Regarding the critical interactions in the theory of classical fields, we have studied equations of nonrelativistic fields and of semirelativistic fields and related inequalities appearing in the analysis of those equations. As for the nonlinear Shr"odinger equations, we proved blow-up of solutions in non-gauge invariant setting.Moreover, we have improved the Br\'ezis-Gallou"et argument by renormalizing higher order energy to cover higher order nonlinearities in the global Cauchy problem for nonrelativistic and semirelativistic equations

In this study, we consider the inverse and direct bifurcation problems of nonlinear eigenvalue problems. For the direct problems, we establish the precise asymptotic formulas for the eigenvalue problems which have biological and physical background. For the inverse bifurcation problems, we consider the typical inverse problem for elliptic equations to understand well the structure of inverse problems. In particular, we concentrate on the study of the global structure of bifurcation curves for some nonlinear ordinary differential equations. We apply these precise asymptotic properties to the typical inverse bifurcation problem and obtained some fundamental and new results in this direction

For the Boussinesq equations, we established the unique existence of the solutions, and obtained the asymptotic behavior up to the second order.Besides, for the stationary Navier-Stokes euqations on two-dimensional whole plane and exterior domains, we introduced a new assumption on the symmetry of domains, external force and the boundary value, and showed the existence of solutions which decay at infinity.Further, under a weaker assumption on the symmetry, we showed the global asymptotic stability of the stationary solutions under arbitrary perturbations in the L2-space, together with the speed of decay measured by various norms

Via variational approached, we study nonlinear elliptic problems. In particular, we develop a new variational approach and we construct concentrating solutions for several singular perturbation problems, where uniqueness and non-degenerary of solutions of limit problems are not known and the Lyapunov-Schmidt reduction method is not applicable.We also introduce new variational approaches based on the scaling properties the problems, which enable us to study ground states and other solutions for several nonlinear elliptic equations and systmes

In this research, we studied the structure of solutions of nonlinear ellipticpartial differential equations arising in mathematical models of quantum phenomena and pattern formation problems of the mathematical biology. Especially, we investigated the precise asymptotic behavior of eigenvalues of the Laplacian with the mixed boundary condition on a thin domain and the precise asymptotic behavior of the least energy of a nonlinear variational problem associated with the Bose-Einstein condensation. We also studied the structure of global minimizers for 3-component FitzHugh-Nagumo system and the parameter range of existence and non-existence of non-constant steady patterns for several nonlinear rection-diffusion systems, e.g., the prey-predator model with a cross-diffusion effect

Mass resonance phenomenon of systems of nonlinear Schr"odinger equations is studied. We have found an appropriate Lagrangean for the system of Schr"odinger equations with quadratic interactions, by which we are able to give a clear explanation of the roles of mass resonance phenomenon as well as of complex conjugate in wavefunctions.A variational setting has been introduced for the analysis on standing waves associated with mass resonance. We proved the existence and (essential) uniqueness of ground states.Moreover, examples of interactions were found for the blow-up solutions with arbitrarily small Cauchy data under mass resonance condition

This research is concerned with a free boundary problem for reaction-diffusion equations in mathematical ecology. This problem models the invasion or migration of a certain biological species. Our main interest is to study the evolution of the population density and habitat of the species. The population density is described by a reaction-diffusion equation and the boundary (or a part of the boundary) of the habitat is controlled by a free boundary condition of Stefan type. We could obtain theoretical understanding on asymptotic behaviors of solutions for free boundary problems of various types: whether the species vanishes eventually or the species persists with spreading free boundary. Moreover, we got precise results on the spreading speed of the free boundary

We studied nonlinear elliptic equations arising in various fields of mathematical physics by means of variational analysis, ordinary differential equations, and viscosity techniques. We studied orbital stability of standing waves, explicit blow-up solutions, and exponential decay of ground states for systems of nonlinear Schr"odinger type equations

Blow-up phenomenon of positive solutions to the Cauchy problem for nonlinear heat equation of Fujita type is studied. The blow-up phenomenon was first observed and proved by Hiroshi Fujita about half a century ago. There is a large literature on the subject, especially on the formation of blow-up solutions in configuration space up to the blow-up time. We have proved that the blow-up time is characterized by means of the spherical average of the Cauchy data. We removed the assumptions of uniformity and isotropy on the Cauchy data, which were necessary in the former works by Lee and Ni (Trans. Ams, 1992) and Gui and Wang (JDE 1995). We described how the ODE structure controls the blow-up phenomenon

We proved existence of non-constant stationary solutions to nonlinear reaction-diffusion equations arising in some pattern formation problems, for example, an Allen-Chan modelwith non-homogeneous environment and a Chemotaxis model with saturation effect

In this study, we investigated the precise asymptotic properties of the eigenfunctions and eigenvalues of nonlinear elliptic equations, and studied the inverse eigenvalue problems associated with thedirect problems. We clarify the global and local structures of the bifurcation curves for the equations with several types of nonlinear terms. As for inverse problems, we mainly studied the inverse bifurcation problems for logistic type equations. By using the theory of ordinary differential equations and the asymptotic expansion formulas for bifurcation curves, we determined the unknown nonlinear terms by the asymptotic behaviors of the bifurcation curves

This research is concerned with the Navier-Stokes equations on either the whole plane or two-dimensional exterior domains. It was shown that, if there exists a small stationary external force with strong symmetry, the equation has a small stationary solution decaying rapidly at infinity. It was also shown that, if the stationary solution above is sufficiently small, it is stable under initial perturbation without restriction on the size. The Navier-Stokes equaions in an infinite layer is also studied. It is shown that, if the equation is treated in the Besov spaces, nontrivial solutions with no external forces exist if p is infinite, and that these solutions correspond to the Poiseuille flows.

Various types of nonlinear PDEs (nonlinear elliptic equations, nonlinear diffusion equations, nonlinear wave equations, nonlinear Schrodinger equations) arising in physics and engineering were synthetically studied from the viewpoint of the theory of nonlinear evolution equations by using the techniques from the theory of nonlinear functional analysis, the theory of functions of a real variable, the theory of ordinary differential equations and the calculus of variations.

This research project is concerned with the mathematical formulation of non-uniformity of species in mathematical ecology such as the segregation of two competing species and the spreading of invasive species. These phenomena are described by reaction-diffusion equations with population densities as unknown functions. We have obtained satisfactory results on the structure of positive steady-states for two-species models with nonlinear diffusion and the mechanism of spreading and vanishing for free boundary problems in invasion models.

We study nonlinear problems via variational approaches. Especially (1) we study singular perturbation problems for nonlinear Schrodinger equations and systems. We introduce a new purely variational method which enables us to construct concentrating solutions in a very general setting. (2) We study nonlinear elliptic equations and systems in various settings. We give a new variational construction of radially symmetric ground states. We also study stability and instability of solutions. (3) We also study highly oscillatory solutions in 1-dimensional singular perturbation problems. We give characterization and existence result.

Analysis for the heat convectin problems for the system of Oberbeck-Boussinesq equations in the horizontal strip domain. The existence of bifurcation curve of roll-type solutions is proved for the ten times Rayleigh number of critical Rayleigh number by a computer assisted proof . The second bifurcation point of stationary roll-type solutions is determined by a computer assisted proof. The hexagonal-type and rectangle-type solutions of 3-dimensional problems are also proved for the existence by a computer assisted proof at least for rather small Rayleigh numbers.
The cocoon bifurcation for the Michelson system is analyzed and proved for the existence bf infinitely many bifurcations of heteroclinic orbits to the saddle-node periodic orbit by a topological method and a computer assisted proof.
The driven-cavity problem of 2-dimensional Navier-Stokes equation is solved for rather large Reynolds numbers compared to the existing verified result by a Newton type computer assited proof.

We study nonlinear elliptic partial differential equations and Hamiltonian systems via variational meth-ods. We put emphasis on singular perturbation problems.
1. We study the existence of high frequency solutions-families of solutions whose numbers of spikes or layers increase to ∞ as the singular perturbation parameter ε goes to 0. We give the existence and the characterization of such families for 1 dimensional elliptic problems including nonlinear Schrodinger equations, Allen-Cahn equations, Fisher equations and Girerer-Meinhardt systems. Especially for Girerer-Meinhardt systems, we introduce and analyze a limit equation using adiabatic invariants. We also give a precise estimate of the number of positive solutions of nonlinear Schrodinger equations.
2. We also study a singular perturbation problem for -ε^2△μ+V(χ)μ =g(μ) in R^N. Under very general conditions on g(μ), which is related to the work of Berestycki, Gallouet-Kavian, we prove the existence of a concentrating solution for N=1,2.
3. We also study the prescribed energy problem for singular first order Hamiltonian systems. We suc-ceed to obtain the existence of periodic orbit under conditions which generalize the strong force condition of Gordon. We remark that our condition is given as a property of the energy surface S={(q, p);H(q, p) =E} not on the Hamiltonian H(q, p).

(i) L^∞-energy Method, developed in this research, is applied to the nonlinear parabolic equations with nonlinear terms involving the time derivative to show the existence of the unique local solution. The verification for the uniqueness was difficult for the existing methods because of the lack of regularity. However this method makes it possible by assuring the high regularity of solutions. Furthermore this method turns out to be very effective also for nonlinear parabolic systems for chemotaxis and systems with the hysteresis effect by the fact that it can assure the existence an uniqueness of solution under much weaker conditions than ever
(ii) The infinite dimensional global attractor is constructed in L^2, which attracts all orbits for the initial boundary value problem for the quasi-linear parabolic equation governed by the p-Laplacian. The infinite dimensional global attractor is never observed for the semilinear parabolic equations, so this very new observation seems to be very important. On the other hand, the existence of the exponential attractor with finite fractal dimension , which attracts all orbits starting from some special class of initial data exponentially, is shown for some special quasilinear parabolic equations involving Laplacian and p-Laplacian., whence follows the finite dimensionality of the global attractor. These observations suggest that in contrast with semilinear equations, there should exist some structure in quasilinear parabolic equations which controls the finite-dimensionality and infinite-dimensionality of global attractors, which gives a very interesting future object ton study.
iii) It is shown that for Cauchy problem and periodic problem for the abstract evolution equation governed by time-dependent subdifferential operators, if the sequence of approximating subdifferential operators converges to the original one, then the corresponding approximating solutions converge to the solution of the original equation. As for the periodic problem, it is very meaningful to give an affirmative answer to the open problem left long.

We study the spectral analysis of Stokes equations based on the recent development of the real analysis, Fourier analysis and functional analysis and its application to the Naveir-Stokes equations in several different situations arising from the mathematical physics.
1) We studied an inncompressible viscous flow past a rigid body, which is mathematically described by Oseen equations. We studied the decay properties of the Oseen semigroup in the exterior domain and showed global in time stability Navier-Stokes flow past a rigid body
2) We studied an inncompressible viscous flow in a perturbed half-space which describes for example flow past high buildings. We studied an optimal decay properties of solutions to the Stokes equations in a perturbed half-space and proved a global in time unique existence of solutions to the Navier-Stokes equations in a perturbed half-space with small initial data.
3) We proved the maximal regularity of solutions to the Stokes equation with the Neumann boundry condition in a bonded domain. We use some recent development of the operator-valued Fourier analysis by Weis and Denk-Hieber-Pruss. Our method is very simple compared with previous results and seems to be applicable to linear evoulution equations of parabolic type. Moreover, we proved local in time unique existence of strong solutions with arbitrary initial data and global in time unique existence of strong solutions with some small initial data of the free boundary problem of Navier Stokes equations which describes the transient motion of an isolated volume of viscous incompressible fluid
4) We studied an inncompressible viscous flow past a rotating rigid body. This problem was already studied by Galdi and Galdi and Silvestre in the L_2 framework. Our main contribution is to show the decay estimate of the continuous semigroup associated with linearized problem. The main difficulty comes from first order differential system with polynomially growing coefficients which is not subordinated by the Laplacian. We developed new technique to investigate the high frequency part of the spectrum.

We study the spectral analysis of Stokes equations based on the recent development of the real analysis, Fourier analysis and functional analysis and its application to the Naveir-Stokes equations in several different situations arising from the mathematical physics.
1) We studied an inncompressible viscous flow past a rigid body, which is mathematically described by Oseen equations. We studied the decay properties of the Oseen semigroup in the exterior domain and showed global in time stability Navier-Stokes flow past a rigid body
2) We studied an inncompressible viscous flow in a perturbed half-space which describes for example flow past high buildings. We studied an optimal decay properties of solutions to the Stokes equations in a perturbed half-space and proved a global in time unique existence of solutions to the Navier-Stokes equations in a perturbed half-space with small initial data.
3) We proved the maximal regularity of solutions to the Stokes equation with the Neumann boundry condition in a bonded domain. We use some recent development of the operator-valued Fourier analysis by Weis and Denk-Hieber-Pruss. Our method is very simple compared with previous results and seems to be applicable to linear evoulution equations of parabolic type. Moreover, we proved local in time unique existence of strong solutions with arbitrary initial data and global in time unique existence of strong solutions with some small initial data of the free boundary problem of Navier Stokes equations which describes the transient motion of an isolated volume of viscous incompressible fluid
4) We studied an inncompressible viscous flow past a rotating rigid body. This problem was already studied by Galdi and Galdi and Silvestre in the L_2 framework. Our main contribution is to show the decay estimate of the continuous semigroup associated with linearized problem. The main difficulty comes from first order differential system with polynomially growing coefficients which is not subordinated by the Laplacian. We developed new technique to investigate the high frequency part of the spectrum.

対称臨界性原理とは、「Banach空間X上で定義された汎関数Jに対し、ある群Gの作用に関して不変な部分空間X_G上でのJの臨界点が、X全体でのJの臨界点を与える」という原理である。
この原理は、Jの汎関数(フレッシェ)微分を、劣微分作用素を含むかなり一般的な多価作用素Aに置き換えても成立することが、本研究により示されている。
さらに、作用素Aが、必ずしも変分構造を有していない場合に対して拡張することが可能(AがG-共変であれば十分)であり、時間発展を伴う発展方程式に対して有用であろうことが期待されていた。
(1)非線形放物型方程式、波動方程式、シュレンディンガー方程式等への応用考えるとき、時間に関する微分作用素d/dtがどのような空間でG-共変となるかを調べることは重要であるが、今回L2(0,T;X),X=L2(Ω),L2(Ω)xL2(Ω),H10(Ω)xL2(Ω),などでそのG-共変性(G=O(N))が確かめられた。これらの空間は、上記の諸方程式を抽象発展方程式に帰着させるときに現れる基本的な空間であり、これは、今後の発展方程式への応用研究において重要な知見である。
(2)放物型方程式の時間大域解の漸近挙動を解析する際、コンパクト性は強力なな道具を提供する。一般の非有界領域では欠如しているソボレフの埋蔵定理にかかわるコンパクト性が、回転対称性を有する関数からなる部分空間においては、恢復するという事実に基づき、ある種の回転対称性を有する非有界領域におけp-Laplace作用素と爆発項を含む非線形放物型方程式の対称大域解のW1,p-有界性が確かめられた。

In this project, we have studied the structure of solutions for the following two types of equations : (a) reaction diffusion systems with nonlinear diffusion in mathematical biology and (b) semilinear diffusion equations describing phase transition phenomena
The first problem in mathematical biology is given by a system of differential equations with quasilinear diffusion of the form
u_t=Δ[φ(u,v)u]+au(1-u-v), v_t=Δ[ψ(u,v)v]+bv(1+du-v),
under homogeneous Dirichlet boundary conditions. Here u and v denote population densities of prey and predator species, respectively. It is well known that the corresponding stationary problem has a positive steady-state under a suitable condition. Our main interest is to derive useful information on profile and stability of each positive steady-state. In case φ(u,v)=1 and 4,φ(u,v=1+β u, we have shown that the stationary problem has at least three positive solutions if β is sufficiently large and some other conditions are imposed. Moreover, stability or instability of each positive solution is also investigated.
The second problem is given by u_t=ε^2u_<xx>+u(1-u)(u-a(x)) with homogeneous Neumann boundary condition, where 0<a(x)<1. When ε is sufficiently small, it is known that this problem admits various kinds of steady-state solutions. In particular, we are interested in steady state with transition layers and spikes. Here transition layer for a solution means a part of u(x) where u(x) drastically changes from 0 to 1 or 1 to 0 in a very short interval. Such oscillating solutions have been studied by Ai-Chen-Hastings and our group, independently. It has been proved that any transition layer appears only in a neighborhood of x such that a(x)=1/2 and that any spike appears only in a neighborhood of x such that a(x) takes its local maximum or minimum. We have also established more information on profiles of multi-transition layers and multi-spikes, their location and the relationship between profile and stability of steady-state solution with transition layers.

1. Kurata studied the existence and qualitative properties of optimal solutions to several optimization problems for nonlinear elliptic boundary value problems arising in mathematical biology and nonlinear heat conduction phenomena. Kurata also proved the existence of multiple stable patterns in population growth model with Allee effect, symmetry breaking phenomena of the least energuy solution to nonlinear Schroedindger equation and asymptotic profile of radial solution with vortex to 2-dimensional nonlinear Schroedinger equation.
2. Okada studied numerial simulation and numerical analysis of nonlinear paratial differential equations. Especially, he constructed boundary spline function by using Newton extrapolation polynomials.
3. Sakai studied Hele-Shaw free boundary problem in the case that initial data has a cusp and found sufficient conditions to specify the typical pheneomena.
4. Isozaki discovered the relationship between the hyperbolic geometry and inverse problem. He also studied the inverse conductivity problem with discontinuous inclusions and found a numerical algorithm to detect discontinuities.
5. Jimbo continued his research on the study of solution structure of the Ginzburg-Landau equation arising in superconductivity under heterogeneous environments. He also studied the spectrum of elliptic operator associated with the Maxwell equation and proved characterization of eigenvalues and proved a perturbation formula by using weak forms.
6. Tanaka studied concentration phenomena of solutions and clustered solutions for nonlinear elliptic singular perturbation problems. Especially, he constructed high frequency solution to nonlinear Schroedinger equations and multi-clustered high energy
Solutions to a phase transition prolem.

We study the existence and multiplicity of solutions of nonlinear differential equations via variational methods. In particular, we study singular perturbation problems.
1.We study the existence and multiplicity of solutions of nonlinear scalar field equations : -Δu+V(x)u=f(u) in R^N. Usually in such a problem global conditions on nonlinearity f(u)(ex.global Ambrosetti-Rabinowitz condition) are required to ensure the existence of solutions. In this study we tried to obtain an existence result without such global assumptions and we find that it is possible if we require sufficiently fast decay of the potential V(x).
2.We also study singular perturbation problem : -Δu+λ^2a(x)u=|u|^<p-1>u in R^N, where a(x)【greater than or equal】0. As a limit problem as λ→∞, a Dirichlet boundary value problem -Δu=|u|^<p-1>u, u|_<∂Ω>=0 in Ω≡{x ∈R^N;a(x)=0} appears. We assume Ω consists of several bounded connected components Ω_1,【triple bond】, Ω_κ and for given solutions u_i(x) of the Dirichlet problem in Ω_i, we try to find a solution u_λ(x) in R^N whose limit is u_i(x) in Ω_i (connecting problem). We succeed to find a solution joining Mountain Pass solutions without non-degeneracy conditions. Also we show that there are infinitely many sign-changing solutions that are connectable with Mountain Pass solutions.
3.For 1-dimensional Allen-Cahn equations and Schrodinger equaitons, we study the characterization of a family of solutions in the setting of singular perturbation. More precisely, we consider a family of solutios with increasing number of layers or spikes. We give a characterization of such a family using "limit enery function" or "envelop function". Conversely for addmissible patterns we construct corresponding families of solutions via variational methods.

We studied nonlinear elliptic eigenvalue problems with several parameters. Our main concern was to clarify the asymptotic properties of eigenvalue parameters and associated eigenfunctions by using variational methods and singular perturbation approaches when a parameter was very large.
We studied two-parameter ordinary differential equations with two pure power nonlinear terms. We established several asymptotic formulas for eigenvalues by using several variational approaches. We also studied the two-parameter problems which were related to the simple pendulum problems. We established the precise asymptotic formulas for the solutions with boundary layers when a parameter was very large under the Dirichlet boundary conditions. The formulas obtained here were totally different from those for the associated one-parameter simple pendulum problems.
For problems with one-parameter, we studied the problems which were related to the simple pendulum problems. We first studied ordinary differential equations and established precise asymptotic formulas for the boundary layers when a parameter was large under the Dirichlet boundary conditions. We next extended this result to the nonlinear elliptic eigenvalue problems in a smooth bounded domain. We studied nonlinear elliptic eigenvalue problems with pure power nonlinearity and established the asymptotic formula for the eigenvalues in L-2 framework. We also treated the perturbed simple pendulum problems in a bounded domain. It is known that if parameter is large, then an associated solution is nearly flat inside the domain. We have succeeded to establish the precise asymptotic formulas for the interior behavior of the solutions to understand precisely how flat the solutions were inside a domain when a parameter was very large. The formulas obtained here were exactly represented by using the nonlinear term.

In a joint work with Yoshihiro Shibata, we obtained a sufficient condition on time-independent external forces for the unique existence of a stationary solution in a certain class of the Navier-Stokes equation in exterior domains of dimension n greater than or equal to 3 by using the duality between the Lorentz spaces and real interpolation. Our class is a natural generalization of the so-called physically reasonable solutions, and our suffirient condition gives a unified view for the case with zero velocity at infinity and the case with non-zero velocity at infinity.
Next, in a joint work with Yuko Enomoto and Yoshihiro Shibata, we verified the stability in the weak-Ln space of the stationary solution above for time-evolution under small initial perturbation in the weak-Ln space, and showed that the smallness above can be taken uniformly in the velocity at infinity of the stationary solution.
Furthermore, by using real interpolation for sublinear operators, we generalized these results for time-dependent external forces, and obtained a sufficient condition for the unique existence of the corresponding time-periodic or almost periodic solutions. We also showed the stability of these solutions in weak-Ln spaces under perturbations on the external forces and initial data uniform in the velocity at infinity of the solutions.
On the other hand, as a preparation for generalized the results above for general unbounded domains, we generalized the Lp-theory on the boundary value problem for the Stokes equation in a layer domain, in a joint work with Takayuki Abe for higher-order Sobolev spaces and Besov spaces, and obtained a sufficient condition on the external forces for the unique existence of the solution of the boundary value problem. In particular, we showed that the uniqueness of the solution fails in the case p=infinity, and that the Poiseuille flow can be characterized as the solution with zero as the external forces and boundary values.

Keller and Segel derived the mathematical model describing chemotactic aggregation of cellular slime molds which move toward high cocentrations of chemical substance. So this model is called as the Keller-Segel system. Chirdress and Percus conjectured that there exists a threshold number (8π) such that if an initial value is smaller than 8π, the solutions exist globally in time, on the other hand if an initial value is greater than 8π, the chemotactic collapse can occure. Our objective is to solve this conjecture. We study the Keller-Segel system in R^2 because this system is rarely studied in hole space. Especially we treated the self-similar solution and solved almost positively the Chirdress and Percus conjectuer. For this purpose we encountered several problems and solved these. We enumerate these problems : (1)determine decay order of the self-similar solutions (2)derive the Lieuville type theorem (3)reduction to one elliptic equation from an system of two elliptic equations (4)show the radial symmetry of solutions by using the moving plane method (5)determine the global blanch of solutions (6)solve the Chirdress and Percus conjectuer.

(1)"L^∞-energy method" is invented. This assures the high differentiablity of solutions of quasilinear parabolic equations. By this method, the existence of W^<1. ∞>-solutions for a general doubly nonlinear parabolic equations and the open problem : "porous medium equations admit C^∞-solutions?" is solved affirmatively. Recent studies suggest that this gives a quite powerful tool for various problems.
(2)"The theory of nonmonotone perturbations for subdifferentials " is extended to Banach space setting. By this theory, we can treat the existence and regularity of solutions for degenerate parabolic equations in a more natural way than Galerkin' s method and open problems, left unsolved in the usual way, were solved.
(3)A Concentration Compactness (CC) theory with partial symmetry is given. The usual CC theory is known to be useful to analyze the problem with lack of compactness. On the other hand, the high symmetry such as the radial symmetry often recovers the compactness. It is studied how the partial symmetry not enough to recover compactness is reflected to CC theory. By this theory, the existence of nontrivial solutions is proved for some quasilinear elliptic equations in infinite cylindrical domains.
(4)The classical "Principle of Symmetric Criticality (PSC)" by R.Palais assures that under suitable conditions, critical points in the subspace with the symmetry give real critical points in the whole space, but is restricted to the system with variational structures. PSC is extended to a more general theory which covers the elliptic systems without full symmetry or evolution equations including time evolution terms.
(5)A new degree theory is established. It can teat mutivuled operators including subdifferential operators and cover nonlinear PDE with various multivaluedness nature.
(6)The theory of nonmonotone perturbations for subdifferentials is ameliorated to cover the initial-boundary value problems and time periodic problems for magneto-micropolar fluid equations.

1.Stability of the Oseen flow in the n-dimensional exterior domain (n>2).
2.Stability of the Couette flow and the Poiseuille flow in the infinite layer.
3.Rate of convergence of the non-stationary flow to the stationary flow of compressible viscous fluid.
4.Resolvent estimate of solutions to the Stokes equation with Neumann boundary condition.

1. Kurata studied the following:
(1) breakdown of the monotonicity of the minimizer to a one-dimensional Cahn-Hilliard energy with inhomogeneous weight and the existence of non-topological solution to a nonlinear elliptic equation arising from Chern-Simons-Higgs theory.
(2) optimal location of a obstacle in an optimization problem for the first Dirichlet eigenvalue to Schrodinger operator.
2. Jimbo studied the existence of stable vortex solutions and the non-existence of permanent current in a convex domain to Ginzburg-Landau equation with magnetic effect. Tanaka constructed solutions with complex patterns to inhomogeneous Allen-Cahn equation and nonlinear Schrodinger equation. Murata studied the structure of positive solutions to elliptic equation of skew-product type and classifies the Martin boundary and Martin kernel completely.
3. Mochizuki studied the inverse spectrum problem for Dirac operator and Sturm-Liouville operator by interior datas. Sakai studied the asymptotic behavior of the moving boundary for Hale-Shaw flow when the initial region has an angle in details.

In our project we have mainly discussed the stationary and non-stationary problems for the following reaction diffusion systems with quasilinear diffusion terms:
(E) u_l = Δ[(1 + αv + γu)u] + uf (u, v), v_l = Δ[(1 + βv + δv)v] + vg (u, v).
This is a well-known system which models the habitat segregation phenomenon between two species. In (E) u, v denote the population densities and f, g represent the interaction between u and v such as Lotka-Volterra competition type or prey-predator type.
(1) Non-stationary problem. When the system has a cross-diffusion effect, the existence result of global solutions was restricted to the two dimensional case. We have proved that, if α, γ > 0 and β = δ = 0, then (E) admits a unique global solution without any restrictions on the space dimension and the amplitude of initial data. Our strategy is to decouple the system and study reaction-diffusion equations separately. We combine parabolic fundamental estimates with energy estimates of solutions of parabolic equation with self-diffusion. This method is also valid for the case δ > 0; so that the global existence is shown when the space dimension is less than six.
(2) Stationary problem. From the view-point of mathematical biology, it is very important to study positive stationary solutions and to know their number. We have tried to get some conditions for the multiplicity of such positive solutions. In particular, the multiple existence is established if interactions are very large in case of competition model with linear diffusion or if one of cross-diffusion is very large in case of prey-predator model.

1. (1) We studied the nonlinear two-parameter problem -u"(x) + λu(x)^q = μu(x)^p,u(x) > 0,x ∈(0,1),u(0) = u(1) = 0. Here 1 < q < p are constants and λ,μ > 0 are parameters. We established precise asymptotic formulas with exact second term for variational eigencurve μ(λ) as λ→∞. We emphasize that the critical case concerning the decaying rate of the second term is p = (3q - 1)/2 and this kind of criticality is new.
(2) We considered the nonlinear two-parameter problem u"(x) + μu(x)^p = λu(x)^q, u(x) > 0, x ∈ I = (0,1), u(0) = u(1) = 0, where 1 < q < p < 2q + 3 and λ,μ > 0 are parameters. We established the three-term spectral asymptotics for the eigencurve λ = λ(μ) as μ→∞ by using a variational method on the general level set due to Zeidler. The first and second terms of'λ(μ) do not depend on the relationship between p and q. However, the third term depends on the relationship between p and q, and the critical case is p = (3q-1)/2.
2. We considered the nonlinear eigenvalue problem -Δu = λf(u), u > 0 in Ω,u = 0 on ?∂Ω, where Ω=B_R={x ∈ R^N : |x| <R} or A_<a,R> = {x ∈ R^N : a< |x| <R} (N【greater than or equal】2) and λ>0 is a parameter. It is known that under some conditions on f and g, the corresponding solution u_λ develops boundary layers when λ>> 1. We established the asymptotic formulas for the width of the boundary layers with exact second term and the estimate of the third term.
3. We considered several elliptic partial differential equations and parabolic systems related to nonlinear eigenvalue problems and obtained some existence results and qualitative properties of the solutions.

We study the existence problems for nonlinear differential equations via variational methods. We mainly dealt with nonlinear elliptic problems and Hamiltonian systems.
1. We study the existence and multiplicity of positive solutions of nonlinear scalar field equations in unbounded domains. In particular, we are concerned with equations which depend on the space variable x and we investigate the effects of the inhomogeneity (dependence on the space variable x) on the set of solutions of the scalar field equation. We find a very delicate dependence ― very small inhomogeneity induces a big change in the set of soluitons ― and we find an example of nonlinear scalar field equation which has 4 positive solutions after very small but not zero pert perturbation.
2. We also consider the singular perturbation problems for nonlinear elliptic problems. We get 2 results : (a) for 1-dimensional setting we introduce a new finite dimensional reduction and we succeed to prove the existence of solutions with a cluster of interior or boundary layers for inhomogeneous Allen-Cahn type equations. We also succeed to prove the existence of solutions with a cluster of spikes for nonlinear Schrodinger equations. (b) We give a mountain pass characterization of positive solutions for a wide class of nonlinear elliptic equations. As an application, we show the existence of a spike solution for a wide class of nonlinear elliptic equations including asymptotically linear equations.
3. For Hamiltonian systems we deal with singular Hamiltonian systems with 2-body type singularities. In case the potential V(q) has more than 3 strong force type singularities we find a family of very complex solutions which are related with symbolic dynamical systems. We also deal with the case the set S of singularity is not a point and it has a positive volume. We consider the case where V(q) 〜 - 1/ dist (q, S)^α and we find the existence of non-collision solutions for all positive α > 0, that is, even for weak force case 0 < α < 2.

The following is the abstract for the main results obtained under this research project.
1. In the research of Hamiltonian systems, Ito generalized the notion of complete integrability for Hamiltonian systems to that for general vector fields. He proved that the integrability of an analytic vector field is equivalent to the existence of a convergent normalizing transformation near an equlibrium point that are non-resonant and elliptic. It gives an answer to the Poincare center problem.
2. In the research of ergodic theory, Morita studied the zeta function associated with two dimensional scattering billiards problem. He succeeded in extending it meromorphically to a half plane with its real part greater than some negative constant.
3. In the research of bifurcation theory of dynamical systems, Kokubu studied the generalization of Conley index theory to slow-fast systems which are singularly perturbed vector fields. He defined transition matrices when the slow variables are of dimension one, and obtained a general method for proving the existence of periodic or heteroclinic orbits.
4. By using variational method for singular Hamiltonian systems, Tanaka proved the existence of orbits such as (1) scattering type ; (2) periodic orbits under the class of perturbation of type -1/γ^2 ; (3) unbounded and chaotic motions for systems whose potential have two singular points.
5. In the research of symplectic/contact geometry, Ono succeeded in constructing the Floer homology with integer coefficients. Nakai studied 1st order PDE's from the viewpoint of foliation theory and Web geometry. In particular, he defined affine connections for those PDE's with finite type, and used them to study the singularities associated with the foliation defined by their solutions.

T.Nagi, Y.Naito and K.Yoshida studied the Keller-Segel system which is the mathematical model describing chemotactic aggregation of cellular slime molds which move toward high cocentrations of chemical substance. Naito and Yoshida with N.Muramoto studied the self-similar solution to the Keller-Segel system. Then the Keller-Segel system is reduced to an elliptic equation with an parameter σ, and obtained solutions of two types, one of which is in a low critical lebel, the other is in a high cirtical lebel. When the parameter is large, there is no self-similar radial solution. This means two solutions are connected by a global blanch. Nagai with T.Senba and T.Suzuki considered the location of blow-up points. These results are announced at each conference or workshop e.g, IMS Workshop on Reaction-Diffusion Systems at HongKong (December 6-10, 1999), The Third World Congress of Nonlinear Analysis at Catania (July 19-26, 2000). H.Usami treated the oscillation problem to the second order ordinary differential equations and the asymptotic properties of the variational eigenvalues. Y.Mizuta completed the results by Koskela concerning the uniqueness property for Sobolev functions.

1. Kurata studied the following :
(1) estimates of the second and third derivatives of fundamental solutions to magnetic Schrodinger operators with non-smooth potentials and the Calderon-Zygmund property of certain operators.
(2) estimates of the heat kernel of magnetic Schrodinger operators.
(3) exiestence and further properties of the optimal configuration to several optimization problems for the first Dirichlet eigenvale. Especially, we find a symmetry-breaking pheneomena of the optimal configuration for certain symmetric domains. We also studied the regularity of the free boundary associated with the optimal configuration.
2. Jimbo studied the non-existence of stable non-constant solution to Ginzburg-Landau equation with magnetic effect.
3. Tanaka studied discontinuous phenomena for solutions under the perturbation to nonlinear ellitic equation -Δu+u=u^p.
4. Murata studied the structure of positive solutions to elliptic and parabolic equations of second order on non-compact Riemannian manifolds.
5. Mochizuki studied the blow-up and the asymptotic behavior of solutions to KPP equation and inverse spectrum problem for Sturm-Liouville operator by interior datas.
6. Ishii showed the convergence of geometric approximation method for the Gauss curvature flow.
7. Sakai studied the condition of the existence of a measure which has a fine support and makes the same potential outside the polygon in two-dimensional case.

(1) Eigenvalue Problems of Elliptic Equations : Two-parameter eigenvalue problems for semilinear elliptic equations are studied. We establish asymptotic properties of (variational) eigenvalues and eigenfunctions. Two-parameter Ambrosetti-Prodi problems are also studied. We investigate the relation between parameters and the number of solutions.
(2) Positive Solutions of Elliptic Equations : Semilinear second-order elliptic euations are considered in unbounded domains. We establish multiplicity results for positive solutions and uniqueness theorems for positive solutions.
(3) Positive Solutions of Quasilinear Ordinary Differential Equations : Quasilinear ordinary differential equations whose leading term is one-dimensionai pseudo-Laplacian are considered. We obtain asynrptotic representations of positive solutions. As an application of these results, we show existence of several types of positive solutions of exterior Dirichlet problems for quasilinear elliptic equations.
(4) Mathematical Models Describing Aggregation Phenomena of Molds : We consider self-similar solutions of parabolic systems introduced by Keller and Segel to describe aggregation phenomena of molds due to chemotaxis. We clarify the relation between parameters and the number of self-similar solutions.
(5) Nonnegative Nontrivial Solutions of Quasilinear Elliptic Equations and Elliptic Systems : We establish necessary and/or sufficient conditions for quasilinear elliptic equations, as well as quasilinear elliptic systems, to possess nontrivial nonnegative entire solutions. Several Liouville type theorems are also obtained.

Elliptic Equations (1) Concerning the equation (E) - △u = |u|ィイD1q-2ィエD1u x ∈Ω, u(x) = 0 x ∈∂Ω we obtained the following results.
Let Ω = RィイD1NィエD1＼BィイD2R1ィエD2, BィイD2RィエD2 = {x ∈ IRィイD1NィエD1 ; |x|【less than or equal】 R }, 2ィイD1*ィエD1<q< +∞ (2ィイD1*ィエD1 is the critical exponent for Sobolev's embedding HィイD31(/)0ィエD3 (Ω) ⊂ LィイD1qィエD1 (Ω) ), then (E) admits a radially symmetric solution in HィイD11ィエD1 (Ω) ∩ LィイD1qィエD1 (Ω). This fact has been conjectured from the duality between bounded domains and exterior domains.
(II) Consider the equation : (E)ィイD2λィエD2 -△u = λu + |u|ィイD1q-2ィエD1u x ∈Ω, u(x) = 0 x ∈∂Ω (1) Let Ω = ΩィイD2dィエD2 × λRィイD1N-dィエD1, (ΩィイD2dィエD2 is a bounded domain in IRィイD1dィエD1), q = 2ィイD1*ィエD1, d【greater than or equal】 1, N 【greater than or equal】 4, then for all λ ∈ (0, λィイD21ィエD2), λィイD21ィエD2 = infィイD2v∈HィイD31(/)0ィエD3 (Ω)ィエD2‖∇ィイD2uィエD2‖LィイD42ィエD4ィイD12ィエD1/‖u‖LィイD42ィエD4ィイD22ィエD2 > 0, (E)ィイD2λィエD2 has a nontrivial solution, which gives a generalization of the well-known result of Brezis-Nirenberg to unbounded cylinders. (2) Let Ω = ΩィイD2dィエD2 x RィイD1N-dィエD1 and let ΩィイD2dィエD2 be a d-dimensional annulus. ・ If q 【greater than or equal】 NィイD2dィエD2 = 2 (N -d+1)/(N-d+1-2) , then (E)ィイD2λィエD2 admits no nontrivial weak solution.
・ If q < NィイD2dィエD2, then (E)ィイD2λィエD2 admits a nontrivial weak solution.
These results reveal the fact that the d-dimensional symmetry reduces the effective dimension by (d-1).
(III) Consider (E)ィイD21ィエD2 -Δu + u = a(x) |u|ィイD1q-2ィエD1u + f(x) x ∈ IRィイD1NィエD1, 2 < q < 2ィイD1*ィエD1 o < a(x), |a(x) - 1| 【less than or equal】 CeィイD1λ|x|ィエD1, λ > 0 It is shown that if ‖f‖ィイD2H-1(RィイD1NィエD1)ィエD2 is sufficiently small, then (E)ィイD21ィエD2 has at least two positive solutions. Furthermore, we found that for the case where f = 0 and q < 2ィイD1*ィエD1 is close enugh to 2ィイD1*ィエD1,the multiplicity of positive solutions depends upon the topological property (su as category) of the set {x ∈Ω ; u(x) = maxィイD2x∈ΩィエD2 }.The analysis of this phenomenon will be an interesting subject to study in future.
Parabolic Equations (I) It has been well known that weak solutions of porous medium equations enjoy the Holder continuity. However, the existence of smooth (local) solutions has been left as an open problem for long time. Otani-Sugiyama gave an affirmative answer to this open problem, by developing the LィイD1∞ィエD1-energy method, which was introduce by themselves to show the local existence of WィイD11,∞ィエD1-solutions for more general doubly nonlinear parabolic equations. This is the most fascinating result among our results obtained in this reseach project.
(II) It was left as an unsolved problem to determine the asymptotic behabiour of solutions of (P) uィイD2tィエD2, -Δu = |u|ィイD12ィイD1*ィエD1-2ィエD1u x∈Ω, u(x) = 0 x∈∂Ω. To this problem, the following partial answer was obtained. 「Let Ω = {x ∈ RィイD1nィエD1 : |x|< 1 } and the solution u (x.t) be positive, radially symmetric and monotone decreasing with respect to r = |x|. Then u blows up in a finite time or becomes a global solution and satisfies the following property : 「There exists a sequence {tィイD2nィエD2 } such that |∇u (x,tィイD2nィエD2)|ィイD12ィエD1 - CoィイD1δィエD1(0) (u - x), |u (x,tィイD2nィエD2)|ィイD12ィエD1 - CoィイD1σィエD1(0) (u - x). 」 This result give some information about the problem above to some extent. However, since strong technical condtions are assumed. We need further in vestigation to solve this problem in a natural setting.

1. Kurata studied the following :
(1) unique continuation theorem and an estimate of zero set of solutions to Schrodinger operators with singular magnetic fields.
(2) finiteness of the lower spectrum of uniformly elliptic operators singular potentials.
(3) Liouville type theorem for Ginzburg-Landau equation and existence and its profile of the least energy solution to nonlinear Schrodinger equation with magnetic effect.
(4) existence of non-topological solution to a nonlinear elliptic equation arising from Chern-Simons-Higgs theory
2. Jimbo studied existence and zero set of stable non-constant solution to Ginzburg-Landau equation.
3. Tanaka studied Hamilton system, uniquness and non-degeneracy of positive solution to a nonlinea elliptic equation, and the construction of multi-bump solutions.
4. Murata studied uniqueness of non-negative solution to parabolic equation.
5. Mochizuki studied global existence and blow-up of solutions to reaction-diffusion systems.
6. Ishii studied dynamics of hypersurfaces and homogenization of Hamilton-Jacobi equation.
7. Sakai studied Hale-Shaw flow in the case that initial domain has a corner.

(1) T.Kori investigated the theory of the index of a gauge coupled Dirac operator with Grassmannian boundary condition, especially he gave a direct method of calculations not using the Atiyah-Patodi-Singer theory for those problems on the four dimensional hemisphere.
(2) T.Kori.proved a formula about the chiral anomaly of gauge coupled Dirac operators. Here he proved that the index of a gauge coupled Diracoperator on S^4 is equal to the index of the geometric Dirac operator on the hemisphere with a Grassmannian boundary condition comming from the vector potential, that is, the effect by the gauge is absorbed in the boundary condition. This result will be published in the Proceeding of the conference on Geometric Aspects of Partial Differential Equations as one volume of AMS Contemporary Mathematics series.
(3) The problem of extension of spinors from the boundary to the interior or the exterior as zero mode spinors is solved. By this an analogy of the Laurent expansion theorem for zero mode spinors is obtained. Thus the concepts of meromorphic spinors and their residues are introduced. He proved the residue theorem on a domain in S^4. Many theorems that are counterparts of what are known in complex function theory are expected to hold in our framework of spinor analysis. This will be our next project.

変分的手法により非線型微分方程式の解の存在問題の研究を行った.特に本年度は(1)R^Nにおけるnonlinear scalar field equation,(ii)ハミルトン系の非有界軌道の存在等を主に扱った.(i)R^Nにおけるnonhlinear scalar field equationに関しては,まず軸対称な空間依存性をもつ方程式-Δu+V(|x|)u=u^pの正値解の一意性を考察し,Kwongによる一意性の結果の非常に簡略化された証明を得ることができた.またその一意性の応用として周期ポテンシャルをもつ非線型楕円型方程式-Δu+V(x)u=u^pのあるクラスに対してmulti-bump solutionが存在することを,特に無限個の正値解が存在することを示すことができた.
(ii)ハミルトン系に関しては,2体問題型のポテンシャルV(q)〜-1/(|q|^α)(α>0)に対して無限から来て無限に飛びさる軌道の変分的な構成を考え,与えられたH>0をtotal energyとしてもち,さらに与えられた入射角,出射角をもつ軌道の存在を空間次元Nに関する制限なしでstrong force条件(α>2)の下で示した.ここで空間次元が2のときは回転数を有効に利用することができ比較的容易に証明はなされるが,N 3の場合は異なりR^N＼{0}上のループ空間のtopologyに関する考察が必要不可欠となることに注意して頂きたい.なおH=0のときは古典力学における放物軌道に対応し,非常に興味ある問題であるが,その存在は今後の課題としたい.
(iii)上記の(i),(ii)以外にもMoser-Trudinger型の不等式の最良指数についても研究を行い,Ogawa,Ozawaにより導入されたスケール不変なMoser-Trudinger型の不等式は有界領域の場合と異なり最良指数を達成しないことを示した.

The following is the abstract for the main results obtained under this research project.
1. In the research of integrable systems, Miyaoka proved that all isoparametric hypersurfaces in the sphere with six principal curvatures are homogeneous. For the proof, she used the isospectrality of the family of the shape operators on the focal set of isoparametric hypersurfaces. This is a remarkable result to solve the conjecture by Yau, and shows close connection between the theory of hypersurfaces and that of integrable systems.
2. In the research of ergodic theory, using transfer operators method, Morita obtained Fredholm determinant representation for the Selberg zeta function Z(s) of closed geodesics on hyperbolic Riemann surface with finite area. He investigated the spectral properties of the transfer operators, and then obtained some analytic information of Z(s).
3. By using variational methods, Tanaka studied unbounded solutions of singular Hamiltonian systems of the two-body type. Namely he proved the existence of a hyperbolic-like solutions for a class of singular potentials with the strong force alpha > 2. Also, he studied the prescribed energy problem for Hamiltonian systems of the same form with alpha = 2, and showed variationally the existence of periodic solutions on the zero energy surface. This led to an existence theorem of closed geodesics on noncompact Riemannian manifold.
4. In the research of symplectic geometry, Ono solved the Arnold's conjecture for general closed symplectic manifold by constructing Floer homology for periodic Hamiltonian and Gromov-Witten invariant. He generalized this approach further and studied constructions of Floer homology for Lagrangian intersection.
5. In the research of complex dynamical systems, Shiga showed some similar properties between limit sets of Kleinian group and Julia sets in theory of complex dynamical systems.

今年度の研究成果は、"cross-diffusion"と呼ばれる拡散項をもつLotka-Volterra型モデルに対する定常解集合の研究と、退化型拡散項(p-Laplacian)をもつ放物型方程式の解のダイナミックスの研究の二つに分けられる。
1.数理生態学における"biodiffusion"のなかには"cross-diffusion"と呼ばれる重要な非線形拡散がある。同一の領域で生存競争している2種以上の生物の固体密度を未知関数として定式化すると、"cross-diffusion"の効果により、拡散係数が固体密度にも依存するような準線形拡散方程式系となる。このようなモデルは1979年に提起され、数値実験では分岐やパターンの形成などの興味深い現象が見られるにもかかわらず、理論的な解析は十分ではない。我々の研究グループは数年前から正値定常解集合の解明に取り組み、正値解が存在するための十分条件や必要条件を見いだしている。今年度は解の多重性に関して非線形拡散がいかなる影響を及ぼすかを調べ、写像度の理論と分岐理論を組み合わせて、正値定常解が2個以上存在する状況を新たに発見した。
2.p-Laplacianを含む拡散方程式にたいしてChafee-Infanteタイプの非線形項を付け加え、解の挙動、定常解集合の構造、安定性を研究した。空間次元1のケースに限定されるが、定常解集合の構造を完全に解明することができた。とくにp-Laplacianの退化性のため、定常解集合の構造は非退化のときと全く異なり、非可算集合となる。さらに、解のプロフィール、解の分岐構造、解の個数、安定性について今まで知られていなかった情報が得られた。今後は、空間次元の高いときの解集合の構造も調べたい。

計画調書の研究目的にかかげた目標に関連した主たる成果は以下の通りである。
1.非線形項が境界で特異性を有する半線形楕円型方程式 -Δu(χ)=Κ(χ)u^β(χ)/(1-|χ|)^α χ∈B={χ∈R^N;|χ|<1}に対して、変分的手法により以下の結果を得た。
(1)β+1【greater than or equ
非自明古典解(C^2(B)∩C^1(B^^-)に属する解)は存在しない。
(2)0<α<min(β+1,(β+1)/2+1),α<2^*=(N+1)(N-2) ならば、非自明古典解が存在する。
(3)0<β【less than or equal】1,β+1【less than or equal】α<(β+1)/2+1 ならば、Holder連続な非自明解が一意的に存在する。これらの成果は、従来の結果を大幅に改良したもので、その全貌がほぼ解明されたと言える。しかしながら、1<β,(β+1)/2+1【less than or equal】α<β+1 の場合の
2.非有界領域における弱解に対するPohozaev型の不等式が、星状領域の外部領域及び柱状領域に対して確立され、準線形楕円型方程式の弱解の非存在に応用された。この結果、解の存在・非存在に関して、星状領域の内部と外部との双対性が明らかにされ、この分野における重要な知見が得られた。
3.Pohzaev型の(不)等式に依らない、正値解の非存在の為の新たな手法の端緒が開かれた。これは、領域は平行移動不変性と正値解の一意性の議論を組み合わせた議論によるもので、正値解の一意性がよく調べられている、固有値問題、sub-linear(sub-principal)caseに対して有力な道具を提供するものである。この手法のより一般的な場合への拡張が期待される。
その他、これに関連する周辺の成果も多数得られている。

To solve non-symmetric linear systems derived from the discretization of singular pertur-bation problems, we propose a generalized SOR method with multiple relaxation parameters, that is the improved SOR method with orderings and study its theory and practical use.
In the case of tridiagonal matrices, optimal choices of the parameters are examined : It is shown that the spectral radius of the iterative matrix is reduced to zero for a pair of parameter values which are computed from the pivots of the Gaussian elimination applied to the system. A proper choice of orderings and starting vectors for the iteration is also proposed.
We apply the above method to two-dimensional cases, and propose the "adaptive improved block SOR method with orderings" for block tridiafonal matrices. The point of this method is to change the multiple relaxation parameters not only for each block but also for each iteration. If special multiple relaxation parameters are selected and used with this method for an n * n block tridiagonal matrix whose block matrices are all n * n matrices, then this iterative method converges at most n^2 iterations.
We also proposed the improved SSOR method with orderings, which converges at most only one iteration for a tridiagonal system, and n iterations for a block tridiagonal system.
The generalized convergence theorems to the improved SOR method with orderings are also considered, and we study necessary and sufficient conditions for a matrix to be a generalized diagonally dominant.
Using the notation 'basic LUL factorization' of matrices, we give some techniques to obtain special multiple relaxation parameters such that the spectral radius of the iterative matrix is zero for the Hessenberg matrices and a class of matrices.

ハミルトン系の周期解,ホモクリニック解および非線型楕円型方程式の解の存在問題を変分的手法により研究し,次の研究実績をあげることができた.
1.特異なハミルトン系に対する周期解の存在問題は,従来2体問題に関連したラグランジュ系に対してのみ考察されていた.本研究においては,より一般的なハミルトン系で特異点をもつものに対して周期解の存在を考え,ミニマックス法と有限次元近似をあわせて用いることにより,特異なハミルトン系のクラスで周期解の存在が保証されるものを得ることができた.近年,ハミルトン系の周期解の存在問題はsymplectic幾何学の視点からも重要であることが認識され,盛んに研究が行われている.特異なハミルトン系に対してはenergy surface{(p,q);H(p,q)=h}はnon-compactとなり,non-compact集合に対してsymplecticな不変量を導入する問題と密接に関連するものと思われる.この関連を研究するため現在prescribed energy problemを初めてとして研究を続行している.また特異なハミルトン系に対する周期解の多重性も重要な問題である.この問題についても現在研究を続行している.
2.ホモクリニック解の存在については,non-compactなリーマン多様体上である種のラグランジュ系を考え,ミニマックス法により,その存在を得た.この結果はR^Nの場合であっても,ホモクリニック解の新しい存在結果を与えていると思われる.
3.非線型楕円型方程式に関してはR^N(N【greater than or equal】3)上でΔu+K(|x|)u^<(N+2)/(N-2)>=0を考察した.この方程式は微分幾何学における山辺の問題と関連した重要な方程式である.ここでは特に球対称解u(|x|)の存在を考察し,変分的手法により,その存在を非常に一般的なK(|x|)に対して示した.従来,球対称解の存在問題はシューチング法で扱われることが多いが,変分的手法を導入することによりより一般的な存在結果を得ることができた

By using some properties of potential-kernels of logarithmic type, we gave a definitive solution of the following well-known problem. "Does the totality of convolution kernels satisfying the domination principle coincide with the closure of the set of Hunt convolution kernels?" In this connection, we proved that a potential-kernel is spectral synthetic if it satisfies the domination principle. Applying to the theory of potential-kernels, we obtain that with a given potential-kernel satisfying the domination principle, its resolvent formed by nice potential-kernels is associated. Suggested by the sweeping-out process, we worked out an arc-variation to investigate the analytic capacity. In the study of the classical harmonic function theory, it is remarkable to determine domains on which non-zero subharmonic functions are not integrable.
Potential-kernels of logarithmic type possess recurrent semi-group,s which is closely related with the probability theory. In the study of the probability theory, we obtain a criterion of the transiency of Ornstein-Uhlenbeck type processes and results concerning optimal diffusion processes.

極小曲面の安定性を調べるのに重要な第2変分から定まるヤコビ作用素の研究とかかわって以下の結果を得た。
1、3次元リ-マン多様体が局所的に等質であるための条件を曲率テンソルを用いてあらわした。これはI、M、Singerが問題としていたところのものに対する3次元の場合の部分的解等を与えている。(大和一夫)
2、Lie接触構造の中で特に共形多様体の接球束上の構造に対して正規接続を共形接続から具体的に構成した。(佐藤肇)
3、ハミルトン系のホモクリニック軌道あるいは周期軌道の存在を変分を用いて研究した。モ-ス理論及び関数解析的な手法を用いることにより2〜3の存在定理を得た。(田中和永)
4、解析的偏微分方程式のGevrey族空間での可解性を研究したものによって定まる様子を詳しく調べた。特に指数が非正の場合のGevrey族空間での可解性の研究が偏微分方程式において始めて取り扱われた。(三宅正武)
5、領域の形状とそこの調和関数の境界挙動の関係を可積分性の観点から調べた。(鈴木紀明)
6、有限な全曲率を持つ種数Oの3次元コ-クリッド空間の極小曲面のガウス写像からきまるシュレジンガ-作用素のO固有値に対応する固有関数の代数的な構成方法を与えた。これを使って種数Oの場合のヤコビ作用素のindexは一般にはガウス写像の写像度dを使って2dーlとなることがわかった。(江尻典雄)
7、高全次元の場合の極小曲面のindexには単位法束上の解析が興味ある対象であり上の結果達と深い関係が期待される。

研究目的にかかげた目標に関する次の幾つかの興味ある成果が得られた。(1)我々の先行研究によって、p-Laplacianを主要項に持つ準線形放物型方程式u_t=△_p u+uに対する初期値境界値問題に対して、全ての解軌道を引き付ける「大域アトラクター」が、L^2で構成され、さらにそれが無限次元を持つ事実が知られていたが、これはかなり特殊な状況であり、非線形楕円型方程式に関するLyusternik-Scnirelman理論からも、その無限次元性は導出できるという難点があった。uをαu-b(x)|u|^q uとしても、大域アトラクターの存在とその無限次元性が導かれることが示された。これは、より一般的な非線形項f(x, u)に対しても、同様な結果が成立することを示唆する、重要な発見である。(2)多孔質媒質中を流れる流体(溶媒)の速度及び温度と流体中の溶質の濃度の振舞いを記述する、2または3次元有界領域におけるBrin kman-Forchheimer方程式の時間大域解の存在と一意性が、H^1に属する初期値に対して、示された。これによって、この方程式に対する、大域アトラクターの構成の出発点がクリアーされたことになる。また、よく知られているように、3次元空間におけるナビエ・ストークス方程式の一意的時間大域解の存在問題が未解決大問題である事実と比較すると、非常に興味深い知見を与えている

We carried out the investigation about the structure of solutions of nonlinear parabolic and elliptic equations. Our main results are as follows : Next, we studied the existence and uniqueness of solutions with moving singularities for a nonlinear parabolic partial differential equation. We also showed that there exists a solution with a moving singularity that changes its type suddenly., and made clear the asymptotic behavior of singular solutions that converges to a singular steady state. We also studied a chemotaxis system, and made clear the structure of self-similar solutions that blows up by concentrating to a point in finite time.For a reaction-diffusion system, which is called a Gierer-Meinhardt system, we studied the mathematical structure of pattern formation, and also made clear the behavior of time-dependent solutions

　変分的手法により非線型楕円型方程式の解の存在問題、Hamilton系の研究を行った。特に本年度はRNにおける非線型楕円形方程式の正値解の存在、多重度に関して進歩があった。　具体的にはRNにおける非線型楕円型方程式－Δu＋u＝a(x)up＋f(x)　in RN　　　　　　　　　(＊)正値解の存在、多重度を扱った。特に解が方程式に非常にデリケートに依存していることを示す次のような存在結果を得た。係数a(x)はコンパクト集合を除いて１、コンパクト集合上で０と１の間の値をとる連続関数とする。このとき非常に小さい、しかし０でない正の関数f(x)に対して、(＊)は少なくとも４つの正値解をもつ。そのうち２つの解はf(x)に連続的に依存しf(x)→0すると－Δu＋u＝a(x)upの解に収束する。しかし他のふたつはf(x)→0としても－Δu＋u＝a(x)upの解に強収束せず、無限遠にエネルギーの中心部が平行移動してゆく解ω(・－yf)(｜yf｜→∞　as f(x)→0)としてのプロファイルをもつ。(Calculus of Variations and Partial Differential Equationsより出版予定)。通常、微分方程式の研究においては方程式の係数等に解が連続的に依存する場合が研究されているが、(＊)のような簡単な方程式であっても微小な摂動f(x)により不連続な依存性が現れることを示しており、興味深いと思われる。また同様のアイデアに基づいたもう一編の論文では、より一般的な非線型項g(x,u)を扱いCao、Jeanjean、Hirano、Zhouらの結果を拡張している。(Nonlinear Analysis:T.M.A.より出版予定)。　またHamilton系に関してはmulti-bump solutionと呼ばれる記号力学系に対応した複雑な解軌道の構成をNehari多様体を用いた変分的な方法により行った。

　変分的手法により非線形微分方程式の解の存在問題の研究を行った。特に本年度は(i)２体問題型のハミルトン型に対する非有界軌道の存在、(ii)２体問題型のハミルトン型に対する周期軌道の存在と関連するnon-compact 多様体上の閉測地線の存在(iii)Moser-Trudinger 型の不等式の最良指数についての研究を行った。(i) ハミルトン型に関して２体問題型のポテンシャル〈I〉Ｖ(q) 〈/I〉～－1／｜〈I〉q 〈/I〉｜〈SUP〉α〈/SUP〉（α＞０）に対して無限から来て無限に飛びさる軌道の変分的な構成を考え、与えられた〈I〉Ｈ〈/I〉＞０をtotal energy としてもち、さらに与えられた入射角、出射角をもつ軌道の存在を空間次元〈I〉Ｎ〈/I〉に関する制限なしでstrong force 条件（α＞２）の下で示した。ここで空間次元が２のときは回転数を有効に利用することができ比較的容易に証明はなされるが、〈I〉Ｎ〈/I〉≧３の場合は異なりＲ〈SUP〉N〈/SUP〉＼｛０｝上のループ空間のtopology に関する考察が必要不可欠となることに注意して頂きたい。なお〈I〉Ｈ〈/I〉＝０のときは古典力学における放流軌道に対応し、非常に興味ある問題であるが、その存在は今後の課題としたい。(ii) またハミルトン系の周期解に関しても２体問題型ポテンシャル〈I〉Ｖ（ｑ）〈/I〉～～－1／｜〈I〉q 〈/I〉｜〈SUP〉α〈/SUP〉を考え、α＝２の場合を扱った。α＝２の場合はいわゆるstrong force とweak force の境界の場合でありprescribed energy problem は今までほとんど研究されていない。Total energy が０の周期解の存在を特に考え、その存在を示した。また関連する問題としてnon-compact Riemman 多様体（〈I〉Ｓ〈SUP〉n〈/SUP〉〈/I〉×Ｒ、〈I〉ｇ〈/I〉）上の閉測地線の存在を示した。(iii) 楕円形型方程式に関してはMoser-Trudinger 型の不等式の最良指数について研究を行い、Ogawa, ozawa により導入されたスケール不変なMoser-Trudinger 型の不等式は有界領域の場合と異なり最良指数を達成しないことを示した。

研究成果は（ⅰ）非線型楕円型方程式に関する存在結果および（ⅱ）ハミルトン系に対する周期軌道の存在問題に関する結果の２つに大別される。　まず非線型楕円型方程式に関してはRNにおけるscaler field equation -Δu+V(x)u=up, u∈H1(RN)の生値解について考察した。ここでは、ポテンシャルV(x)がxについて周期的な場合を主な対象とした。このような場合、ground stateと呼ばれるenergyが最小の解ω0 (x)が存在することはconcentration-compactness method等によりよく知られている。本研究では、より高いenergy levelの生値解の存在について考察し、ω0 (x)の重ね合わせた形の解u(x)～∑Nj=1ω0 (x-lj),(│lI-lj│>>1)の存在をポテンシャルV(x)に対する適当な条件の下で示した。非線型方程式に対する解の重ね合わせの方法はごく最近変分的に見直され、注目を集めているが、その存在のための条件は具体的にcheckしにくいものであった。ここではそのような具体例を初めて与えている。　次にハミルトン系の周期解に関しては、ポテンシャルV(q)が特異性V(q)=-1/│q│aをもつ場合に与えられたtotal energy Hをもつ周期解の存在について研究を行った。　この問題においては、特異性の指数αが重要な役割を果たすことがわかっている。02のときはH>0に対してのみ解の存在が期待でき、それぞれの場合に研究が行われている。ここではα=2の場合を考え、H=0に対する周期解の存在を考察した。α>2あるいは0<α<2の場合と異なり、α=2の場合は摂動に対して極めてsensitiveであるが周期解の存在を保証する十分広い摂動のクラスを求めるのに成功した。また対応する状況下でnon-compact Riemann多様体上の閉測地線の存在を議論した。研究成果の発表1998　(with A. Amrosetti) On Keplerian N-body type problems, in“Nonlinear Analysis and continuum mechanics”(G. Battazo, G.P. Galdi, E. Lanconelli, P. Pacci ed.) Springer, 1998, 15-25.1997　Multiple positive solutions for some nonlinear elliptic systems, Topological Methods in Nonlinear Analysis, 10 (1997), 15-45.