2017年度秋季総合分科会(於:山形大学)

SUMMARY: The study of nonharmonic Fourier series was initiated by Paley and Wiener (1934). We take up the problems of the stability of Riesz basis properties of complex exponential systems in \(L^2[-\pi ,\pi ]\). We consider the sequence \(\{\lambda _{n}\}\) with perturbations of some subsequence of integers and investigate whether the system \(\{e^{i\lambda _{n}t}\}\) becomes a Riesz basis in \(L^2[-\pi ,\pi ]\). We use some stability theorems and the criterion obtained by B. S. Pavlov for \(\{e^{i\lambda _{n}t}\}\) to become a Riesz basis. Finally, we refer to the existences of the bases which are not Riesz bases.

SUMMARY: This talk is concerned with solvability of Vlasov–Poisson equations in a half-space with external magnetic field. In 2013, local-in-time existence of solutions to the equation was proved by Skubachevskii. However, the result shows the fact that the existence time \(T\) is exponentially small, which means the plasma can reach a wall and melt it in the extremely short time. The purpose of this talk is to an obtain existence result for the equation with a very large time \(T\). Moreover this talk provides global-in-time solvability for the equation with a more strict condition for the magnetic force whose direction is horizontal to the wall.

SUMMARY: In this talk, we consider a backward self-improvement property of the reverse Hölder inequality with increasing support \begin{equation*} \left ( \frac {1}{\mu (B)} \int _{B} u^{s} \, d \mu \right )^{1 / s} \leq C \left ( \frac {1}{\mu (2B)} \int _{2B} u^{p} \, d \mu \right )^{1 / p}, \end{equation*} where \(B = B(x_{0}, R)\) and \(2B = B(x_{0}, 2R)\) are balls in a metric measure space \((X, d, \mu )\), \(u\) is a nonnegative function on \(2B\) and \(0 < p < s \leq \infty \) and \(C\) are positive constants.

SUMMARY: Let \((\Omega , \mathcal {A}, \mu ), \mu (\Omega )=+\infty ,\) be an infinite measure space and \(L_0=L_0(\Omega , \mathcal {A}, \mu )\) be the space of all real valued measurable functions on \((\Omega , \mathcal {A}, \mu )\). We introduce the closed linear subspace \(M_0\) of \(L_0\) generated by the integrable step functions. We give a characterization of \(M_0\).

SUMMARY: The Vitali theorem for uniformly integrable functions is fundamental in Lebesgue integration theory and contains other important convergence theorems for abstract Lebesgue integral. The purpose of this talk is to formulate Vitali type theorems for the Choquet integral and its symmetric and asymmetric extensions with respect to a nonadditive measure. The bounded convergence theorem and the dominated convergence theorem for Choquet integrals are obtained as corollaries to our Vitali type theorems.

SUMMARY: Fixed points of spherically nonspreading mappings in complete geodesic metric spaces with curvature bounded above.

In this talk, we propose the concepts of spherically nonspreading mappings and firmly spherically nonspreading mappings in complete \(\textup {CAT}(1)\) spaces and obtain fixed point and convergence theorems for them. The resolvent of a proper lower semicontinuous convex function is a typical example of these mappings. As applications, we study the problem of minimizing convex functions in such spaces.

SUMMARY: Throughout this talk, let \(H\) be a Hilbert space and let \(f : H\rightarrow (-\infty ,\infty ]\) be a proper lower semicontinuous convex function. We assume that \(f\) is bounded below. The proximal point algorithm is an approximation method for finding a minimizer of \(f\). In this talk, we consider the convergence rate of the proximal point point algorithm.

SUMMARY: We shall characterize the uniform non-squareness of the \(\psi \)-direct sum \((X_1\oplus \cdots \oplus X_N)_{\psi }\) of Banach spaces \(X_{1},\ldots ,X_{N},\) where \(\psi \) is a convex function on the N-simplex \(\Delta _{N}\) satisfying certain conditions. To do this we shall introduce a new class of convex functions.

SUMMARY: We will talk about the redefinition of \(\tau \)-distance.

SUMMARY: In this talk, we prove attractive point theorems for nonlinear mappings. Using the ideas of attractive points and acute points, we also prove weak and strong convergence theorems for nonlinear mappings by some iterative methods.

SUMMARY: In this talk, we extend the concept of mixed monotone mappings and then we consider certain fixed point theorems for a pair of mappings in metric spaces with a partial ordering. As an application, we study existence of solutions for the following fourth-order two-point boundary value problems for elastic beam equations.

SUMMARY: We discuss the boundedness of the commutators \([b,T]\) and \([b,I_{\rho }]\) on generalized Morrey spaces with variable growth condition, where \(T\) is a Calderón–Zygmund operator, \(I_{\rho }\) is a generalized fractional integral operator and \(b\) is a function in generalized Campanato spaces with variable growth condition.

SUMMARY: We consider some variations on wavelet reconstruction formulae. An alternative formula was considered by Lebedeva and Postnikov in 2014. One of the aims of the talk is to give a multidimensional version of their formula.

SUMMARY: In this talk, we discuss the boundedness of the fractional maximal operator between weighted \(L^{p}\)-spaces with different weights. The \(B_{p}\)-condition which is introduced by Pérez is also necessary and sufficient condition for the boundedness of the fractional maximal operator. As application of this theorem is related to the classical weighted inequality of Fefferman–Stein.

SUMMARY: In this talk, we first introduce the \(d\)-dimensional weighted Hausdorff content with arbitrary weight on \({\mathbb R}^n\). Then we establish the Fefferman–Stein type inequalities for the fractional maximal operator with the weighted Hausdorff content. Further, we discuss the boundedness of the fractional maximal operator on Choquet–Lorentz spaces.

SUMMARY: We give the sharp boundedness result for bilinear pseudo-differential operators in \(L^p \times L^q\) to \(L^r\), \(1/p+1/q=1/r\le 1\), in the case that the symbols satisfy the Hörmander condition with \(0\le \rho <1\).

SUMMARY: This work is concerned with the question that “how far does small chemotactic interaction perturb the Fisher–KPP dynamics?”. A chemotaxis system with logistic source was studied by e.g., Winkler (2010, 2014) and Zheng (2016). However, there are still many open problems about the chemotaxis system. On the other hand, the Fisher–KPP system has been studied extensively. Thus the development of this work will enable us to see new properties of solutions for the chemotaxis system. The main result of this talk gives convergence of solutions for the chemotaxis system to solutions for the Fisher–KPP system on bounded convex domains.

SUMMARY: This talk is concerned with global existence of solutions to a Keller–Segel–(Navier–)Stokes system with singular sensitivity. In the fluid-free case, Winkler established global existence of classical solutions under some condition in 2011, and Fujie showed that the global solutions are bounded in 2015. However, a Keller–Segel system with singular sensitivity coupled with a Navier–Stokes equation has not been studied. The main result of this talk gives that the same condition assumed in Winkler’s result (2011) leads to global existence in the system.

SUMMARY: The existence and uniqueness problem for a degenerate parabolic equation with dynamic boundary condition is discussed. Follows from the previous works by A. Damlamian (1977), the abstract theory of evolution equation, governed by the subdifferential of proper, lower-semicontinuous and convex functional, can be applied. Then, the suitable setting of function spaces and duality mapping is needed. One of the key point is the assumption of growth condition for the maximal monotone graph which characterizes the original degenerate diffusion.

SUMMARY: In this talk, we treat some parabolic problem with related to the quasi-variational inequality. The unknown functions \(u=u(t,x)\) and \(\sigma =\sigma (t,x)\) describe the displacement and stress, respectively in the one-dimensional interval \((0,L)\). The system stands for the hardening problem that the materials are harden by plasticity. That is derived from the perfect plasticity model introduced by Duvaut–Lions. In the perfect plasticity model, the function which stands for threshold value in the plastic deformation is a constant. In this talk, we discuss the solvability for the above model under the situation that threshold function depending upon time or unknown function.

SUMMARY: This talk deals with nonlinear diffusion equations under Neumann boundary conditions in a unbounded domain with smooth bounded boundary. Recently, Kurima–Yokota (2017) proved existence of solutions to these equations with growth conditions for diffusion terms. The present work asserts that we can solve the original problem by passing to the limit in the approximate problem without growth conditions in a unbounded domain.

SUMMARY: The well-posedness for a system of partial differential equations and dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk \(\Omega \) and on the boundary \(\Gamma \). The Poisson equation for the chemical potential, the Allen–Cahn equation for the order parameter in the bulk \(\Omega \) are considered as auxiliary conditions for solving the Cahn–Hilliard equation on the boundary \(\Gamma \).

SUMMARY: We discuss a new class of doubly nonlinear evolution equations governed by time-dependent subdifferentials in uniformly convex Banach spaces, and establish an abstract existence result of solutions. Also, we give some applications to nonlinear PDEs with gradient constraint for time-derivatives.

SUMMARY: In this talk, we consider coupled system of nonlinear PDEs. The system consists of an Allen–Cahn type equation with singular diffusion in a bounded spatial domain \( \Omega \), and another Allen–Cahn type equation on the smooth boundary \( \partial \Omega \). The coupled PDEs are transmitted via the dynamic boundary condition. The objective of this study is to achieve a mathematical treatment to analyze the systems for singular diffusion equations and the dynamic boundary conditions. Now, the results concerned with the well-posedness of the system, involved in the representation of solution and comparison principal and the continuous association between solutions to our system and those in regular systems, are reported in forms of some Main Theorems.

SUMMARY: In this talk, we consider a system of one-dimensional elliptic type boundary value problems, denoted by (S\(_\infty \)). The system corresponds to a one-dimensional steady-state problem for the mathematical model of grain boundary motion, proposed by [Kobayashi et -al, Phys. D, 140 (2000), 141–150], and one of characteristics is in the point that the inhomogeneous Dirichlet type boundary condition is imposed for the crystalline orientation. On this basis, we set the objectives of the talk as follows: (A) to show the structures of all steady-state solutions, including physically-important ones; (B) to clarify the base-structure of steady-states with the physical importance; (C) the verification of the SBV-regularity for the steady-state solutions.

SUMMARY: We consider a system of partial differential equations describing a mass conservation law for moisture in a porous medium. This type of systems can be found in concrete carbonation process and already proposed and studied by Kumazaki–Aiki. In the system the relationship between the relative humidity and the degree of saturation is described by a play operator. In this talk we consider a real time control problem for the above system. The aim of the problem is to control a solution of the system by putting a multi-valued operator into the differential equation. Here, I will discuss about a physical background for the control and establish the existence of a solution to the control problem.

SUMMARY: We consider a stationary problem of a certain reaction diffusion system arising from a nuclear reactor model, which consists of two unknown functions representing the neutron density and the temperature in nuclear reactors. In Gu–Wang (1994, 1996), they studied this problem with some boundary conditions (homogeneous Dirichlet–Dirichlet conditions and homogeneous Neumann–Robin conditions) and prove the solvability and the uniqueness of ordered positive solution. In this talk, we impose Robin and power type nonlinear boundary conditions on the problem and show the existence and the uniqueness results similar to the previous results. We rely on Krasnoselskii’s type fixed point theorem due to lack of the variational structure.

SUMMARY: We consider the following complex Ginzburg–Landau equation, (CGL)\(_-\): \[ u_t(t, x) - (\lambda + i\alpha )\Delta u - (\kappa + i\beta )|u|^{q - 2}u - \gamma u = f(t, x) \quad \mbox {on}\ [0, T) \times \Omega , \] where \(\lambda , \kappa > 0\); \(\alpha , \beta , \gamma \in \mathbb {R}\); \(i\) denotes the imaginary unit; \(T > 0\); \(f: [0, T) \times \Omega \to \mathbb {C}\) is a given external force and \(\Omega \) is general, possibly unbounded domain. Our approach to (CGL)\(_-\) is to regard our equation as a parabolic equation in a product Hilbert space \(({\rm L}^2(\Omega ))^2\) over \(\mathbb {R}\) with \(-\lambda \Delta u\) being a principal term, and when \(\kappa > 0\), our nonlinear term should be treated as a non-monotone perturbation. For a general domain \(\Omega \), it is difficult to handle such kind of perturbations, because of the lack of compactness.

SUMMARY: We consider the initial value problem (CP) for degenerate parabolic-elliptic systems with variable coefficients. The systems are coupled with strongly degenerate parabolic equations and elliptic equations. Strongly degenerate parabolic equations are regarded as a linear combination of the time-dependent conservation laws (quasilinear hyperbolic equations) and the porous medium type equations (nonlinear degenerate parabolic equations). Thus, this equation has both properties of hyperbolic equations and those of parabolic equations. In this talk, we formulate entropy solutions to (CP) and show the existence and uniqueness of the solutions.

SUMMARY: In this talk, we propose a mathematical model for moisture swelling process in concrete materials. Moisture swelling process appear in, for instance, frost damage in concrete materials which is a nonlinear phenomenon to give rise to crack inside of concrete. Our model consists of a diffusion equation for moisture in a one microscopic hole of concrete and a free boundary problem for the front of moisture region. In this talk, we discuss the existence and uniqueness of a solution of this problem.

SUMMARY: In this talk, we deal with existence of solutions to Vlasov–Poisson systems in a half-space with external magnetic force horizontal to a wall. In 2013, Skubachevskii gives local-in-time solvability to the system. Moreover, in 2017, global-in-time solutions were obtained by effectively using the magnetic force whose direction is horizontal to the wall. This talk provides an existence result for the system where the magnetic force has angle error in the vertical direction.