://www.zbmath.org/1483.310222022-05-16T20:40:13.078697Z"Hardin, Douglas"https://www.zbmath.org/authors/?q=ai:hardin.douglas-p"Saff, Edward B."https://www.zbmath.org/authors/?q=ai:saff.edward-b"Shu, Ruiwen"https://www.zbmath.org/authors/?q=ai:shu.ruiwen"Tadmor, Eitan"https://www.zbmath.org/authors/?q=ai:tadmor.eitanSummary: We investigate the large time behavior of \(N\) particles restricted to a smooth closed curve in \(\mathbb{R}^d\) and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz \(s\)-energy with \(s>1\). We show that regardless of their initial positions, for all \(N\) and time \(t\) large, their normalized Riesz \(s\)-energy will be close to the \(N\)-point minimal possible energy. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.Minimizing capacity among linear images of rotationally invariant conductorshttps://www.zbmath.org/1483.310232022-05-16T20:40:13.078697Z"Laugesen, Richard S."https://www.zbmath.org/authors/?q=ai:laugesen.richard-snyderSummary: Logarithmic capacity is shown to be minimal for a planar set having \(N\)-fold rotational symmetry (\(N \ge 3\)), among all conductors obtained from the set by area-preserving linear transformations. Newtonian and Riesz capacities obey a similar property in all dimensions, when suitably normalized linear transformations are applied to a set having irreducible symmetry group. A corollary is Pólya and Schiffer's lower bound on capacity in terms of moment of inertia.Sharp bounds for the anisotropic \(p\)-capacity of Euclidean compact setshttps://www.zbmath.org/1483.310242022-05-16T20:40:13.078697Z"Li, Ruixuan"https://www.zbmath.org/authors/?q=ai:li.ruixuan"Xiong, Changwei"https://www.zbmath.org/authors/?q=ai:xiong.changwei.1|xiong.changweiSummary: We prove various sharp bounds for the anisotropic \(p\)-capacity \(\mathrm{Cap}_{F , p}(K)\) (\(1 < p < n\)) of compact sets \(K\) in the Euclidean space \(\mathbb{R}^n\) (\(n \geq 2\)). Our results are mainly the anisotropic generalizations of some isotropic ones in [\textit{M. Ludwig} et al., Math. Ann. 350, No. 1, 169--197 (2011; Zbl 1220.26020); \textit{J. Xiao}, Ann. Henri Poincaré 17, No. 8, 2265--2283 (2016; Zbl 1345.83014); Adv. Math. 308, 1318--1336 (2017; Zbl 1361.31008); Adv. Geom. 17, No. 4, 483--496 (2017; Zbl 1387.53024)]. Key ingredients in the proofs include the inverse anisotropic mean curvature flow (IAMCF), the anisotropic Hawking mass and its monotonicity property along IAMCF for certain surfaces, and the anisotropic isocapacitary inequality.The \(L_p\) Minkowski problem for the electrostatic \(\mathfrak{p} \)-capacity for \(\mathfrak{p} \geqslant n\)https://www.zbmath.org/1483.310252022-05-16T20:40:13.078697Z"Lu, Xinbao"https://www.zbmath.org/authors/?q=ai:lu.xinbao"Xiong, Ge"https://www.zbmath.org/authors/?q=ai:xiong.geSummary: Sufficient conditions are given for the existence of solutions to the discrete \(L_p\) Minkowski problem for \(\mathfrak{p} \)-capacity when \(0<p<1\) and \(\mathfrak{p}\geqslant n\).Continuity of generalized Riesz potentials for double phase functionals with variable exponentshttps://www.zbmath.org/1483.310262022-05-16T20:40:13.078697Z"Ohno, Takao"https://www.zbmath.org/authors/?q=ai:ohno.takao"Shimomura, Tetsu"https://www.zbmath.org/authors/?q=ai:shimomura.tetsuSummary: In this note, we discuss the continuity of generalized Riesz potentials \(I_{\rho}f\) of functions in Morrey spaces \(L^{\Phi,\nu(\cdot)}(G)\) of double phase functionals with variable exponents.Hyperbolic B-potentials: properties and inversionhttps://www.zbmath.org/1483.310272022-05-16T20:40:13.078697Z"Shishkina, E. L."https://www.zbmath.org/authors/?q=ai:shishkina.elina-leonidovnaSummary: The paper is devoted to the study of the fractional integral operator which is a negative real power of the singular wave operator generated by Bessel operator, its properties and its inverse using weighted distributions.
For the entire collection see [Zbl 1470.47003].On Green's function for Laplace's equation in a rigid tubehttps://www.zbmath.org/1483.310282022-05-16T20:40:13.078697Z"Martin, P. A."https://www.zbmath.org/authors/?q=ai:martin.paul-andrew|martin.philippe-a|martin.paulo-a|martin.philippa-aSummary: A classical problem from potential theory (a point source inside a long rigid tube) is revisited. It has an extensive literature but its resolution is not straightforward: standard approaches lead to divergent integrals or require the discarding of infinite constants. We show that the problem can be solved rigorously using classical methods.Multiplicity of solutions for some \(p(x)\)-biharmonic problemhttps://www.zbmath.org/1483.310292022-05-16T20:40:13.078697Z"Ghanmi, Abdeljabbar"https://www.zbmath.org/authors/?q=ai:ghanmii.abdeljabbarSummary: This paper deals with the study of some class of non-homogeneous problems involving the \(p(x)\)-biharmonic operator. Using direct variational methods, the existence of nontrivial solution is obtained. The multiplicity of solutions is obtained by combining Ekeland's variational principle with the Mountain pass theorem. Finally, the Fountain theorem is applied to prove the existence of infinetely many solutions for the given problem.Classification criteria for regular treeshttps://www.zbmath.org/1483.310302022-05-16T20:40:13.078697Z"Nguyen, Khanh"https://www.zbmath.org/authors/?q=ai:nguyen.khanh-hoan|nguyen.khanh-hieu|nguyen.khanh-p|nguyen.khanh-quoc|nguyen.khanh-ngocSummary: We give characterizations for the parabolicity of regular trees.Collision avoidance of multiagent systems on Riemannian manifoldshttps://www.zbmath.org/1483.310312022-05-16T20:40:13.078697Z"Goodman, Jacob R."https://www.zbmath.org/authors/?q=ai:goodman.jacob-r"Colombo, Leonardo J."https://www.zbmath.org/authors/?q=ai:colombo.leonardo-jesusMean exit time for the overdamped Langevin process: the case with critical points on the boundaryhttps://www.zbmath.org/1483.310322022-05-16T20:40:13.078697Z"Nectoux, Boris"https://www.zbmath.org/authors/?q=ai:nectoux.borisSummary: Let \((X_t)_{t\geq 0}\) be the overdamped Langevin process on \(\mathbb{R}^d\) i.e. the solution of the stochastic differential equation
\[
dX_t=-\nabla f(X_t)\,dt+\sqrt{h}\,dB_t.
\]
Let \(\Omega\subset\mathbb{R}^d\) be a bounded domain. In this work, when \(X_0=x\in\Omega\), we derive new sharp asymptotic equivalents (with optimal error terms) in the limit \(h\to 0\) of the mean exit time from \(\Omega\) of the process \((X_t)_{t\geq 0}\) (which is the solution of \(\left(-\frac{h}{2}\Delta+\nabla f\cdot\nabla\right)w=1\) in \(\Omega\) and \(w=0\) on \(\partial\Omega\)), when the function \(f\to\Omega\to\mathbb{R}\) has critical points on \(\partial\Omega\) Such a setting is the one considered in many cases in molecular dynamics simulations. This problem has been extensively studied in the literature but such a setting has never been treated. The proof, mainly based on techniques from partial differential equations, uses recent spectral results from
[\textit{D. Le Peutrec} and the author, SIAM J. Math. Anal. 52, No. 1, 581--604 (2020; Zbl 1430.35175)] and its starting point is a formula from the potential theory. We also provide new sharp leveling results on the mean exit time from \(\Omega\).Some function theoretic properties of nonlinear resistive networkshttps://www.zbmath.org/1483.310332022-05-16T20:40:13.078697Z"Hattori, Tae"https://www.zbmath.org/authors/?q=ai:hattori.tae"Kasue, Atsushi"https://www.zbmath.org/authors/?q=ai:kasue.atsushi"Ohkubo, Motoki"https://www.zbmath.org/authors/?q=ai:ohkubo.motokiSummary: We consider nonlinear resistive networks. The equivalence of the Liouville property, the Khas'minskii condition and the weak maximum principle for operators of Laplacian with potential is proved, and a number of criteria for these properties are given. The parabolicity of networks is also discussed.Polyharmonic functions for finite graphs and Markov chainshttps://www.zbmath.org/1483.310342022-05-16T20:40:13.078697Z"Hirschler, Thomas"https://www.zbmath.org/authors/?q=ai:hirschler.thomas"Woess, Wolfgang"https://www.zbmath.org/authors/?q=ai:woess.wolfgangSummary: On a finite graph with a chosen partition of the vertex set into interior and boundary vertices, a \(\lambda\)-polyharmonic function is a complex function \(f\) on the vertex set which satisfies \((\lambda \cdot I- P)^nf(x) = 0\) at each interior vertex. Here, \(P\) may be the normalised adjacency matrix, but more generally, we consider the transition matrix \(P\) of an arbitrary Markov chain to which the (oriented) graph structure is adapted. After describing these ``global'' polyharmonic functions, we turn to solving the \textit{Riquier} problem, where \(n\) boundary functions are preassigned and a corresponding ``tower'' of \(n\) successive Dirichlet type problems is solved. The resulting unique solution will be polyharmonic only at those points which have distance at least \(n\) from the boundary. Finally, we compare these results with those concerning infinite trees with the end boundary, as studied by \textit{J. M. Cohen} et al. [Am. J. Math. 124, No. 5, 999--1043 (2002; Zbl 1025.31003)], and more recently, by \textit{M. A. Picardello} and the second author [Potential Anal. 51, No. 4, 541--561 (2019; Zbl 1429.31007)].
For the entire collection see [Zbl 1473.53004].Infinite Schrödinger networkshttps://www.zbmath.org/1483.310352022-05-16T20:40:13.078697Z"Nathiya, Narayanaraju"https://www.zbmath.org/authors/?q=ai:nathiya.narayanaraju"Smyrna, Chinnathambi Amulya"https://www.zbmath.org/authors/?q=ai:smyrna.chinnathambi-amulyaSummary: Finite-difference models of partial differential equations such as Laplace or Poisson equations lead to a finite network. A discretized equation on an unbounded plane or space results in an infinite network. In an infinite network, Schrödinger operator (perturbed Laplace operator, \(q\)-Laplace) is defined to develop a discrete potential theory which has a model in the Schrödinger equation in the Euclidean spaces. The relation between Laplace operator \(\Delta \)-theory and the \(\Delta_q\)-theory is investigated. In the \(\Delta_q\)-theory the Poisson equation is solved if the network is a tree and a canonical representation for non-negative \(q\)-superharmonic functions is obtained in general case.Discrete harmonic functions in the three-quarter planehttps://www.zbmath.org/1483.310362022-05-16T20:40:13.078697Z"Trotignon, Amélie"https://www.zbmath.org/authors/?q=ai:trotignon.amelieSummary: In this article we are interested in finding positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane -- resolution of a functional equation via boundary value problem using a conformal mapping -- to the three-quarter plane applying the strategy of splitting the domain into two symmetric convex cones. We obtain a simple explicit expression for the algebraic generating function of harmonic functions associated to random walks avoiding a quadrant.Homogenization of symmetric Dirichlet formshttps://www.zbmath.org/1483.310372022-05-16T20:40:13.078697Z"Tomisaki, Matsuyo"https://www.zbmath.org/authors/?q=ai:tomisaki.matsuyo"Uemura, Toshihiro"https://www.zbmath.org/authors/?q=ai:uemura.toshihiroSummary: We consider a homogenization problem for symmetric jump-diffusion processes by using the Mosco convergence and the two-scale convergence of the corresponding Dirichlet forms. Moreover, we show the weak convergence of the processes.Spectral clustering revisited: information hidden in the Fiedler vectorhttps://www.zbmath.org/1483.310382022-05-16T20:40:13.078697Z"DePavia, Adela"https://www.zbmath.org/authors/?q=ai:depavia.adela"Steinerberger, Stefan"https://www.zbmath.org/authors/?q=ai:steinerberger.stefanSummary: We study the clustering problem on graphs: it is known that if there are two underlying clusters, then the signs of the eigenvector corresponding to the second largest eigenvalue of the adjacency matrix can reliably reconstruct the two clusters. We argue that the vertices for which the eigenvector has the largest and the smallest entries, respectively, are unusually strongly connected to their own cluster and more reliably classified than the rest. This can be regarded as a discrete version of the Hot Spots conjecture and should be a useful heuristic for evaluating 'strongly clustered' versus 'liminal' nodes in applications. We give a rigorous proof for the stochastic block model and discuss several explicit examples.A survey on multiscale mollifier decorrelation of seismic datahttps://www.zbmath.org/1483.350762022-05-16T20:40:13.078697Z"Blick, C."https://www.zbmath.org/authors/?q=ai:blick.christian"Eberle, S."https://www.zbmath.org/authors/?q=ai:eberle.simon|eberle.sarahSummary: In this survey paper, we present a multiscale post-processing method in exploration. Based on a physically relevant mollifier technique involving the elasto-oscillatory Cauchy-Navier equation, we mathematically describe the extractable information within 3D geological models obtained by migration as is commonly used for geophysical exploration purposes. More explicitly, the developed multiscale approach extracts and visualizes structural features inherently available in signature bands of certain geological formations such as aquifers, salt domes etc. by specifying suitable wavelet bands.Generalized nonlocal Robin Laplacian on arbitrary domainshttps://www.zbmath.org/1483.350792022-05-16T20:40:13.078697Z"Oussaid, Nouhayla Ait"https://www.zbmath.org/authors/?q=ai:oussaid.nouhayla-ait"Akhlil, Khalid"https://www.zbmath.org/authors/?q=ai:akhlil.khalid"Aadi, Sultana Ben"https://www.zbmath.org/authors/?q=ai:ben-aadi.sultana"Ouali, Mourad El"https://www.zbmath.org/authors/?q=ai:el-ouali.mouradSummary: In this paper, we prove that it is always possible to define a realization of the Laplacian \(\Delta_{\kappa ,\theta }\) on \(L^2(\Omega )\) subject to nonlocal Robin boundary conditions with general jump measures on arbitrary open subsets of \({\mathbb{R}}^N\). This is made possible by using a capacity approach to define an admissible pair of measures \((\kappa ,\theta )\) that allows the associated form \({\mathcal{E}}_{\kappa ,\theta }\) to be closable. The nonlocal Robin Laplacian \(\Delta_{\kappa ,\theta }\) generates a sub-Markovian \(C_0\)-semigroup on \(L^2(\Omega )\) which is not dominated by the Neumann Laplacian semigroup unless the jump measure \(\theta\) vanishes.Bôcher's theorem with rough coefficientshttps://www.zbmath.org/1483.350832022-05-16T20:40:13.078697Z"Taylor, Michael"https://www.zbmath.org/authors/?q=ai:taylor.michael-eugeneLet \((\sigma,p)\in \mathbb R\times (1,\infty)\). The \(L_p\)-Sobolev space is defined by \(H^{\sigma,p}(\mathbb R^n)=\{\Lambda^{-\sigma}f:f\in L_p(\mathbb R^n)\}\), where \((\Lambda^{-\sigma}f)_ { }\widehat{ }\ (\xi)=(1+|\xi|^2)^{-\sigma/2}\hat{f}(\xi)\). \(\delta_p\) is the delta function supported at \(p\in\mathcal{O}\), an open bounded connected subset of \(\mathbb R^n\) with smooth boundary \(\partial\mathcal{O}\). The author assumes that \(u\) is a positive continuously differentiable function on \(\mathcal{O}\setminus\{p\}\), the solution of \(Lu:=\displaystyle\sum_{j,k}\partial_j(a^{jk}(x)\partial_k u)=0\), where the coefficients \(a^{jk}\) are real-valued functions satisfying \(\displaystyle\sum_{j,k}a^{jk}(x)\xi_j\xi_k\ge \lambda|\xi|^2\) for some \(\lambda>0\). Furthermore, the author supposes that \(\nabla a^{jk}\in H^{\varepsilon,n}(\mathcal{O})\cap L_r(\mathcal{O})\) for \(\varepsilon>0\) and \(r>n\). Then, he states that there exist continuously differentiable functions \(h\) on \(\mathcal{O}\) and \(A>0\) such that \(Lh=0\) on \(\mathcal{O}\) and \(u(x)=AV_p(x)+h(x)\), where \(V_p\) satisfies \(LV_p=-\delta_p\) on \(\mathcal{O}\) and \(V_{p\restriction_{\partial\mathcal{O}}}=0\).
Reviewer: Mohammed El Aïdi (Bogotá)Some coefficient estimates on real kernel \(\alpha\)-harmonic mappingshttps://www.zbmath.org/1483.350872022-05-16T20:40:13.078697Z"Long, Bo-Yong"https://www.zbmath.org/authors/?q=ai:long.boyong"Wang, Qi-Han"https://www.zbmath.org/authors/?q=ai:wang.qihanSummary: We call the solution of a kind of second order homogeneous partial differential equation as real kernel \(\alpha\)-harmonic mappings. For this class of mappings, we explore its Heinz type inequality. Furthermore, for a subclass of real kernel \(\alpha\)-harmonic mappings with real coefficients, we estimate their coefficients. At last, we study the extremal function of Schwartz type lemma for the class of real kernel \(\alpha\)-harmonic mappings.The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacityhttps://www.zbmath.org/1483.350932022-05-16T20:40:13.078697Z"Akman, Murat"https://www.zbmath.org/authors/?q=ai:akman.murat"Gong, Jasun"https://www.zbmath.org/authors/?q=ai:gong.jasun"Hineman, Jay"https://www.zbmath.org/authors/?q=ai:hineman.jay-lawrence"Lewis, John"https://www.zbmath.org/authors/?q=ai:lewis.john-l"Vogel, Andrew"https://www.zbmath.org/authors/?q=ai:vogel.andrew-lIn this interesting paper, the authors consider two potential-theoretic problems in convex geometry.
The first main result of the paper proves a Brunn-Minkowski inequality for a nonlinear capacity, \(\text{Cap}_{\mathcal{A}}\), where \(\mathcal{A}\)-capacity is associated with a nonlinear elliptic PDE whose structure is modeled by the \(p\)-Laplace equation. More precisely, let \(1<p<n\), \(0<\lambda<1\), and \(E_1\), \(E_2\) be convex compact sets with positive \(\mathcal{A}\)-capacity. Then
\[
[\text{Cap}_{\mathcal{A}}(\lambda E_1+(1-\lambda)E_2)]^{\frac{1}{n-p}}\ge\lambda[\text{Cap}_{\mathcal{A}}(E_1)]^{\frac{1}{n-p}}+(1-\lambda)[\text{Cap}_{\mathcal{A}}(E_2)]^{\frac{1}{n-p}}.
\]
Furthermore, the authors show that if equality holds for some \(E_1\) and \(E_2\), then under certain conditions on \(\mathcal{A}\), the two sets must be homothetic.
The key ingredients of the proof include the fact that \(\{x \ : \ u(x)>t\}\) is convex for \(0<t<1\), if \(u\) is a nontrivial \(\mathcal{A}\)-harmonic capacitary function for a compact, convex set \(E\). The authors also use a maximum principle argument of [\textit{R. M. Gabriel}, J. Lond. Math. Soc. 30, 388--401 (1955; Zbl 0068.08303)], and the proof of equality is inspired by ideas in [\textit{A. Colesanti} and \textit{P. Salani}, Math. Ann. 327, No. 3, 459--479 (2003; Zbl 1052.31005); \textit{M. Longinetti}, SIAM J. Math. Anal. 19, No. 2, 377--389 (1988; Zbl 0647.31001)].
Subsequently, the authors consider a Minkowski problem for a measure associated with a compact convex set \(E\) with nonempty interior and its \(\mathcal{A}\)-harmonic capacitary function in the complement of \(E\). More precisely, denoting with \(\mu_E\) this measure, the authors consider the problem of, given a finite Borel measure \(\mu\) on \(\mathbb{S}^{n-1}\), find necessary and sufficient conditions for the existence of a set \(E\) as above, with \(\mu_E=\mu\). The authors prove that necessary and sufficient conditions for existence are the same conditions in the classical Minkowski problem for volume, and also in the work [\textit{D. Jerison}, Acta Math. 176, No. 1, 1--47 (1996; Zbl 0880.35041)], which addresses electrostatic capacity. Here, the authors are inspired by ideas from [\textit{A. Colesanti} et al., Adv. Math. 285, 1511--1588 (2015; Zbl 1327.31024); \textit{D. Jerison}, Acta Math. 176, No. 1, 1--47 (1996; Zbl 0880.35041); \textit{J. L. Lewis} and \textit{K. Nyström}, J. Eur. Math. Soc. (JEMS) 20, No. 7, 1689--1746 (2018; Zbl 1397.35088); \textit{M. Venouziou} and \textit{G. C. Verchota}, Proc. Symp. Pure Math. 79, 407--422 (2008; Zbl 1160.35022)]. Finally, using the Brunn-Minkowski inequality result from the first part of this paper, the authors prove that this problem has a unique solution, up to translation when \(p\neq n-1\), and translation and dilation when \(p=n-1\).
Reviewer: Mariana Vega Smit (Bellingham)Regularity properties of the cubic biharmonic Schrödinger equation on the half linehttps://www.zbmath.org/1483.351972022-05-16T20:40:13.078697Z"Başakoğlu, Engin"https://www.zbmath.org/authors/?q=ai:basakoglu.enginSummary: In this paper we study the regularity properties of the cubic biharmonic Schrödinger equation posed on the right half line. We prove local well-posedness and obtain a smoothing result in the low-regularity spaces on the half line. In particular we prove that the nonlinear part of the solution on the half line is smoother than the initial data obtaining a full derivative gain in certain cases. Moreover, in the defocusing case, we establish global well-posedness and global smoothing in the higher order regularity spaces by making use of the global-wellposedness result of \textit{T. Özsarı} and \textit{N. Yolcu} [Commun. Pure Appl. Anal. 18, No. 6, 3285--3316 (2019; Zbl 1479.35816)] in the energy space. Also this paper improves the well-posedness result of Özsarı and Yolcu [loc. cit.] in the case of cubic nonlinearity.Modal approximation for strictly convex plasmonic resonators in the time domain: the Maxwell's equationshttps://www.zbmath.org/1483.352422022-05-16T20:40:13.078697Z"Ammari, Habib"https://www.zbmath.org/authors/?q=ai:ammari.habib-m"Millien, Pierre"https://www.zbmath.org/authors/?q=ai:millien.pierre"Vanel, Alice L."https://www.zbmath.org/authors/?q=ai:vanel.alice-lSummary: We study the possible expansion of the electromagnetic field scattered by a strictly convex metallic nanoparticle with dispersive material parameters placed in a homogeneous medium in a low-frequency regime as a sum of \textit{modes} oscillating at complex frequencies (diverging at infinity), known in the physics literature as the \textit{quasi-normal modes} expansion. We show that such an expansion is valid in the static regime and that we can approximate the electric field with a finite number of modes. We then use perturbative spectral theory to show the existence, in a certain regime, of plasmonic resonances as poles of the resolvent for Maxwell's equations with non-zero frequency. We show that, in the time domain, the electric field can be written as a sum of modes oscillating at complex frequencies. We introduce renormalised quantities that do not diverge exponentially at infinity.Melas-type bounds for the Heisenberg Laplacian on bounded domainshttps://www.zbmath.org/1483.353112022-05-16T20:40:13.078697Z"Kovařík, Hynek"https://www.zbmath.org/authors/?q=ai:kovarik.hynek"Ruszkowski, Bartosch"https://www.zbmath.org/authors/?q=ai:ruszkowski.bartosch"Weidl, Timo"https://www.zbmath.org/authors/?q=ai:weidl.timoSummary: We study Riesz means of the eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on bounded domains in \(\mathbb{R}^3\). We obtain an inequality with a sharp leading term and an additional lower order term, improving the result of \textit{A. M. Hansson} and \textit{A. Laptev} [Lond. Math. Soc. Lect. Note Ser. 354, 100--115 (2008; Zbl 1157.58008)].Multipoint Padé approximation of the psi functionhttps://www.zbmath.org/1483.410042022-05-16T20:40:13.078697Z"Sorokin, V. N."https://www.zbmath.org/authors/?q=ai:sorokin.vladimir-nSummary: The Newton interpretation of the psi function by rational functions with free poles is studied. A discrete formula for the Rodrigues determinants is obtained and the limit distribution of their zeros is found. The corresponding equilibrium problem of the theory of logarithmic potential is obtained.Hardy-Sobolev inequalities for Sobolev functions in central Herz-Morrey spaces on the unit ballhttps://www.zbmath.org/1483.460252022-05-16T20:40:13.078697Z"Mizuta, Yoshihiro"https://www.zbmath.org/authors/?q=ai:mizuta.yoshihiro"Shimomura, Tetsu"https://www.zbmath.org/authors/?q=ai:shimomura.tetsuSummary: Our aim in this paper is to establish Hardy-Sobolev inequalities for Sobolev functions and generalized Riesz potentials in central Herz-Morrey spaces on the unit ball. As an application, we obtain norm inequalities for Green potentials.Harmonic maps relative to \(\alpha \)-connections of statistical manifoldshttps://www.zbmath.org/1483.530302022-05-16T20:40:13.078697Z"Uohashi, Keiko"https://www.zbmath.org/authors/?q=ai:uohashi.keikoSummary: In this paper, we study harmonic maps relative to \(\alpha \)-connections, but not necessarily standard harmonic maps. A standard harmonic map is defined by the first variation of the energy functional of a map. A harmonic map relative to an \(\alpha \)-connection is defined by an equation similar to a first variational equation, though it is not induced by the first variation of the standard energy functional. In this paper, we define energy functionals of maps relative to \(\alpha \)-connections of statistical manifolds. Next, we show that, for harmonic maps relative to \(\alpha \)-connections, the Euler-Lagrange equations are induced by first variations of energy functionals relative to \(\alpha \)-connections.
For the entire collection see [Zbl 1470.00021].Lowest-order equivalent nonstandard finite element methods for biharmonic plateshttps://www.zbmath.org/1483.651772022-05-16T20:40:13.078697Z"Carstensen, Carsten"https://www.zbmath.org/authors/?q=ai:carstensen.carsten"Nataraj, Neela"https://www.zbmath.org/authors/?q=ai:nataraj.neelaSummary: The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the \(C^0\) interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side \(F \in H^{-2}(\Omega)\) replaced by \(F \circ (JI_{\mathrm{M}})\) and then are quasi-optimal in their respective discrete norms. The smoother \(JI_{\mathrm{M}}\) is defined for a piecewise smooth input function by a (generalized) Morley interpolation \(I_{\mathrm{M}}\) followed by a companion operator \(J\). An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [\textit{C. Carstensen} et al., ESAIM, Math. Model. Numer. Anal. 49, No. 4, 977--990 (2015; Zbl 1327.65211)] without data oscillations. This paper extends the work [\textit{A. Veeser} and \textit{P. Zanotti}, SIAM J. Numer. Anal. 56, No. 3, 1621--1642 (2018; Zbl 1412.65191)] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.One parameter family of rationally extended isospectral potentialshttps://www.zbmath.org/1483.810812022-05-16T20:40:13.078697Z"Yadav, Rajesh Kumar"https://www.zbmath.org/authors/?q=ai:yadav.rajesh-kumar"Banerjee, Suman"https://www.zbmath.org/authors/?q=ai:banerjee.suman"Kumari, Nisha"https://www.zbmath.org/authors/?q=ai:kumari.nisha"Khare, Avinash"https://www.zbmath.org/authors/?q=ai:khare.avinash"Mandal, Bhabani Prasad"https://www.zbmath.org/authors/?q=ai:mandal.bhabani-prasadSummary: We start from a given one dimensional rationally extended shape invariant potential associated with \(X_m\) exceptional orthogonal polynomials and using the idea of supersymmetry in quantum mechanics, we obtain one continuous parameter \((\lambda)\) family of rationally extended strictly isospectral potentials. We illustrate this construction by considering three well known rationally extended potentials, two with pure discrete spectrum (the extended radial oscillator and the extended Scarf-I) and one with both the discrete and the continuous spectrum (the extended generalized Pöschl-Teller) and explicitly construct the corresponding one continuous parameter family of rationally extended strictly isospectral potentials. Further, in the special case of \(\lambda=0\) and \(-1\), we obtain two new exactly solvable rationally extended potentials, namely the rationally extended Pursey and the rationally extended Abraham-Moses potentials respectively. We illustrate the whole procedure by discussing in detail the particular case of the \(X_1\) rationally extended one parameter family of potentials including the corresponding Pursey and the Abraham Moses potentials.