2018年度秋季総合分科会(於:岡山大学)

SUMMARY: The Lebesgue integral is used to aggregate an infinite number of inputs into a single output value, and gives a continuous aggregation process as a result of the Lebesgue convergence theorem.

The Choquet, Šipoš, Sugeno, and Shilkret integrals may be considered as a nonlinear aggregation functional \(I\colon \mathcal {M}(X)\times \mathcal {F}^+(X)\to [0,\infty ]\), where \(\mathcal {M}(X)\) is the set of all nonadditive measures \(\mu \colon \mathcal {A}\to [0,\infty ]\) on a measurable space \((X,\mathcal {A})\), and \(\mathcal {F}^+(X)\) is the set of all \(\mathcal {A}\)-measurable functions \(f\colon X\to [0,\infty ]\). For this functional, its continuity corresponds to a convergence theorem of integral, which means that the limit of the integrals of a sequence of functions is the integral of the limit function. Many attempts have thus been made to formulate the monotone, the bounded, the dominated, and the Vitali convergence theorems for the Choquet, Šipoš, Sugeno, and Shilkret integrals, all of which are nonlinear integrals determined by the \(\mu \)-decreasing distribution function \(G_\mu (t):=\mu (\{f\geq t\})\).

The purpose of this talk is to present a unified approach to those convergence theorems of such distribution-based nonlinear integrals, in other words, an approach that does not depend on the types of nonlinear integrals. A crucial ingredient is a perturbation of functional that manages the change in the functional value \(I(\mu ,f)\) when the integrand is slightly shifted from \(f\) to \(f+\varepsilon \) and the \(\mu \)-decreasing distribution function is slightly shifted from \(G_\mu (f)\) to \(G_\mu (f)+\delta \).

SUMMARY: We consider the nonlinear Schrödinger equations with inverse-square potentials (NLS) \[ i\dfrac {\partial u}{\partial t} = \Bigl ( -\Delta + \dfrac {a}{|x|^{2}}\Bigr )u + g_{0}(u) \quad \textrm {in}\; \mathbb {R}\times \mathbb {R}^{N}, \] where \(i=\sqrt {-1}\), \(N\ge 3\), and \(a\ge a(N)=-(N-2)^{2}/4\). The condition of \(a\) is derived from the selfadjointness of \(P_{a}:=-\Delta +a|x|^{-2}\) (in the sense of form-sum). For instance, we suppose \(g_{0}(u):=\pm |u|^{p-1}u\) or \(g_{0}(u):=\pm u\,(|x|^{-\gamma }*|u|^{2})\); so that (NLS) conserves the charge and energy. The perturbed operator \(P_{a}\) has a lot of interesting properties both in physical and mathematical sides. In this talk we solve the Cauchy problems for (NLS) in the energy class and analyze the global solutions to (NLS). If \(a>a(N)\), the energy class is just equal to \(H^{1}(\mathbb {R}^{N})\) (\(L^{2}\)-type Sobolev space). But if \(a=a(N)\), the energy class is a little wider than \(H^{1}(\mathbb {R}^{N})\). Thus it is difficult to apply the usual contraction principle to solve (NLS). Hence we need to apply another approach: energy methods. If we have time, we generalize the perturbed potential \(a|x|^{-2}\) as inverse-square singular potentials \(V(x)\): \(V(\mu x)=\mu ^{-2}V(x)\) (\(\mu >0\)); for example, \(V(x)=(b\cdot x)|x|^{-3}\).

SUMMARY: This paper is the part III of the series of the articles of the axiomatic method of measure and integration. In this paper, we define the Lebesgue measure on \(R\)\(^d, (d\geq 1)\) by prescribing the complete system of axioms. Then we prove the uniquness and existence theorem of the Lebesgue measure. This is a new result.

SUMMARY: This paper is the part IV of the series of the articles of the axiomatic method of measure and integration. In this paper, we define the Lebesgue integral of the Lebesgue measurable functions on \(R\)\(^d, (d\geq 1)\). Then we study the method of calculation of the Lebesgue integral. Further we clarify the convergence properties of the Lebesgue integral completely. These facts are the new results.

SUMMARY: The author discussed the oriented and non-oriented volumes in the \begin{math} C^0 \end{math} class. It was found, however, an example which violates the property, “The non-oriented volume exceeds the oriented volume”. This definition estimates the desired value of the non-oriented volume too small, if it exists. So this talk proposes a new definition of the non-oriented volume to avoid this counter example. Our new definition gives an upper estimate for the desired value, if it does exist. We may conclude that it is not the fault of this definition, even if the property above does not hold.

SUMMARY: Minor dimensional measures were proposed by many authors for a subset of an euclidean space. These measures, however, did not assure the property, “The value of product set is given by the product of the value of it’s component”. This talk points out that this property holds for Mincowski content, though it is not sigma-additive.

SUMMARY: In this talk we discuss some convergence theorems for Pan and Lehrer integrals. A fuzzy measure is a monotone set function, and this may be non-additive. There are several integrals with respect to fuzzy measures and some of them are defined using distribution functions. For such integrals, a unified discussion was given by J. Kawabe. Our target Integrals: Pan and Lehrer integrals are not defined using distribution functions. We give some sufficient conditions for some convergence theorems.

SUMMARY: In this talk, we discuss a convergence theorem for the algebraic product inclusion-exclusion integral, which is a non-linear integral with respect to a non-additive measure. We show a monotone uniform convergence theorem for this integral under the assumption that the measure is continuous from below.

SUMMARY: We will talk about contractive conditions on mappings on metric spaces.

SUMMARY: The equiliburium problem is one of the nonlinear problems including various problems in convex analysis, and a large number of researchers has been investigating it in the setting of Hilbert and Banach spaces. In this work, we deal with this problem defined on a complete geodesic space and consider the existence of the resolvent operator and its properties with several results concerning the approximation to the solutions of this problem.

SUMMARY: Convex minimization problem is one of the nonlinear optimization problems. We study this problem by using various approches in the setting of a complete CAT(1) space. It is known that the set of fixed points of the resolvent of a convex function coincides with its minimizers. Hence we find a fixed point of the resolvent instead of a minimizer of the convex function. To find a solution to this problem, we consider Mann type iteration for two resolvents, and prove its convergence property. We introduce this theorem and several properties of the resolvents.

SUMMARY: We prove the existence and uniqueness of solutions of boundary value problems for differential equations of order \(\alpha \) where \(3<\alpha \leq 4\).

SUMMARY: In this talk, we study attractive points of normally generalized hybrid mappings. Using the idea of attractive points, we prove weak convergence theorems for the mappings. We also prove some convergence theorems for nonlinear mappings.

SUMMARY: We construct a subset of \(\mathbb {R}^d\) which asymptotically and omnidirectionally contains arithmetic progressions but has Assouad dimension 1. More precisely, we say that \(F\) asymptotically and omnidirectionally contains arithmetic progressions if we can find an arithmetic progression of length \(k\) and gap length \(\Delta >0\) with direction \(e\in S^{d-1}\) inside the \(\epsilon \Delta \) neighbourhood of \(F\) for all \(\epsilon >0\), \(k\geq 3\) and \(e\in S^{d-1}\). Moreover, the dimension of our constructed example is the lowest-possible because we prove that a subset of \(\mathbb {R}^d\) which asymptotically and omnidirectionally contains arithmetic progressions must have Assouad dimension greater than or equal to 1. We also get the same results for arithmetic patches, which are the higher dimensional extension of arithmetic progressions.

SUMMARY: In this talk, we consider the weak-boundedness of the fractional Orlicz maximal operators \(M_{B,\alpha }\). If \(\alpha =0\), the weak-boundedness for the Orlicz maximal operator \(M_{B}\) is characterized by condition \( B(t)\leq Ct^{p}\). Condition \(B(t)\leq Ct^{p}\) characterizes the weak-boundedness for the fractional cases too. Amazingly, the condition unifies the Hardy–Littlewood–Sobolev type and the Sawyer type: Given the appropriate conditions of indices, the weak-boundedness of the Hardy–Littlewood–Sobolev type is equivalent to the weak-boundedness of the Sawyer type.

SUMMARY: Let \(\mathbb {R}^n\) be the \(n\)-dimensional Euclidean space. Let \(b\in \mathrm {BMO}(\mathbb {R}^n)\) and \(T\) be a Calderón–Zygmund singular integral operator. In 1976 Coifman, Rochberg and Weiss proved that the commutator \([b,T]=bT-Tb\) is bounded on \(L^p(\mathbb {R}^n)\) (\(1<p<\infty \)), that is, \( \|[b,T]f\|_{L^p}=\|bTf-T(bf)\|_{L^p}\le C\|b\|_{\mathrm {BMO}}\|f\|_{L^p}, \) where \(C\) is a positive constant independent of \(b\) and \(f\). For the fractional integral operator \(I_{\alpha }\), Chanillo proved the boundedness of \([b,I_{\alpha }]\) in 1982. These results were extended to Orlicz spaces by Fu, Yang and Yuan. In this talk we discuss the boundedness of the commutator \([b,I_{\rho }]\) on Orlicz spaces, where \(I_{\rho }\) is a generalized fractional integral operator and \(b\) is a function in generalized Campanato spaces.

SUMMARY: In this talk we give the characterization of pointwise multipliers on weak Orlicz spaces. To do this we first prove a generalized Hölder’s inequality for the weak Orlicz spaces. Next, to characterize the pointwise multipliers, we use the fact that all pointwise multipliers from a weak Orlicz space to another weak Orlicz space are bounded operators

SUMMARY: It is well known as the Hardy–Littlewood–Sobolev theorem that the fractional integral operators \(I_{\alpha }\) on the Euclidean space \(\mathbb {R}^n\) is bounded from \(L_p\) to \(L_q\) for \(1<p<q<\infty \), \(0<\alpha <n\) and \(-n/p+\alpha =-n/q\).

In martingale theory, based on the result by Watari (1964), Chao and Ombe (1985) proved the boundedness of the fractional integrals for \(H_p\), \(L_p\), \(\mathrm {BMO}\) and Lipschitz spaces of the dyadic martingales. These fractional integrals were defined for more general martingales by Sadasue (2011). On the other hand, martingale Morrey spaces and their generalization were introduced by Nakai and Sadasue (2012) and Nakai,Sadasue and Sawano (2013), respectively, and the boundedness of fractional integrals as martingale transforms were established. In this talk we investigate the boundedness of fractional integrals on martingale Orlicz spaces.

SUMMARY: We consider the bilinear estimates of a product of functions in homogeneous Besov spaces showed by Bony. It is seen that if we change the condition of indices denoting differential orders, then we can find examples of functions that never satisfy the bilinear estimates. Such examples can be constructed due to those used in the ill-posedness problem of the Navier–Stokes equations, such as Bourgain–Pavlović and Yoneda.

SUMMARY: In the theory of orthogonal polynomials, (non-trivial) probability measures on the unit circle are parametrized by the Verblunsky coefficients. Baxter’s theorem asserts that such a measure is absolutely continuous and has positive density with summable Fourier coefficients if and only if its Verblusnky coefficients are summable. This talk presents a version of Baxter’s theorem in the matrix case from a view point of the Nehari problem.

SUMMARY: In this talk, we investigate the composition of the fractional maximal operators with the \(d\)-dimensional weighted Hausdorff content \(H^{d}_{\omega }\). In the special case, it is shown that the iterated Hardy–Littlewood maximal operators \(M^{N}\) is bounded from \(L^p(M\omega )\) to \(L^p(\omega )\) with an arbitrary weight \(\omega \) and any \(N\in {\mathbb N}\).

SUMMARY: A sparse bound for local smoothing operators related to maximal Riesz means is given.

SUMMARY: In this talk we consider a chemotaxis-haptotaxis system with signal-dependent sensitivity. In the previous works by Cao (2016), Tao (preprint) and Tao–Winkler (2014) it was shown that the system without signal-dependent sensitivity possesses a global classical solution some conditions; however, the system with signal-dependent sensitivity has not been studied. The purpose of this talk is to establish globlal existence and boundedness in the system with signal-dependent sensitivity.

SUMMARY: This talk deals with a nonlocal Cahn–Hilliard system on an unbounded domain with smooth bounded boundary. In the case of bounded domains, this system has been studied by using a Faedo–Galerkin approximation scheme considering a compactness. However, in the case of unbounded domains, the compactness breaks down. The present work establishes existence and energy estimates of weak solutions for the above system on an unbounded domain.

SUMMARY: We consider the initial boundary value problem for the viscous Cahn–Hilliard equation. In 2014, Bui, et al. proved the existence of strong solutions under the condition that the nonlinear term \(\varphi (u)\) satisfies \(\varphi (u)u\geq 0\) for all \(u\in \mathbb {R}\) and Sobolev subcritical growth condition. In this talk, we exclude these conditions by decomposing the nonlinear term \(\varphi (u)\) into the sum of a monotone function and a locally Lipschitz perturbation and show the existence of global solutions. In physics, \(\varphi (u)=u^3-u\) is often used as a typical example. However, the previous result cannot cover this case. Our framework can cover not only this case but also more general cases \(\varphi (u)=|u|^{p-2}u-|u|^{q-2}u\) with \(p>q\geq 2\).

SUMMARY: We consider the uniform boundedness for global solutions of a reaction diffusion system arising from a nuclear reactor model with the Robin boundary condition, which consists of two real-valued unknown functions. Since this system has no variational structure, we cannot apply the standard methods relying on the Lyapunov functional in order to obtain a priori estimates of global solutions. To cope with this difficulty, we make use of the weighted \(L^1\) norm characterized by the first eigenfunction of Laplacian with the Robin boundary condition.

SUMMARY: We are concerned with the existence of time periodic solutions for 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 bounded domains with smooth boundaries. Our approach to (CGL) is to regard it as a parabolic equation governed by \(-\lambda \Delta u + \kappa |u|^{q-2}u\). Since (CGL) has a monotone perturbation \(-i\alpha \Delta u\) and a non-monotone one \(i\beta |u|^{q-2}u\), it is hard to directly apply general theories for periodic problems of parabolic equations.

SUMMARY: In this talk we discuss an initial boundary value problem describing a real experiment related to the Soret effect on a bounded domain in a plane. Our aim is to present a numerical method for the problem by using a dummy variable, since the shape of the domain is rather complex. Here, we shall show the idea and prove a existence and uniqueness of a solution to the approximation problem given by the dummy variable.

SUMMARY: In this talk, we consider a system of parabolic type PDEs. Each constituent system is based on the mathematical model of grain boundary motion, proposed by [Kobayashi-et. al, Physica D., 140 (2000), 141–150], and the principal part of the system consists of a quasilinear diffusion equation of singular type, subject to the dynamic boundary condition. The objective of this study is to obtain a uniform mathematical method for the models of grain boundary motions including dynamic boundary conditions. On this basis, we here address three issues. The first is to show the existence of solutions to the systems, including the rigorous expressions of solutions. The second is to show the continuous associations among the different systems. The final is to show the large time behavior of solutions. The three issues will be demonstrated in forms of the Main Theorems of this talk.

SUMMARY: We consider the existence and uniqueness of solutions to the initial value problem for a scalar conservation law with a flux function which is discontinuous with respect to the unknown function. For the initial value problem, we generalize the shock admissibility condition introduced by Oleinik. Using the wave front tracking method constructed by shock waves which satisfy the generalized shock admissibility condition, we prove the existence of weak solutions to the initial value problem. Moreover, for the initial value problem, we generalize the well-posedness theory introduced by Liu and Yang. Using the generalized theory, we derive an \(L^1\) contractive estimate concerned with weak solutions to the initial value problem.

SUMMARY: Systems for parabolic-hyperbolic conservation laws are interesting research object in the sense of mathematics and applications. In a mathematical point of view, the systems have both properties of hyperbolic equations and those of parabolic equations. Therefore, it has discontinuous solutions in general. From this, the unified well-posedness theory is not given. In application point of view, the systems can be applied to many mathematical models (fluid dynamics, traffic flow, aggregation phenomena, crowd dynamics and so on). In this talk, we formulate initial value problems for the systems with anisotropic diffusion terms and discuss the well-posedness for the problem.

SUMMARY: We study the definition of the semigroup generated by the Dirichlet Laplacian of fractional order on an arbitrary open set and its properties. It will be shown that we obtain the boundedness, the smoothing effects and the maximal regularity estimates in the homogeneous Besov spaces as well as whole space case.

SUMMARY: We deal with initial-boundary problems for Vlasov–Poisson equations in a half-space with magnetic. In 2013, Skubachevskii gives local-in-time solvability to the system. Moreover, in 2017, existence result with waeker condition were obtained by effectively using the magnetic force whose direction is horizontal to the wall. This talk provides an existence result for the equation where the magnetic force has angle error in the vertical direction.

SUMMARY: In this talk, we discuss the parabolic problem form the hardening phenomena. The unknown functions \(u\) and \(\sigma \) describe the displacement and stress, respectively in the one-dimensional interval \((0,L)\). Our problem means the hardening problem that the materials are harden by plasticity. That is derived from the hardening model by Visintin (2006), and the perfect plasticity model by Duvaut–Lions (1976).

In the perfect plasticity model, the function that is threshold value in the plastic deformation, is a constant. In this talk, we discuss the solvability for the above model with the threshold function depending upon time or unknown function, based on the idea of Duvaut–Lions (1976). The problem equipped with the constraint set depend on the unknown function, is called quasi-variational inequality. The solvabilities of quasi-variational inequality have been dealt with in some papers.

SUMMARY: In this talk, we propose a mathematical model describing water swelling in porus materials. Water swelling is a important issue to investigate frost damage which is a nonlinear phenomenon to give rise to crack on the concrete surface. Our model consists of a diffusion equation for water content in a one microscopic hole inside of concrete and a ordinaly differential equation describing the growth rate of the front of the water content region. In this talk, we discuss the existence and uniqueness of a time local and global solution for this problem.

SUMMARY: In this talk, we consider a class of optimal control problems for state problems of one-dimensional parabolic PDE systems. Each state problem is denoted by \( \mathrm {(S)}_\varepsilon \), with \( \varepsilon > 0 \), and is associated with the phase-field model of grain boundary motion, proposed by [Kobayashi et al.; Phys. D, 140 (2000), 141–150]. In this regard, each optimal control problem is denoted by \( \mathrm {(OCP)}_\varepsilon \), with \( \varepsilon > 0 \), and it is prescribed as a minimization problem of a cost. Additionally, the problems \( \mathrm {(S)}_\varepsilon \) and \( \mathrm {(OCP)}_\varepsilon \) are supposed to admit limiting profiles as \( \varepsilon \downarrow 0 \), and then, the limiting problems are supposed to contain no little singularityies. In this talk, the main interest is in the case when \( \varepsilon > 0 \) (regular case), and the mathematical results concerned with the existence of the optimal control when \( \varepsilon > 0 \); (b) the necessary condition for the regular optimal control; (c) limiting observation as \( \varepsilon \downarrow 0 \); will be reported as the main theorems of this talk.

SUMMARY: In this talk, we show the existence of solutions to the following double quasi-variational evolution equations governed by time-dependent subdifferentials in a uniformly convex Banach space \(V^*\):
\( \qquad \partial _* \psi ^t(u;u'(t)) + \partial _* \varphi ^t(u; u(t))+g(t,u(t)) \ni f(t) \mbox { in } V^* \mbox { for a.a.}\ t\in (0,T).\)
Here, the time-dependent function \( \psi ^t (v;z)\) is proper, lower semi-continuous (l.s.c.), and convex in \(z\in V\). Also, \(\varphi ^t(v;z)\) is a time-dependent, non-negative, continuous convex function in \(z\in V\). Note that \((t,v) \in [0,T]\times C([0,T];H)\) is a parameter that determines the convex functions \( \psi ^t (v;\cdot )\) and \(\varphi ^t(v;\cdot )\) on \(V\). In addition, the subdifferentials \( \partial _* \psi ^t (v;z) \) of \( \psi ^t (v;z) \) with respect to \(z\in V\) is a multivalued operator in \(V^*\), and \( \partial _* \varphi ^t (v;z) \) of \( \varphi ^t(v;z) \) with respect to \(z\in V\) is a single-valued linear operator in \(V^*\).

SUMMARY: We consider a Cauchy problem of an abstract evolution inclusion on a real Hilbert space associated with subdifferentials of time-dependent proper l.s.c. convex functions. The abstract evolution inclusion, which is treated in this paper, contains not only convex functions but also inner products of the Hilbert space depending upon unknown functions. Especially, we call such structures for convex functions and inner products quasi-variational structures for convex functions and inner products. The main purposes of this paper are to show the existence of global-in-time solutions to the Cauchy problem, which has the quasi-variational structures.