Recent zbMATH articles in MSC 31https://zbmath.org/atom/cc/312024-02-15T19:53:11.284213ZWerkzeugSpectral decimation for a graph-directed fractal pairhttps://zbmath.org/1526.280042024-02-15T19:53:11.284213Z"Cao, Shiping"https://zbmath.org/authors/?q=ai:cao.shiping"Qiu, Hua"https://zbmath.org/authors/?q=ai:qiu.hua"Tian, Haoran"https://zbmath.org/authors/?q=ai:tian.haoran"Yang, Lijian"https://zbmath.org/authors/?q=ai:yang.lijianSummary: We introduce a graph-directed pair of planar self-similar sets that possess fully symmetric Laplacians. For these two fractals, due to Shima's celebrated criterion, we point out that one admits the spectral decimation by the canonic graph approximation and the other does not. For the second fractal, we adjust to choosing a new graph approximation guided by the directed graph, which still admits spectral decimation. Then we make a full description of the Dirichlet and Neumann eigenvalues and eigenfunctions of both of these two fractals.Boundary behaviour of open, light mappings in metric measure spaceshttps://zbmath.org/1526.300292024-02-15T19:53:11.284213Z"Cristea, Mihai"https://zbmath.org/authors/?q=ai:cristea.mihai.1|cristea.mihaiThe article is devoted to the study of mappings of metric spaces. The author considers mappings that satisfy a Poletsky-type weight inequality. The properties of the mappings are largely determined by the properties of the corresponding weight. Therefore, the author introduces a certain function that controls the behavior of this weight and assumes that the weight \(p\)-modulus of the family of paths passing through the point is equal to zero. Under these conditions, the author formulates and proves several results concerning the local and boundary behavior. In particular, he obtains results on the continuous extension of mappings to a point of a totally disconnected set, a theorem on the normality (equicontinuity) of families of such mappings, gives a partial description of the cluster set of mappings and the set of their asymptotic limits.
Reviewer: Evgeny Sevost'yanov (Zhitomir)Laplacian growth on branched Riemann surfaceshttps://zbmath.org/1526.300322024-02-15T19:53:11.284213Z"Gustafsson, Björn"https://zbmath.org/authors/?q=ai:gustafsson.bjorn"Lin, Yu-Lin"https://zbmath.org/authors/?q=ai:lin.yu-linThis interesting book is devoted to the Laplacian growth on Riemann surfaces. The main topics of this book are the Laplacian growth of simply connected domains in the complex plane and recent extensions obtained by its authors to the branched Riemann surfaces. There are several applications of these results in Physics, such as those related to the string equation and Hamiltonian systems.
From the classical point of view, the Laplacian growth or the Hele-Shaw problem describes the evolution (the expanding or shrinking) of a domain in the complex plane, whose boundary is evolving in the normal direction with a velocity proportional to the harmonic measure of the boundary.
The book is dedicated to the memory of Makoto Sakai, who made valuable and deep contributions to the study of extremal problems for analytic functions on Riemann surfaces, the theory of quadrature domains, and other areas of the potential theory.
The book is divided into ten chapters followed by a glossary, a list of references and an index.
In the first chapter the authors introduce the Laplacian growth and make an overview of the main topics of the book. The main subject is related to the loss of univalence of the conformal map from the unit disk onto a non-star-shaped domain, and to the construction of branched Riemann surfaces.
The second chapter is devoted to the Polubarinova-Galin and Löwner-Kufarev equations. It starts with basic results in the univalent case. The dynamics and subordination are also discussed. The last part of this chapter is devoted to the connection between the Polubarinova-Galin and Löwner-Kufarev equations in the non-univalent case. The Polubarinova-Galin equation is given by
\[
\mathrm{Re }\, \left[\dot{f}(\xi ,t)\overline{\xi f(\xi,t)}\right]=q(t),\ \ \xi \in \partial \mathbb D,
\]
where \(f(\cdot, t)\) is a conformal map of the unit disk \(\mathbb D\), such that \(f(0,t)=0\) and \(f'(0,t)>0\). In addition, \(q(t)\) is a given real-valued function representing the strength of the source/sink in the classical Hele-Shaw problem. The Löwner-Kufarev equation has the form
\[
\dot{f}(\xi ,t)=\xi f'(\xi ,t)P(\xi,t),\ \ \xi \in \mathbb{D},
\]
where
\[
P(\xi,t)=\frac{1}{2\pi i}\int _{\partial \mathbb D}\frac{q(t)}{|f'(z,t)|^2}\frac{z+\xi }{z-\xi }\frac{dz}{z},\ \ \xi \in \mathbb D,
\]
and \(f\) has a similar meaning as above.
In the third chapter the authors introduce and study weak solutions of the Polubarinova-Galin equation and their properties in terms of partial balayage related to quadrature formulas for subharmonic functions. The inverse balayage and results related to more general Laplacian evolutions are also presented.
The next chapter is concerned with weak and strong solutions on Riemann surfaces. In this chapter the authors extend the notions of Laplacian growth and partial balayage to Riemann surfaces. Useful examples are also presented.
Chapter five is devoted to global simply connected weak solutions. Theorem 5.1, which is the fundamental result of this chapter, asserts that starting with a simply connected domain in the complex plane with an analytic boundary, the corresponding Laplacian growth can be continued at any time as an evolution family of simply connected domains on some branched Riemann surface.
Chapter six studies the general structure of solutions of the Polubarinova-Galin equation and of the Löwner-Kufarev equation, when the derivative of the conformal map \(f\) is a rational function. The direct approach and the approach based on quadrature Riemann surfaces are used in the analysis developed in this chapter.
Chapter seven presents various interesting examples of Laplacian evolutions of a cardioid. There are studied both cases of the univalent and nonunivalent evolutions.
Chapter eight is concerned with the analysis of the Polubarinova-Galin equation and Laplacian growth in the context of the string equation. To this end, the string equation for univalent conformal maps is investigated.
Chapter nine deals with the Laplacian evolutions within a Hamiltonian framework, especially with the dependence of the conformal map on harmonic moments and other parameters in Hamiltonian descriptions.
Chapter ten presents a deep analysis of the Polubarinova-Galin equation in the context of the string equation for some rational functions.
This book is a valuable contribution to the modern theory of Laplacian growth. It contains many useful and interesting results, together with a rigorous analysis of all treated problems.
Reviewer: Mirela Kohr (Cluj-Napoca)Critical measures on higher genus Riemann surfaceshttps://zbmath.org/1526.300542024-02-15T19:53:11.284213Z"Bertola, Marco"https://zbmath.org/authors/?q=ai:bertola.marco"Groot, Alan"https://zbmath.org/authors/?q=ai:groot.alan"Kuijlaars, Arno B. J."https://zbmath.org/authors/?q=ai:kuijlaars.arno-b-jSummary: Critical measures in the complex plane are saddle points for the logarithmic energy with external field. Their local and global structure was described by Martínez-Finkelshtein and Rakhmanov. In this paper we start the development of a theory of critical measures on higher genus Riemann surfaces, where the logarithmic energy is replaced by the energy with respect to a bipolar Green's kernel. We study a max-min problem for the bipolar Green's energy with external fields \(\mathrm{Re} \; V\) where \(dV\) is a meromorphic differential. Under reasonable assumptions the max-min problem has a solution and we show that the corresponding equilibrium measure is a critical measure in the external field. In a special genus one situation we are able to show that the critical measure is supported on maximal trajectories of a meromorphic quadratic differential. We are motivated by applications to random lozenge tilings of a hexagon with periodic weightings. Correlations in these models are expressible in terms of matrix valued orthogonal polynomials. The matrix orthogonality is interpreted as (partial) scalar orthogonality on a Riemann surface. The theory of critical measures will be useful for the asymptotic analysis of a corresponding Riemann-Hilbert problem as we outline in the paper.On Cauchy problem solution for a harmonic function in a simply connected domainhttps://zbmath.org/1526.310012024-02-15T19:53:11.284213Z"Shirokova, E. A."https://zbmath.org/authors/?q=ai:shirokova.elena-a"Ivanshin, P. N."https://zbmath.org/authors/?q=ai:ivanshin.pyotr-nSummary: Here we present an investigation of the Cauchy problem solvability for the Laplace equation in a simply connected plane domain. The investigation is reduced to solution of two singular integral equations. If the problem is resolvable, its solution can be restored via the integral Cauchy formula. Examples of the solvable and unsolvable problems are presented. The construction involves the auxiliary approximate conformal mapping.Uniqueness results for solutions of continuous and discrete PDEhttps://zbmath.org/1526.351312024-02-15T19:53:11.284213Z"Malinnikova, Eugenia"https://zbmath.org/authors/?q=ai:malinnikova.eugeniaSummary: We give an overview of some recent results on unique continuation property ``at infinity'' for solutions of elliptic and dispersive PDE and their discrete counterparts. The proofs of most of the results are given in previous works written with coauthors.
For the entire collection see [Zbl 1519.00033].The Heinz type inequality, Bloch type theorem and Lipschitz characteristic of polyharmonic mappingshttps://zbmath.org/1526.351412024-02-15T19:53:11.284213Z"Chen, Shaolin"https://zbmath.org/authors/?q=ai:chen.shaolin|chen.shaolin.1Summary: Suppose that \(f\) satisfies the following: (1) the polyharmonic equation \(\varDelta^m f = \varDelta (\varDelta^{m - 1} f) = \varphi_m\) \((\varphi_m \in \mathcal{C} (\overline{\mathbb{B}^n}, \mathbb{R}^n))\), (2) the boundary conditions \(\varDelta^0 f = \varphi_0, \varDelta^1 f = \varphi_1, \dots, \varDelta^{m - 1} f = \varphi_{m - 1}\) on \(\mathbb{S}^{n - 1}\) (\(\varphi_j \in \mathcal{C} (\mathbb{S}^{n - 1}, \mathbb{R}^n)\) for \(j \in \{0, 1, \dots, m - 1\}\) and \(\mathbb{S}^{n - 1}\) denotes the boundary of the unit ball \(\mathbb{B}^n\)), and (3) \(f (0) = 0\), where \(n \geq 3\) and \(m \geq 1\) are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mappings satisfying the polyharmonic equation with the above Dirichlet boundary value conditions. Furthermore, we establish a Bloch type theorem of mappings satisfying the above polyharmonic equation, which gives an answer to an open problem in [\textit{S. Chen} and \textit{S. Ponnusamy}, Indag. Math., New Ser. 30, No. 6, 1087--1098 (2019; Zbl 1443.31002)]. Additionally, we show that if \(f\) is a \(K\)-quasiconformal self-mapping of \(\mathbb{B}^n\) satisfying the above polyharmonic equation, then \(f\) is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as \(K \to 1^+\) and \(\|\varphi_j\|_\infty \to 0^+\) for \(j \in \{1, \dots, m\}\).Singular value decomposition of the wave forward operator with radial variable coefficientshttps://zbmath.org/1526.352292024-02-15T19:53:11.284213Z"Moon, Minam"https://zbmath.org/authors/?q=ai:moon.minam"Hur, Injo"https://zbmath.org/authors/?q=ai:hur.injo"Moon, Sunghwan"https://zbmath.org/authors/?q=ai:moon.sunghwanSummary: Photoacoustic tomography (PAT) is a novel and promising technology in hybrid medical imaging that involves generating acoustic waves in the object of interest by stimulating electromagnetic energy. The acoustic wave is measured outside the object. One of the key mathematical problems in PAT is the reconstruction of the initial function that contains diagnostic information from the solution of the wave equation on the surface of the acoustic transducers. Herein, we propose a wave forward operator that assigns an initial function to obtain the solution of the wave equation on a unit sphere. Under the assumption of the radial variable speed of ultrasound, we obtain the singular value decomposition of this wave forward operator by determining the orthonormal basis of a certain Hilbert space comprising eigenfunctions. In addition, numerical simulation results obtained using the continuous Galerkin method are utilized to validate the inversion resulting from the singular value decomposition.Fixed angle inverse scattering for sound speeds close to constanthttps://zbmath.org/1526.353202024-02-15T19:53:11.284213Z"Ma, Shiqi"https://zbmath.org/authors/?q=ai:ma.shiqi"Potenciano-Machado, Leyter"https://zbmath.org/authors/?q=ai:potenciano-machado.leyter"Salo, Mikko"https://zbmath.org/authors/?q=ai:salo.mikkoSummary: We study the fixed angle inverse scattering problem of determining a sound speed from scattering measurements corresponding to a single incident wave. The main result shows that a sound speed close to constant can be stably determined by just one measurement. Our method is based on studying the linearized problem, which turns out to be related to the acoustic problem in photoacoustic imaging. We adapt the modified time-reversal method from \textit{P. Stefanov} and \textit{G. Uhlmann} [Inverse Probl. 25, No. 7, Article ID 075011, 16 p. (2009; Zbl 1177.35256)] to solve the linearized problem in a stable way, and we use this to give a local uniqueness result for the nonlinear inverse problem.Accessibility and porosity of harmonic measure at bifurcation locushttps://zbmath.org/1526.370542024-02-15T19:53:11.284213Z"Graczyk, Jacek"https://zbmath.org/authors/?q=ai:graczyk.jacek"Świątek, Grzegorz"https://zbmath.org/authors/?q=ai:swiatek.grzegorzSummary: We study hyperbolic geodesics running from \(\infty\) to a generic point, by the harmonic measure with the pole at \(\infty \), on the boundary of the connectedness locus \(\mathcal{M}_d\) for unicritical polynomials \(f_c(z)=z^d+c\). It is known that a generic parameter \(c\in \partial\mathcal{M}_d\) is not accessible within a John angle and \(\partial\mathcal{M}_d\) spirals round them infinitely many times in both directions. We prove that almost every point from \(\partial\mathcal{M}_d\) is asymptotically accessible by a flat angle with apperture decreasing slower than \((\log \circ \dots \circ \log\operatorname{dist}(c,\partial\mathcal{M}_d))^{-1}\) for any iterate of \(\log \). This is a consequence of an iterated large deviation estimate for exponential distribution. Additionally, for an arbitrary \(\beta >0\), the bifurcation locus is not \(\beta \)-porous on a set of scales of positive density along almost every external ray with respect to the harmonic measure.On (Fejér-)Riesz type inequalities, Hardy-Littlewood type theorems and smooth modulihttps://zbmath.org/1526.420092024-02-15T19:53:11.284213Z"Chen, Shaolin"https://zbmath.org/authors/?q=ai:chen.shaolin"Hamada, Hidetaka"https://zbmath.org/authors/?q=ai:hamada.hidetakaSummary: The purpose of this paper is to develop some methods to study (Fejér-)Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some sharp Riesz type inequalities of pluriharmonic functions on bounded symmetric domains. The obtained results extend the main results in [\textit{D. Kalaj}, Trans. Am. Math. Soc. 372, No. 6, 4031--4051 (2019; Zbl 1422.30002)]. Next, some Hardy-Littlewood type theorems of holomorphic and pluriharmonic functions on John domains are established, which give analogies and extensions of a result in [\textit{G. H. Hardy} and \textit{J. E. Littlewood}, J. Reine Angew. Math. 167, 405--423 (1932; JFM 58.0333.03)]. Furthermore, we establish a Fejér-Riesz type inequality on pluriharmonic functions in the Euclidean unit ball in \(\mathbb{C}^n\), which extends the main result in [\textit{P. Melentijević} and \textit{V. Božin}, Potential Anal. 54, No. 4, 575--580 (2021; Zbl 1460.31005)]. Additionally, we also discuss the Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions. Consequently, we improve and extend the corresponding results in \textit{K. M. Dyakonov} [Acta Math. 178, No. 2, 143--167 (1997; Zbl 0898.30040)], \textit{G. H. Hardy} and \textit{J. E. Littlewood}, [Math. Z. 34, 403--439 (1931; JFM 57.0476.01)], \textit{K. M. Dyakonov} [Adv. Math. 187, No. 1, 146--172 (2004; Zbl 1056.30018)] and \textit{M. Pavlović} [Rev. Mat. Iberoam. 23, No. 3, 831--845 (2007; Zbl 1148.31003)].Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitonshttps://zbmath.org/1526.530392024-02-15T19:53:11.284213Z"Wu, Jia-Yong"https://zbmath.org/authors/?q=ai:wu.jiayong"Wu, Peng"https://zbmath.org/authors/?q=ai:wu.pengLet \(f\) be a smooth function on \((M^n,g)\), an \(n\)-dimensional complete non-compact smooth manifold endowed with a Riemannian metric \(g\), \(\mathrm{Hess}\, f\) be the Hessian of \(f\), \(\mathrm{Ric}\) and \(R\) be the Ricci curvature tensor and the scalar curvature of \(M^n\), respectively. The triple \((M^n,g,f)\) is a shrinker when \(\displaystyle \mathrm{Ric}+\mathrm{Hess}\,f=\frac{g}{2}\).
Let \(\mathcal{H}^f_d(M^n)\) be the linear space of \(f\)-harmonic functions with polynomial growth of degree at most \(d\). Then the authors prove that if \[ \int_{M^n}(4\pi)^{-n/2}e^{-f}dv=e^{\mu}\] and \(R+|\nabla f|^2=f\) (where \(\mu=\mu(g,1)\) is Perelman's entropy functional), and if there is a constant \(c_0\) such that \(R(x).r^2(x,o)\le c_0,\) where \(r(x,o)\) is the distance from \(x\in M^n\) to a fixed point \(o\in M^n\), then \(\dim\mathcal{H}_d^f(M^n)\le C(n)(c_0+1)^{n/2}e^{-\mu}4^d\) for some constant \(C(n)\).
Reviewer: Mohammed El Aïdi (Bogotá)