Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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  • 2020-11-01

    The exponential growth and decay properties for solutions to elliptic equations in unbounded cylinders

    Lidan Wang, Lihe Wang, Chunqin Zhou

    Abstract : In this paper, we classify all solutions bounded from below to uniformly elliptic equations of second order in the form of $Lu(\mathbf{x})=a_{ij}(\mathbf{x})D_{ij}u(\mathbf{x})+b_{i}(\mathbf{x})D_{i}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x})$ or $Lu(\mathbf{x})=D_{i}(a_{ij}(\mathbf{x})$ $D_{j}u(\mathbf{x}))+b_{i}(\mathbf{x})D_{i}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x})$ in unbounded cylinders. After establishing that the Aleksandrov maximum principle and boundary Harnack inequality hold for bounded solutions, we show that all solutions bounded from below are linear combinations of solutions, which are sums of two special solutions that exponential growth at one end and exponential decay at the another end, and a bounded solution that corresponds to the inhomogeneous term $f$ of the equation.

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  • 2021-05-01

    A generalized simple formula for evaluating Radon-Nikodym derivatives over paths

    Dong Hyun Cho

    Abstract : Let $C[0,T]$ denote a generalized analogue of Wiener space, the space of real-valued continuous functions on the interval $[0,T]$. Define $Z_{\vec e,n}:C[0,T]\to\mathbb R^{n+1}$ by \begin{align*} Z_{\vec e,n}(x)=\left(x(0),\int_0^Te_1(t)dx(t),\ldots,\int_0^Te_n(t)dx(t)\right), \end{align*} where $e_1, \ldots,e_n$ are of bounded variations on $[0,T]$. In this paper we derive a simple evaluation formula for Radon-Nikodym derivatives similar to the conditional expectations of functions on $C[0,T]$ with the conditioning function $Z_{\vec e,n}$ which has an initial weight and a kind of drift. As applications of the formula, we evaluate the Radon-Nikodym derivatives of various functions on $C[0,T]$ which are of interested in Feynman integration theory and quantum mechanics. This work generalizes and simplifies the existing results, that is, the simple formulas with the conditioning functions related to the partitions of time interval $[0,T]$.

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  • 2021-03-01

    A van der Corput type lemma for oscillatory integrals with H\"older amplitudes and its applications

    Hussain Al-Qassem, Leslie Cheng, Yibiao Pan

    Abstract : We prove a decay estimate for oscillatory integrals with \linebreak H\"older amplitudes and polynomial phases. The estimate allows us to answer certain questions concerning the uniform boundedness of oscillatory singular integrals on various spaces.

  • 2021-03-01

    Rigidity characterizations of complete Riemannian manifolds with $\alpha$-Bach-flat

    Guangyue Huang, Qianyu Zeng

    Abstract : For complete manifolds with $\alpha$-Bach tensor (which is defined by \eqref{1-Int-2}) flat, we provide some rigidity results characterized by some point-wise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moveover, some Einstein metrics have also been characterized by some $L^{\frac{n}{2}}$-integral inequalities. Furthermore, we also give some rigidity characterizations for constant sectional curvature.

  • 2020-11-01

    Lower order eigenvalues for the bi-drifting Laplacian on the Gaussian shrinking soliton

    Lingzhong Zeng

    Abstract : It may very well be difficult to prove an eigenvalue inequality of Payne-P\'{o}lya-Weinberger type for the bi-drifting Laplacian on the bounded domain of the general complete metric measure spaces. Even though we suppose that the differential operator is bi-harmonic on the standard Euclidean sphere, this problem still remains open. However, under certain condition, a general inequality for the eigenvalues of bi-drifting Laplacian is established in this paper, which enables us to prove an eigenvalue inequality of Ashbaugh-Cheng-Ichikawa-Mametsuka type (which is also called an eigenvalue inequality of Payne-P\'{o}lya-Weinberger type) for the eigenvalues with lower order of bi-drifting Laplacian on the Gaussian shrinking soliton.

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  • 2021-09-01

    On the ratio of biomass to total carrying capacity in high dimensions

    Junyoung Heo, Yeonho Kim

    Abstract : This paper is concerned with a reaction-diffusion logistic model. In \cite{L06}, Lou observed that a heterogeneous environment with diffusion makes the total biomass greater than the total carrying capacity. Regarding the ratio of biomass to carrying capacity, Ni \cite{HN16} raised a conjecture that the ratio has a upper bound depending only on the spatial dimension. For the one-dimensional case, Bai, He, and Li \cite{BHL16} proved that the optimal upper bound is $3$. Recently, Inoue and Kuto \cite{IK20} showed that the supremum of the ratio is infinity when the domain is a multi-dimensional ball. In this paper, we generalized the result of \cite{IK20} to an arbitrary smooth bounded domain in $\mathbb{R}^n, n \geq 2$. We use the sub-solution and super-solution method. The idea of the proof is essentially the same as the proof of \cite{IK20} but we have improved the construction of sub-solutions. This is the complete answer to the conjecture of Ni.

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  • 2021-01-01

    Every abelian group is the class group of a ring of Krull type

    Gyu Whan Chang

    Abstract : Let $Cl(A)$ denote the class group of an arbitrary integral domain $A$ introduced by Bouvier in 1982. Then $Cl(A)$ is the ideal class (resp., divisor class) group of $A$ if $A$ is a Dedekind or a Pr\"ufer (resp., Krull) domain. Let $G$ be an abelian group. In this paper, we show that there is a ring of Krull type $D$ such that $Cl(D) = G$ but $D$ is not a Krull domain. We then use this ring to construct a Pr\"ufer ring of Krull type $E$ such that $Cl(E) = G$ but $E$ is not a Dedekind domain. This is a generalization of Claborn's result that every abelian group is the ideal class group of a Dedekind domain.

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  • 2020-11-01

    Complete $f$-moment convergence for extended negatively dependent random variables under sub-linear expectations

    Chao Lu, Rui Wang, Xuejun Wang, Yi Wu

    Abstract : In this paper, we investigate the complete $f$-moment convergence for extended negatively dependent (END, for short) random variables under sub-linear expectations. We extend some results on complete $f$-moment convergence from the classical probability space to the sub-linear expectation space. As applications, we present some corollaries on complete moment convergence for END random variables under sub-linear expectations.

  • 2020-07-01

    On the $C$-projective vector fields on Randers spaces

    Mehdi Rafie-Rad, Azadeh Shirafkan

    Abstract : A characterization of the $C$-projective vector fields on a Randers space is presented in terms of ${\bf\Xi}$-curvature. It is proved that the ${\bf\Xi}$-curvature is invariant for $C$-projective vector fields. The dimension of the algebra of the $C$-projective vector fields on an $n$-dimensional Randers space is at most $n(n+2)$. The generalized Funk metrics on the $n$-dimensional Euclidean unit ball $\mathbb{B}^n(1)$ are shown to be explicit examples of the Randers metrics with a $C$-projective algebra of maximum dimension $n(n+2)$. Then, it is also proved that an $n$-dimensional Randers space has a $C$-projective algebra of maximum dimension $n(n+2)$ if and only if it is locally Minkowskian or (up to re-scaling) locally isometric to the generalized Funk metric. A new projective invariant is also introduced.

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  • 2021-11-01

    Radial symmetry of positive solutions to a class of fractional Laplacian with a singular nonlinearity

    Linfen Cao, Xiaoshan Wang

    Abstract : In this paper, we consider the following nonlocal fractional Laplacian equation with a singular nonlinearity $$ (-\Delta)^{s}u(x)=\lambda u^{\beta}(x)+a_{0}u^{-\gamma}(x), ~ x\in \mathbb{R}^{n}, $$ where $0

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July, 2022
Vol.59 No.4

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