Recent zbMATH articles in MSC 31https://zbmath.org/atom/cc/312021-11-25T18:46:10.358925ZWerkzeugA low discrepancy sequence on graphshttps://zbmath.org/1472.051362021-11-25T18:46:10.358925Z"Cloninger, A."https://zbmath.org/authors/?q=ai:cloninger.alexander"Mhaskar, H. N."https://zbmath.org/authors/?q=ai:mhaskar.hrushikesh-nSummary: Many applications such as election forecasting, environmental monitoring, health policy, and graph based machine learning require taking expectation of functions defined on the vertices of a graph. We describe a construction of a sampling scheme analogous to the so called Leja points in complex potential theory that can be proved to give low discrepancy estimates for the approximation of the expected value by the impirical expected value based on these points. In contrast to classical potential theory where the kernel is fixed and the equilibrium distribution depends upon the kernel, we fix a probability distribution and construct a kernel (which represents the graph structure) for which the equilibrium distribution is the given probability distribution. Our estimates do not depend upon the size of the graph.Multiplicity of zeros of polynomialshttps://zbmath.org/1472.300052021-11-25T18:46:10.358925Z"Totik, Vilmos"https://zbmath.org/authors/?q=ai:totik.vilmosThe paper grew out of the known result of \textit{P. Erdős} and \textit{P. Túran} [Ann. Math. (2) 41, 162--173 (1940; Zbl 0023.02201)] on zero distributions and bounds for their multiplicities of monic polynomials with all their zeros in \([-1,1]\).
Theorem 1.1. Let \(K\) be a compact set consisting of pairwise disjoint \(C^{1+\alpha}\)-smooth Jordan curves or arcs lying exterior to each other. Given a monic polynomial \(P_n\) of degree at most \(n\) with a zero \(a\in K\) of multiplicity \(m=m(a)\), the following lower bound holds \[ \|P_n\|_k\ge e^{c\frac{m^2}{n}} (\mathrm{cap}\,K)^n, \qquad c>0,\] where \(\mathrm{cap}\,K\) is the logarithmic capacity of \(K\), \(\|\cdot\|_K\) the supremum norm on~\(K\).
In the case when \(K\) is an analytic Jordan curve or arc, the result turns out to be sharp.Pluripotential theory on Teichmüller space. I: Pluricomplex Green functionhttps://zbmath.org/1472.300182021-11-25T18:46:10.358925Z"Miyachi, Hideki"https://zbmath.org/authors/?q=ai:miyachi.hidekiIn the paper [Bull. Am. Math. Soc., New Ser. 27, No. 1, 143--147 (1992; Zbl 0766.30016)], \textit{S. L. Krushkal} announced the following result on the pluricomplex Green function on Teichmüller space:
Theorem. Let \(T_{g, m}\) be the Teichmüller space of Riemann surfaces of analytically finite-type \((g,m)\), and let \(d_T\) be the Teichmüller distance on \(T_{g, m}\). Then, the pluricomplex Green function \(g_{T_{g, m}} \) on \(T_{g, m}\) satisfies \[g_{T_{g, m}}(x, y)\log \tanh d_T(x, y)\] for \(x, y \in g_{T_{g, m}}\).
The author has a programm to investigate of the pluripotential theory on Teichmüller space. In this first one of a series of works, the author gave an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space. In comparison with the original approach by Krushkal, the strategy is more direct here. He first shows that the Teichmüller space carries a natural stratified structure of real-analytic submanifolds defined from the structure of singularities of the initial differentials of the Teichmüller mappings from a given point. Then he gives a description of the Levi form of the pluricomplex Green function using the Thurston symplectic form via Dumas' symplectic structure on the space of holomorphic quadratic differentials [\textit{D. Dumas}, Acta Math. 215, No. 1, 55--126 (2015; Zbl 1334.57020)].Potential theory on minimal hypersurfaces. I: Singularities as Martin boundarieshttps://zbmath.org/1472.300272021-11-25T18:46:10.358925Z"Lohkamp, Joachim"https://zbmath.org/authors/?q=ai:lohkamp.joachimThe author develops a detailed potential theory on (almost) minimizing hypersurfaces applicable to large classes of linear elliptic second-order operators.
Let \(H\) be an (almost) minimizing hypersurface containing the singularity set \(\Sigma \subset H\). By \(\mathcal S\)-uniformity, we can regard \(H \setminus \Sigma\) as a generalized convex set and \(\Sigma\) as its boundary. Then the author proves a generalized boundary Harnack inequality, and use it to deduce other interesing results concerning Martin theory, the Dirichlet problem and Hardy inequalities.Potential theory on Sierpiński carpets. With applications to uniformizationhttps://zbmath.org/1472.310012021-11-25T18:46:10.358925Z"Ntalampekos, Dimitrios"https://zbmath.org/authors/?q=ai:ntalampekos.dimitriosThe goal of this book is to establish a uniformization result for planar Sierpiński carpets. In this purpose it continues a tradition of research in the uniformization of metric spaces, as the author expounds in the seven-page introduction (Chapter 1).
The necessary background is prepared in Chapter 2. Let \(\Omega\subset {\mathbb C}\) be an open Jordan region, \((Q_i)_{i\in {\mathbb N}}\) a family of open Jordan regions compactly contained in \(\Omega\) with disjoint closures. If \(S:=\bar{\Omega}\setminus\cup_{i\in {\mathbb N}}Q_i\) has empty interior and is locally connected, it is called a \textit{Sierpiński carpet}, the \(Q_i\)'s its \textit{peripheral disks}. A key notion for the study is that of a \textit{Sobolev function} on \(S\). Its definition applies the concept of an \textit{upper gradient} for a function. \par Let \(g:S\cap\Omega\rightarrow {\mathbb R}\cup\{-\infty,\infty\}\). A nonnegative sequence \((\rho(Q_i))_{i\in {\mathbb N}}\) is called an \textit{upper gradient} for \(g\) if there exists a family \(\Gamma_0\) of paths in \(\Omega\) with \(\text{mod}(\Gamma_0)=0\) such that for all paths \(\gamma\not\in\Gamma_0\) in \(\Omega\) and \(x, y\in\gamma\cap S\) it holds \(g(x), g(y)\neq\pm\infty\) and \[ |g(x)-g(y)|\le\sum_{i: Q_i\cap\gamma\neq\emptyset}\rho(Q_i)\,. \] Here, the \textit{carpet modulus} \(\text{mod}(\Gamma_0)\) is defined as \(\inf\sum_{i\in {\mathbb N}\cup\{0\}}\sigma(Q_i)^2\) (\(Q_0:={\mathbb C}\setminus \bar{\Omega}\)), where \((\sigma(Q_i))_{i\in {\mathbb N}\cup\{0\}}\) is a nonnegative admissible sequence, meaning that \[ \sum_{i: Q_i\cap\gamma\neq\emptyset}\sigma(Q_i)\ge 1 \] for all \(\gamma\in \Gamma_0\) with \(\mathcal{H}^1 (\gamma\cap S)=0\) (\(\mathcal{H}\) denoting the Hausdorff measure). \par Setting \(M_{Q_i}(g)=\sup_{x\in\partial Q_i}g(x)\), \(m_{Q_i}(g)=\inf_{x\in \partial Q_i}g(x)\), \(\text{osc}_{Q_i}(g)=M_{Q_i}(g)-m_{Q_i}(g)\) (\(i\in {\mathbb N}\)), \(g\) is called a \textit{local Sobolev function} if for every open ball \(B\) relatively compact in \(\Omega\), \[ \sum_{i\in I_B} M_{Q_i}(g)^2 \text{diam}(Q_i)^2 <\infty\;\text{and}\; \sum_{i\in I_B}\text{osc}_{Q_i}(g)^2 <\infty \] (\(I_B:=\{i\in {\mathbb N}: B\cap Q_i\neq\emptyset\}\)) and \((\text{osc}_{Q_i} (g))_{i\in {\mathbb N}}\) is an upper gradient for \(g\). If these inequalities hold for the full sums over \(i\in {\mathbb N}\), the Sobolev property is termed \textit{global}. \par Now, a local Sobolev function \(u\) is called \textit{carpet-harmonic} if for every open set \(V\) relatively compact in \(\Omega\) and every Sobolev function \(\zeta\) supported on \(V\) it holds \[ D_V(u)\le D_V(u+\zeta)\,, \] the so-called Dirichlet energy functional being defined by \(D_V(f)=\sum_{i\in I_V}\text{osc}_{Q_i}(f)^2 \in [0,\infty ]\). The author presents several properties of carpet-harmonic functions, as the Caccioppoli inequality, for instance (Section 8 of Chapter 2).
\par In the sequel we indicate how this notion of harmonicity is applied to obtain the uniformization result. After fixing four points on \(\partial\Omega\), this boundary is decomposed into closed sides \(\Theta_1,\ldots,\Theta_4\), enumerated counterclockwise, where \(\Theta_1\) and \(\Theta_3\) are opposite. Calling a Sobolev function \(g\) on \(S\), continuous up to \(\partial\Omega\), \textit{admissible for the free boundary problem} if \(g|_{\Theta_1}=0\) and \(g|_{\Theta_3}=1\), the author shows that there is a unique carpet-harmonic function \(u\) that minimizes \(D_\Omega (g)\) over all admissible functions \(g\). This \(u\) is continuous up to \(\partial\Omega\), \(u|_{\Theta_1}=0\), and \(u|_{\Theta_3}=1\).
\par By ``integrating'' the ``gradient'' of \(u\) along its level sets, as the author puts it, the \textit{conjugate function} \(v\) of \(u\) is defined (Section 6 of Chapter 3). The function \(v:S \rightarrow [0, D_\Omega (u)]\) is continuous, \(v|_{\Theta_2}=0\), \(v|_{\Theta_4}=D_\Omega (u)\), but it is unclear if this \(v\) is carpet-harmonic. In any case, \(f:=(u,v)\) is the desired uniformization map. It maps \(S\) homeomorphically onto a so-called square Sierpiński carpet, that is, a carpet whose peripheral disks are squares and whose underlying open Jordan region is a rectangle. A condition, however, for this uniformization result is that the peripheral disks \(Q_i\) be uniformly quasiround (there exists \(K_0\ge 1\) such that for each \(Q_i\) there are concentric balls \(B(x,r)\), \(B(x,R)\) with \(B(x,r)\subset Q_i\subset B(x,R)\) and \(\frac{R}{r}\le K_0\)) and uniformly Ahlfors 2-regular (there exists \(K_1>0\) such that for every \(Q_i\) and every \(B(x,r)\) with \(x\in Q_i\) and \(r<\text{diam}(Q_i)\) it holds \(\mathcal{H}^2 (B(x,r)\cap Q_i)\ge K_1 r^2\)). Furthermore, this \(f\) has a property called \textit{carpet-quasiconformality}. Finally, the author presents certain refinements under additional geometric assumptions.Properties of normal harmonic mappingshttps://zbmath.org/1472.310022021-11-25T18:46:10.358925Z"Deng, Hua"https://zbmath.org/authors/?q=ai:deng.hua"Ponnusamy, Saminathan"https://zbmath.org/authors/?q=ai:ponnusamy.saminathan"Qiao, Jinjing"https://zbmath.org/authors/?q=ai:qiao.jinjingLet \(\mathbb{D}\) denote the open unit disk in the complex plane and let \(\rho\) denote the hyperbolic distance on \(\mathbb{D}\). A harmonic mapping \(f:\mathbb{D}\to \mathbb{C}\) is said to be normal if \(f\) is Lipschitz as mapping from the hyperbolic disk to the extended plane endowed with the chordal distance \(\chi\), that is, \(\sup_{z\not=w} \chi(f(z), f(w))/\rho(z,w)<\infty\). The condition turns out to be equivalent to \[ \alpha:=\sup_{z \in \mathbb{D}} (1-|z|)^2 f^\#(z)<\infty, \] where \(f^\#=(|h'|+|g'|)/(1+|f|^2)\) if \(f=h+\overline{g}\) with \(f, g\) holomorphic in \(\mathbb{D}\), and in the case of meromorphic functions \(f\) equivalent to the normality of the family \(\{f \circ \varphi: \varphi \in\mathrm{Aut}(\mathbb{D})\}\). In this paper the authors present necessary, sufficient and also necessary and sufficient conditions for a harmonic \(f\) to be normal, and they discuss a maximum principle in terms of \(\alpha\). Moreover, they show that the existence of an asymptotic value at a point \(\xi\in \partial \mathbb{D}\) already implies the existence of the non-tangential limit at \(\xi\) and they investigate sequences of normal harmonic functions. Finally, a five-point theorem for sense-preserving harmonic functions is proven.The family of level sets of a harmonic functionhttps://zbmath.org/1472.310032021-11-25T18:46:10.358925Z"Ding, Pisheng"https://zbmath.org/authors/?q=ai:ding.pishengThe main result, Theorem 1, characterizes families of level-sets of harmonic functions of two variables without critical points in terms of the curvature \(\kappa\) of a level set \(\gamma_t\). The proof relies on the basic fact (Lemma 3) that a \(C^2\) function \(f\) on a domain \(\Omega\subset \mathbb{R}^2\) without critical points is harmonic if and only if \(\kappa\equiv \frac{D_N^2 f}{D_Nf}\) on \(\Omega\), where \(N=\nabla f/|\nabla f|\) is the unit normal field, \(D_N f\), \(D_N^2 f\) are the first and second directional derivatives of \(f\) along \(N\).
The author also extends Theorem 1 to higher dimensions for the case when the level sets are diffeomorphic to \(\mathbb{R}^{n-1}\) (Theorem 4).Harmonic hereditary convexityhttps://zbmath.org/1472.310042021-11-25T18:46:10.358925Z"Koh, Ngin-Tee"https://zbmath.org/authors/?q=ai:koh.ngin-teeThe article investigates the images of the circles \(\mathbb{T}_r=\{z\in\mathbb{C}:|z|=r<1\}\) and the annulus \(\mathbb{A}_\rho=\{z\in\mathbb{C}:0<\rho<|z|<1\}\) by a harmonic diffeomorphism \(h\) and by an energy-minimal diffeomorphism \(h\) of \(\mathbb{A}_\rho\) onto a doubly connected region \(\Omega\) bounded by two convex Jordan curves in the plane. Conditions on \(h\) under which \(h(\mathbb{T}_r)\) is a strictly convex curve and \(h(\mathbb{D}_r)\) is a strictly convex domain for \(\rho<r<1\) are obtained.Continuity of condenser capacity under holomorphic motionshttps://zbmath.org/1472.310052021-11-25T18:46:10.358925Z"Pouliasis, Stamatis"https://zbmath.org/authors/?q=ai:pouliasis.stamatisA condenser in the complex plane \(\mathbb{C}\) is a pair \((E,F)\) where \(E\) and \(F\) are non-empty disjoint compact subsets of \(\mathbb{C}\). A holomorphic motion of a set \(A \subset \mathbb{C}\), parameterized by a domain \(D \subset \mathbb{C}\) containing \(0\), is a map \(f:D \times A \mapsto \mathbb{C}\) such that \(f(\cdot,z )\) is holomorphic in \(D\) for any fixed \(z\in A\), \(f(\lambda,\cdot ):=f_\lambda (\cdot)\) is an injection for any fixed \(\lambda \in D\) and \(f(0,\cdot )\) is the identity on \(A\). If \((E,F)\) is a condenser with positive capacity and \(f\) is a holomorphic motion of \(E\cup F\) parameterized by a domain \(D\) containing \(0\), then \((f_\lambda (E),f_\lambda (F))\) is also a condenser. In the paper under review the author proves that the capacity of \((f_\lambda (E),f_\lambda (F))\) is a continuous subharmonic function on \(D\). Moreover, he shows that the equilibrium measure of \((f_\lambda (E),f_\lambda (F))\) is continuous with respect to weak-star convergence.
A condenser \((E,F)\) is called a ring if both \(E\) and \(F\) are connected and \(\mathbb{C} \backslash (E\cup F)\) is a doubly connected domain. One way to characterize uniform perfectness is the following. A compact set \(K\subset \mathbb{C}\) is uniformly perfect if and only if the supremum of the equilibrium energy of all the rings that separate \(K\) is finite. Let \(P(K)\) denote this supremum. If \(K\) is a uniformly perfect compact set and \(f\) is a holomorphic motion parameterized by a bounded domain \(D\) containing \(0\), then the author finds an upper and a lower estimate for \(P(f_\lambda (K))\) involving the Harnack distance. The paper is well organized and helps the reader to follow it.Corrigendum to: ``Integral equations method for solving a biharmonic inverse problem in detection of Robin coefficients''https://zbmath.org/1472.310062021-11-25T18:46:10.358925Z"Abdelhak, Hadj"https://zbmath.org/authors/?q=ai:hadj.abdelhak"Saker, Hacene"https://zbmath.org/authors/?q=ai:saker.haceneA typo in the authors' paper [ibid. 160, 436--450 (2021; Zbl 1459.31002)] is corrected.A proof of the Khavinson conjecturehttps://zbmath.org/1472.310072021-11-25T18:46:10.358925Z"Liu, Congwen"https://zbmath.org/authors/?q=ai:liu.congwenThe author gives a complete proof of the validity of the Khavinson conjecture. In order to state the conjecture, let \(h^\infty\) be the space of bounded harmonic functions on the unit ball \(\mathbb{B}^n\) of \(\mathbb{R}^n\), with \(n \geq 3\). For \(x \in \mathbb{B}^n\) we denote by \(C(x)\) the smallest number such that
\[
|\nabla u(x)| \leq C(x)\sup_{y \in \mathbb{B}^n}|u(y)|
\]
for all \(u \in h^\infty\). Similarly, for \(x\in \mathbb{B}^n\) and \(l\in \partial \mathbb{B}^n\), we denote by \(C(x,l)\) the smallest number such that
\[
|\langle\nabla u(x),l \rangle | \leq C(x,l)\sup_{y \in \mathbb{B}^n}|u(y)|
\]
for all \(u \in h^\infty\). As it is well known, both constants are finite. The Khavinson conjecture states that for \(x \in \mathbb{B}^n \setminus \{0\}\) we have
\[
C(x)=C\left(x,\frac{x}{|x|}\right)\, .
\]
The author shows the validity of the conjecture, by considering an equivalent optimization problem and by solving such a problem in terms of the Gegenbauer polynomials.Weak estimates for the maximal and Riesz potential operators in central Herz-Morrey spaces on the unit ballhttps://zbmath.org/1472.310082021-11-25T18:46:10.358925Z"Mizuta, Yoshihiro"https://zbmath.org/authors/?q=ai:mizuta.yoshihiro"Ohno, Takao"https://zbmath.org/authors/?q=ai:ohno.takao"Shimomura, Tetsu"https://zbmath.org/authors/?q=ai:shimomura.tetsuThis nice paper introduces the weak central Herz-Morrey spaces \(WH^{p(\cdot),q,\omega}(\mathbf B)\) and \(WH^{p^\ast(\cdot),q,\omega}(\mathbf B)\) with \( p^\ast(x) = p(x)N/(N-\alpha p(x))\) on the Euclidean unit ball \(\mathbf B\) and shows the boundedness of the generalized maximal operator \(M_\beta\) and the Riesz potential operator \(I_\alpha\) from the non-homogeneous central Herz-Morrey space \(H^{p(\cdot),q,\omega}(\mathbf B)\) to \(WH^{p(\cdot),q,\omega}(\mathbf B)\) (Theorem 3.10) and \(WH^{p^\ast(\cdot),q,\omega}(\mathbf B)\) (Theorem 4.1), respectively.Approximate tangents, harmonic measure, and domains with rectifiable boundarieshttps://zbmath.org/1472.310092021-11-25T18:46:10.358925Z"Mourgoglou, Mihalis"https://zbmath.org/authors/?q=ai:mourgoglou.mihalisThis article discusses the connection between approximate tangents, harmonic measures and domains with rectifiable boundaries. In the first result of the paper it is established that if \(E\subset\mathbb{R}^{n+1}\) is closed, \(0<s<1/3\) and \(\mathcal{T}_m(E)\subset E\) be the set of all points \(x\in E\) such that:
(i) there exists an \(s\)-approximate tangent \(m\)-plane \(V_x\) for \(E\) at \(x\);
(ii) \(E\) satisfies the weak lower Ahlfors-David \(m\)-regularity condition at \(x\).
Then, there exists a countable collection of bounded Lipschitz graphs \(\{\Gamma_j\}_{j\geq 1}\) so that \(\mathcal{T}_m(E)\subset \bigcup_{j\geq 1}\Gamma_j\). In particular, \(\mathcal{T}_m(E)\) is \(m\)-rectifiable.
In the second result of the article it is obtained that if \(0<s<1/\sqrt{90}\), then there exist two countable collections of bounded Lipschitz domains \(\{\Omega_j^\pm\}_{j\geq 1}\) such that \(\Omega_j^+\cap \Omega_j^-=\emptyset\), \(\mathcal{T}_n(E)\cap \Omega_j^+=\mathcal{T}_n\cap\Omega_j^-\) and \(\mathcal{T}_m(E)\subset \bigcup_{j\geq 1}\Omega_j^\pm\).
Further characterizations of the countable collections of bounded Lipschitz domains \(\{\Omega_j^\pm\}_{j\geq 1}\) are provided in the article.Duality between range and no-response tests and its application for inverse problemshttps://zbmath.org/1472.310102021-11-25T18:46:10.358925Z"Lin, Yi-Hsuan"https://zbmath.org/authors/?q=ai:lin.yi-hsuan"Nakamura, Gen"https://zbmath.org/authors/?q=ai:nakamura.gen"Potthast, Roland"https://zbmath.org/authors/?q=ai:potthast.roland-w-e"Wang, Haibing"https://zbmath.org/authors/?q=ai:wang.haibingThe authors show the duality between range and no-response tests for an inverse boundary value problem for the Laplace equation in \(\Omega \setminus \overline{D}\) with an unknown obstacle \(D\) whose closure is contained in \(\Omega\). They consider the boundary value problem
\[
\left\{ \begin{array}{ll} \Delta u=0 & \mbox{in}\ \Omega \setminus \overline{D}\, ,\\
u=0 & \mbox{on}\ \partial D \, ,\\
u=f & \mbox{on}\ \partial \Omega \, . \end{array} \right.
\]
The Cauchy data is the pair made by Dirichlet datum \(f\) and the normal derivative \(\partial_\nu u_{|\partial \Omega}\). The inverse problem consists into identifying the unknown obstacle \(D\), knowing the Cauchy data \(\{f, \partial_\nu u_{|\partial \Omega}\}\).
The authors prove that there is a duality between the range test (RT) and the no-response test (NRT) for the inverse boundary value problem. As an application, they show that either using the RT or NRT, we can reconstruct the obstacle \(D\) from the Cauchy data if the solution \(u\) does not have any analytic extension across \(\partial D\).Correction to: ``On self-adjointness of symmetric diffusion operators''https://zbmath.org/1472.310112021-11-25T18:46:10.358925Z"Robinson, Derek W."https://zbmath.org/authors/?q=ai:robinson.derek-wFrom the text: Due to a typesetting error, the sentence just below equation (31) has been published incorrectly in the original publication [the author, ibid. 21, No. 1, 1089--1116 (2021; Zbl 1469.31029)]. The correct sentence is given.Positive harmonic functions on the Heisenberg group. IIhttps://zbmath.org/1472.310122021-11-25T18:46:10.358925Z"Benoist, Yves"https://zbmath.org/authors/?q=ai:benoist.yvesThis article discusses extremal positive harmonic functions for finitely supported measures on the discrete Heisenberg group. The author establishes that extremal positive harmonic functions are proportional either to characters or to translates of induced from characters.
For Part I, see [the author, ``Positive Harmonic Functions on the Heisenberg group. I'', Preprint, \url{arXiv:1907.05041}].A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected setshttps://zbmath.org/1472.310132021-11-25T18:46:10.358925Z"Björn, Anders"https://zbmath.org/authors/?q=ai:bjorn.anders"Björn, Jana"https://zbmath.org/authors/?q=ai:bjorn.janaThis article establishes that every Sobolev function in \(W^{1,p}_{\mathrm{loc}}(U)\) on a \(p\)-quasiopen set \(U\subset \mathbb{R}^n\) with almost everywhere vanishing \(p\)-fine gradient is almost everywhere constant if and only if \(U\) is \(p\)-quasiconnected. The approach relies on the theory of Newtonian Sobolev spaces on metric measure spaces.Cheeger's energy on the harmonic Sierpinski gaskethttps://zbmath.org/1472.310142021-11-25T18:46:10.358925Z"Bessi, Ugo"https://zbmath.org/authors/?q=ai:bessi.ugoThe author studies the harmonic Sierpinski gasket equipped with Kusuoka's measure. By using properties of the Lyapunov exponent of the gasket, the author gives a different proof of the result that the natural Dirichlet form on the gasket coincides with Cheeger's energy, which was first proved by \textit{P. Koskela} and \textit{Y. Zhou} [Adv. Math. 231, No. 5, 2755--2801 (2012; Zbl 1253.53035)].Corrigendum to: ``Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces''https://zbmath.org/1472.310152021-11-25T18:46:10.358925Z"Björn, Anders"https://zbmath.org/authors/?q=ai:bjorn.anders"Björn, Jana"https://zbmath.org/authors/?q=ai:bjorn.janaCorrigendum to the authors' paper [ibid. 266, No. 1, 44--69 (2019; Zbl 1420.31002)].A uniqueness property for analytic functions on metric measure spaceshttps://zbmath.org/1472.310162021-11-25T18:46:10.358925Z"Łysik, Grzegorz"https://zbmath.org/authors/?q=ai:lysik.grzegorzLet \((X, \rho)\) be a metric space and let \(\mu\) and \(\nu\) be Borel regular measures that are positive on non-empty open sets and finite on bounded sets. Let \(\Omega\) be an open subset of \(X\). A function \(u \in C(\Omega)\) is said to be \((X, \rho, \mu, \nu)\)-analytic on \(\Omega\) if there exist functions \(u_l \in C(\Omega), l \in {\mathbb{N}}\) and \(\varepsilon \in C(\Omega; {\mathbb{R}}_{+})\) such that \[M_{X}(u; x,R) = \sum_{l=0}^{\infty} u_l(x)R^l\] locally uniformly in \(\{(x,R): x \in \Omega, \, 0 \le R < \varepsilon(x) \}\). Here \(M_{X}\) is the solid mean value function with respect to the measures \(\mu\) and \(\nu\) defined for \(u \in C(\Omega)\), \(x \in \Omega\) and \(0< R <\mbox{ dist}_{\rho}(x, \Omega)\) by \[M_{X}(u; x,R)=\frac{1}{\nu(B_{\rho}(x,R))} \int_{B_{\rho}(x,R)} u(y) \, d\mu(y).\]
In his previous paper [Ann. Acad. Sci. Fenn., Math. 43, No. 1, 475--482 (2018; Zbl 1387.26049)] the author proved that this property characterizes real analytic functions in Euclidean spaces.
This paper considers \((X, \rho, \mu, \nu)\)-analytic functions in asymmetric normed spaces and in proper locally uniquely geodesic asymmetric metric spaces, where in addition, for any ball \(B(x,R)=B_{\rho}(x,R)\) with small enough radius \(R\) and \(0 < c < 1\), the mapping that maps the point \(y \in B(x,R)\) to the unique point \(\widetilde{y}\) which lies on the geodesic connecting \(y\) and \(x\) and satisfies \(\rho(x, \widetilde{y}) = c\rho(x, y)\) is a homeomorphism.
The main result proved by the author in this paper is the following uniqueness property: if \((X,\rho)\) is a proper locally uniquely geodesic asymmetric metric space as above or an asymmetric strictly convex normed space over \({\mathbb{R}}^n\) and \(u\) is an \((X, \rho, \mu, \nu)\)-analytic function that vanishes on a non-empty open subset of the connected set \(X\), then \(u\) vanishes on the whole set \(X\).Biharmonic problem with Dirichlet and Steklov-type boundary conditions in weighted spaceshttps://zbmath.org/1472.351322021-11-25T18:46:10.358925Z"Matevossian, H. A."https://zbmath.org/authors/?q=ai:matevossian.hovik-a|matevosyan.o-aSummary: The uniqueness of solutions of a biharmonic problem with Dirichlet and Steklov-type boundary conditions in the exterior of a compact set are studied under the assumption that the generalized solution of this problem has a finite Dirichlet integral with a weight \(| x |^a\). Depending on the parameter \(a\), uniqueness (non-uniqueness) theorems are proved and exact formulas for calculating the dimension of the solution space of this biharmonic problem are found.Asymptotic behavior of integral functionals for a two-parameter singularly perturbed nonlinear traction problemhttps://zbmath.org/1472.351872021-11-25T18:46:10.358925Z"Falconi, Riccardo"https://zbmath.org/authors/?q=ai:falconi.riccardo"Luzzini, Paolo"https://zbmath.org/authors/?q=ai:luzzini.paolo"Musolino, Paolo"https://zbmath.org/authors/?q=ai:musolino.paoloSummary: We consider a nonlinear traction boundary value problem for the Lamé equations in an unbounded periodically perforated domain. The edges lengths of the periodicity cell are proportional to a positive parameter \(\delta \), whereas the relative size of the holes is determined by a second positive parameter \(\varepsilon \). Under suitable assumptions on the nonlinearity, there exists a family of solutions \(\{ u (\varepsilon , \delta , \cdot ) \}_{( \varepsilon , \delta ) \in ]0, \varepsilon^{\prime}[ \times ]0, \delta^{\prime}[}\). We analyze the asymptotic behavior of two integral functionals associated to such a family of solutions when the perturbation parameter pair \((\varepsilon , \delta )\) is close to the degenerate value \((0, 0)\).Multiple results to some biharmonic problemshttps://zbmath.org/1472.352002021-11-25T18:46:10.358925Z"Tang, Xingdong"https://zbmath.org/authors/?q=ai:tang.xingdong"Zhang, Jihui"https://zbmath.org/authors/?q=ai:zhang.jihuiSummary: We study a nonlinear elliptic problem defined in a bounded domain involving biharmonic operator together with an asymptotically linear term. We establish at least three nontrivial solutions using the topological degree theory and the critical groups.Energy scattering for radial focusing inhomogeneous bi-harmonic Schrödinger equationshttps://zbmath.org/1472.353622021-11-25T18:46:10.358925Z"Saanouni, Tarek"https://zbmath.org/authors/?q=ai:saanouni.tarekSummary: This note studies the asymptotic behavior of global solutions to the fourth-order Schrödinger equation
\[
i\dot{u}+\Delta^2 u+F(x,u)=0.
\]
Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to [\textit{B. Dodson} and \textit{J. Murphy}, Proc. Am. Math. Soc. 145, No. 11, 4859--4867 (2017; Zbl 1373.35287)].Existence of the gauge for fractional Laplacian Schrödinger operatorshttps://zbmath.org/1472.354342021-11-25T18:46:10.358925Z"Frazier, Michael W."https://zbmath.org/authors/?q=ai:frazier.michael-w"Verbitsky, Igor E."https://zbmath.org/authors/?q=ai:verbitsky.igor-eSummary: Let \(\Omega\subseteq\mathbb{R}^n\) be an open set, where \(n\geq 2\). Suppose \(\omega\) is a locally finite Borel measure on \(\Omega\). For \(\alpha\in (0,2)\), define the fractional Laplacian \((-\Delta)^{\alpha/2}\) via the Fourier transform on \(\mathbb{R}^n\), and let \(G\) be the corresponding Green's operator of order \(\alpha\) on \(\Omega\). Define \(T(u)=G(u\omega)\). If \(\Vert T\Vert_{L^2(\omega)\rightarrow L^2(\omega)}<1\), we obtain a representation for the unique weak solution \(u\) in the homogeneous Sobolev space \(L^{\alpha/2,2}_0(\Omega)\) of
\[
(-\Delta)^{\alpha/2} u=u\omega+\nu\text{ on }\Omega,\quad u=0\text{ on }\Omega^c,
\]
for \(\nu\) in the dual Sobolev space \(L^{-\alpha/2,2}(\Omega)\). If \(\Omega\) is a bounded \(C^{1,1}\) domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when \(\nu=\chi_{\Omega}\). These estimates are used to study the existence of a solution \(u_1\) (called the ``gauge'') of the integral equation \(u_1=1+G(u_1\omega)\) corresponding to the problem
\[
(-\Delta)^{\alpha/2}u=u\omega\text{ on }\Omega,\quad u\geq 0\text{ on }\Omega,\quad u=1\text{ on }\Omega^c.
\]
We show that if \(\Vert T\Vert<1\), then \(u_1\) always exists if \(0<\alpha<1\). For \(1\leq\alpha<2\), a solution exists if the norm of \(T\) is sufficiently small. We also show that the condition \(\Vert T\Vert <1\) does not imply the existence of a solution if \(1<\alpha<2\). The condition \(\Vert T\Vert\leq 1\) is necessary for the existence of \(u_1\) for all \(0<\alpha\leq 2\).Shape holomorphy of the Calderón projector for the Laplacian in \(\mathbb{R}^2\)https://zbmath.org/1472.450112021-11-25T18:46:10.358925Z"Henríquez, Fernando"https://zbmath.org/authors/?q=ai:henriquez.fernando"Schwab, Christoph"https://zbmath.org/authors/?q=ai:schwab.christophThe authors establish the holomorphic dependence of the Calderón projector for the Laplace equation on a collection of sufficiently smooth Jordan curves in the Cartesian Euclidean plan. To be precise, they establish holomorphy of the domain-to-operator map associated to the Calderón projector.Recent advances in \(L^p\)-theory of homotopy operator on differential formshttps://zbmath.org/1472.580012021-11-25T18:46:10.358925Z"Ding, Shusen"https://zbmath.org/authors/?q=ai:ding.shusen"Shi, Peilin"https://zbmath.org/authors/?q=ai:shi.peilin"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.7|wang.yong.2|wang.yong.6|wang.yong.10|wang.yong|wang.yong.8|wang.yong.3|wang.yong.1|wang.yong.9|wang.yong.5Summary: The purpose of this survey paper is to present an up-to-date account of the recent advances made in the study of \(L^p\)-theory of the homotopy operator applied to differential forms. Specifically, we will discuss various local and global norm estimates for the homotopy operator \(T\) and its compositions with other operators, such as Green's operator and potential operator.State dependent Hamiltonian delay equations and Neumann one-formshttps://zbmath.org/1472.580072021-11-25T18:46:10.358925Z"Frauenfelder, Urs"https://zbmath.org/authors/?q=ai:frauenfelder.urs-adrianThe author proves critical point results for the action functionals involving Hamiltonian terms with a state dependent delay. The studied functionals are related to time dependent perturbations of the symplectic form. Results regarding Arnold type conjecture about lower bounds on the number of periodic orbits for Hamiltonian systems are given too.Geometry of uniform spanning forest components in high dimensionshttps://zbmath.org/1472.600182021-11-25T18:46:10.358925Z"Barlow, Martin T."https://zbmath.org/authors/?q=ai:barlow.martin-t"Járai, Antal A."https://zbmath.org/authors/?q=ai:jarai.antal-aSummary: We study the geometry of the component of the origin in the uniform spanning forest of \(\mathbb{Z}^d\) and give bounds on the size of balls in the intrinsic metric.Some properties of the potential-to-ground state map in quantum mechanicshttps://zbmath.org/1472.813222021-11-25T18:46:10.358925Z"Garrigue, Louis"https://zbmath.org/authors/?q=ai:garrigue.louisThe author considers properties of the map from potential to the ground state in many-body quantum mechanics. External potentials \(v\in L^p + L^\infty\) and interaction potentials \(w \in L^p + L^\infty\) for \(p> \max(2d/3, 2)\) where \(d\) is the dimension of the underlying space are considered. The first result is that the space of binding potentials is path-connected. Then the author shows that the map from potentials to the ground state is locally weak-strong continuous and that its differential is compact. This implies that the Kohn-Sham inverse problem in Density Functional Theory is ill-posed on a bounded set.