Grant amount：\155740000 （ Direct Cost: \119800000 、 Indirect Cost：\35940000 ）

We prove the existence and the uniqueness of a solution and clarify its behavior for evolution equations mainly nonlinear diffusion equations describing evolution of patterns and shapes like crystal growth phenomena. We introduce new notions of a solution which allows shape with singularities for equations having singular structure. We thus establish foundation of mathematical analysis which easily describes real phenomena. Based on these fundamental results, we are able to numerically calculate phenomena which had been difficult to calculate, for example, phenomena of colliding spirals on surfaces of crystals.

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Grant amount：\16250000 （ Direct Cost: \12500000 、 Indirect Cost：\3750000 ）

In this research project we have shown that Lagrangian equivalence among Lagrangian submanifold germs and a certain equivalence relation among the corresponding graph-like wave fronts are the same. Applications of this result include the classical differential geometry, the geometry of the space-time, the differential geometry of singular surfaces and mappings, and so on. On the other hand, a new application of singularity theory to quantum mechanics is also discovered in this research project.

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Grant amount：\3900000 （ Direct Cost: \3000000 、 Indirect Cost：\900000 ）

This research investigates a relation between the scalar conservation equation and the level-set equation based on local interface speed model for a premixed combustion flame. A new model formulation is introduced for the source term of the conservation equation in three-dimensional interface phenomena, which gives the solution of the level-set equation coupled with a re-initialization procedure for the physical interface problems governed by conservation law. It may be an extensional formulation for a diffusive solution of the level-set equation.

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Grant amount：\41080000 （ Direct Cost: \31600000 、 Indirect Cost：\9480000 ）

We proved the basic existence and regularity theorems for the mean curvature flow considered in the framework of geometric measure theory which is called Brakke's mean curvature flow. As for the existence theorem, when given an arbitrary n-dimensional closed set in an n+1-dimensional Euclidean space, we proved the time global existence of the Brakke's mean curvature flow that evolves from the given initial data. For the analysis of the singular set, we proved the regularity theory around triple junction in the one-dimensional case, and showed the strong stability property of the triple junction within the weak topology of measure.

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Grant amount：\13910000 （ Direct Cost: \10700000 、 Indirect Cost：\3210000 ）

Relationships among harmonic functions, solutions to the heat equation, the Green and heat kernels and their defining domains, the influence of the space complexity to the boundary behavior were studied. They were applied to various fields such as non-smooth Euclidean domains, manifolds, varifolds, networks and fractals. In particular, new results were obtained in Harnack principle with exceptional sets, estimates of the principal frequency in terms of capacitary width, sufficient conditions for the global boundary Harnack principle based on the capacitary width of sublevel sets of the Green function, conditions for the parabolic boundary Harnack principle (Intrinsic Ultracontractivity), the critical exponent of a graph domain enjoying the global boundary Harnack principle, and the 0-1 law of the capacity density at infinity and so on.

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Grant amount：\3900000 （ Direct Cost: \3000000 、 Indirect Cost：\900000 ）

I studied spectra of elliptic operators for regularly or singularly deformed domain (Lame operator, Stokes operator, Maxwell operator). (i) I studied polynomial solutions, rational type solutions with their structures of homogeneous Stokes and Elastic equations (with H.Ito, N.Honda), (ii) I obtained spectral Hadamard variational formula of Stokes operator, Maxwell operator for regularly perturbed domain for Dirichlet and Slip type boundary condition (with E. Ushikoshi). I obtained an elaborate behaviors of eigenvalues for Maxwell operator, (iii) I studied elaborate behaviors of eigenvalues of Lame or Maxwell operators in a domain with small hole, (iv) I obtained elaborate behaviors of eigenfrequencies of elastic body composed of several thin rod.

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Grant amount：\3380000 （ Direct Cost: \2600000 、 Indirect Cost：\780000 ）

I have been studying a mathematical model describing phase separation phenomena for the past 15 years. Through the analysis of such model, I realized the importance of a viewpoint to consider surface orientation and surface measure as a pair. In particular, I obtain such a pair as a limiting object when I consider a singular perturbation problem of mean curvature flow. Using this characteristic, I am able to obtain some existence and regularity theorems. The examples of my theorems are existence and regularity theory of mean curvature flow with transport term (joint work with Keisuke Takasao, in review) and existence theorem and characterization of boundary condition of mean curvature flow on a convex domain with Neumann condition (joint work with Masashi Mizuno, in review).

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Grant amount：\16510000 （ Direct Cost: \12700000 、 Indirect Cost：\3810000 ）

In this research project we constructed the notion of 'lightlike curvature' for spacelike submanifolds of Lorentzian space forms by using the notion of 'lightcone Gauss maps'.As an application of the theory of Legendrian singularities, we described the singularities of the lighhtlike hypersurface along a spacelike submanifold. Moreover, we constructed a geometric framework to describe the caustics of world sheets which is an important notion in the theory of general relativity and the brane world scenario. We clarified the relation of the caustics and the wave front propagations iby using the theory of graph-like Legendrian unfoldings.

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Grant amount：\174850000 （ Direct Cost: \134500000 、 Indirect Cost：\40350000 ）

Various phenomena in natural sciences, for example, crystal growth phenomena and fluid motion, etc. are often modeled by nonlinear partial differential equations. We describe evolution of complex shapes and patterns observed there as mathematical phenomena and analyze them by developing the theory of viscosity solutions, variational analysis and real analysis, etc. In evolution of shapes and patterns even if the evolution law is simple and the initial shape is smooth, the solution often develops singularities by forming corners after some time. It is necessary to extend notions of a solution in a suitable way to interpret non-differentiable functions as a solution of differential equations for further analysis. In this project we introduce several new notions of solutions of diffusion equations describing for example crystal growth or fluid motion. We prove the existence of such solutions and analyze their behavior. Moreover, we study relation between discrete and continuous models.

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Grant amount：\15080000 （ Direct Cost: \11600000 、 Indirect Cost：\3480000 ）

A family of smooth surfaces parameterized by time is called mean curvature flow if the velocity of motion at each point and time is equal to its mean curvature vector. We have made fundamental advance of knowledge on the general existence and regularity theory of mean curvature flow which may have singularities, and moreover, on those of geometric time evolution problems in large.

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Grant amount：\4550000 （ Direct Cost: \3500000 、 Indirect Cost：\1050000 ）

For a closed differentiable manifold M, a differential invariant of M, called the Yamabe invariant of M, is one of fundamental subjects in conformal geometry. The studies of the Yamabe problem on singular spaces and Yamabe constants are also important. The results of this Research Theme are the following :
・Aubin's Lemma for the Yamabe constants of infinite covering spaces.
・Positive Mass Theorem for asymptotically flat spaces with singularities.
・Computations of the orbifold Yamabe invariant.
・Proof that any closed 3-manifold with positive flat conformal structure is Kleinian.
・The solvability of the Yamabe Problem on conic spaces.

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Grant amount：\4420000 （ Direct Cost: \3400000 、 Indirect Cost：\1020000 ）

I consider the evolution of the harmonic maps, called the harmonic map flows. In particular, I study the global existence and regularity of weak solutions of the harmonic map flows. I improve the regularity criterion for weak solutions and show the global existence of the dissipative wave equations, with applications to the wave maps.

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Grant amount：\31070000 （ Direct Cost: \23900000 、 Indirect Cost：\7170000 ）

Problems on important functions such as (super, sub)harmonic functions appearing in analysis, geometry and probability are referred to as potential problems. We have investigated potential problems from various view points and unveiled deep properties of important functions in connections with nonsmooth domains, fractals, manifolds and functions spaces.

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Grant amount：\47450000 （ Direct Cost: \36500000 、 Indirect Cost：\10950000 ）

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.

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Grant amount：\18850000 （ Direct Cost: \14500000 、 Indirect Cost：\4350000 ）

For 1) inverse scattering problems, 2) thermography, 3) inverse problems for equations in fluids, some new reconstruction schemes and an framework which integrates several known reconstruction schemes are given.

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Grant amount：\4060000 （ Direct Cost: \3400000 、 Indirect Cost：\660000 ）

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Grant amount：\4030000 （ Direct Cost: \3100000 、 Indirect Cost：\930000 ）

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Grant amount：\46930000 （ Direct Cost: \36100000 、 Indirect Cost：\10830000 ）

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Grant amount：\15680000 （ Direct Cost: \12800000 、 Indirect Cost：\2880000 ）

In this research project, we applied Singularity Theory to some areas in Mathematics such as Differential Geometry, Symplectic Geometry, non-linear Partial Differential Equations etc, so that we have obtained several results. Moreover, we have obtained related results on some boundary areas such as Astrophysics etc. Especially, we applied Singularity Theory to Differential Geometry of submanifolds in several kinds of space forms. Then we constructed new geometries (Horospherical Geometry, Slant Geometry) and induced some new invariants. We also clarified the geometric meanings of these invariants. As a result, we have a geometric characterization of the singularities andthe shape of event horizons

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Grant amount：\14430000 （ Direct Cost: \12600000 、 Indirect Cost：\1830000 ）

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Grant amount：\3500000 （ Direct Cost: \3500000 ）

We obtain the following results and try to publish the papers in some Journal.
(1) Free boundary problem for m-harmonic maps and m-harmonic map flow
We show the existence of the local in time solution of the m-harmonic map flow into smooth compact manifold with free boundary on a closed submanifold of the target manifold, which satisfies the m-harmonic map flow equation in the weak sense and is H"older continuous with its gradient in time-space region up to the boundary of the space domain. The maximal existence time of the solution is estimated below by the m-energy of the initial datum. Also, the singular behavior of the solution at the singular time (maximal existence time) can be characterized by a non-constant m-harmonic maps into the target manifold. The m-harmonic map is defined on m-dimensional sphere, or m-dimensional ball with free boundary, and they are called m-harmonic sphere, or m-harmonic disk, respectively. These solutions are exactly minimal submanifolds in the target manifold.
(2) Finite singularity of the m-harmonic map flow
It is expected that the singular set at the singular time is consist of finitely many points. In the paper, we make device of some formula and try to prove the conjecture. However, we are faced with a serious gap of the proof., which is now studied by us to be overcome. We obtain the formula which says the monotonicity of the scaled energy in the intrinsic way to the m-harmonic Laplace operator and is of its own interest.
(3) A priori estimates for the linearized parabolic system of non-divergence form
We show the a priori estimates in some Sobolev space hold for the linearized parabolic system of the m-harmonic map flow and the existence of a strong solution of the system. The existence result is combined with the Leray-Schauder fixed point theorem aid the reflection method to show the local in time solution of the m-harmonic map flow with free boundary.

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Grant amount：\3700000 （ Direct Cost: \3700000 ）

We have studied the following :
(1)Study of Yamabe Invariants
We estimated and determined the Yamabe invariant of some positive 3-manifolds, by using the inverse mean curvature flow and families of Green's functions. In especial, we classified completely all 3-manifolds with Yamabe invariant greater than that of RP^3. We also studied the positive Yamabe constants of Riemannian products and the behavior of them under magnifying one factor. We are now studying Aubin's type lemma for the positive Yamabe constants of infinite coverings, with some new results.
(2)Study of Conformal/Affine Geometry and group C^*-algebra
We gave new developments on conformal and projective geometry. In particular, we obtained an interesting varitional characterization of affine connections induced from Einstein metrics. We studied on twisted K-theory and groupoid C^*-algebras, and then proved a generalization of Gromov-Lawson theorem for foliated spaces.
(3)Study on Non-linear Analysis in Geometry
We studied on eigenvalue problem on complete manifolds, discrete groups and valiational problems on mean curvature. We then obtained results on non-existence of eigenvalues on complete manifolds of non-positive curvature, and a fixed-point theorem for discrete-group actions on Hadamard spaces.

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Grant amount：\1200000 （ Direct Cost: \1200000 ）

フォッカープランク方程式の解である,有限次元ユークリッド空間上の確率密度関数で、与えられた初期確率分布と終期確率分布を持つ物を考える。これら全てに関して,その一般化されたエネルギーの最小値と,同じ与えられた初期確率分布と終期確率分布を持つ連続セミマルチンゲールに関する一般化、されたエネルギーの最小値が一致する事を示した.
更に,フォッカープランク方程式の解である,有限次元ユークリッド空間上の確率密度関数に関して,その一般化されたエネルギーの最小値と,同じ確率密度関数を持つ連続セミマルチンゲールに関する一般化されたエネルギーの最小値が一致する事を示した.
特にこれにより,フォッカープランク方程式の解である,有限次元ユークリッド空間上の確率密度関数が一般化された有限エネルギー条件を持たす時,1次元周辺分布の確率密度関数が与えられた確率密度関数になるような連続セミマルチンゲールで,ドリフトベクトルはフィードバック型であり,同じ1次元周辺分布を持つ連続セミマルチンゲールの中で一般化されたエネルギーの最小値を取るものの存在がわかった。特に,コスト関数が無限遠方で2次以上の増大度を持つ場合,この連続セミマルチンゲールは,一意で,マルコフ性を持つ事もわかる.

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Grant amount：\3200000 （ Direct Cost: \3200000 ）

We obtain the following results and prepare the papers to be published in some Journal.
(1)Existence and regularity for the evolution of constant mean curvature surfaces in high dimension
In high dimension where the domain dimension is equal to or greater than 3, the mean curvature of the parametric surfaces is given by the m-Laplace operator of the map which is the parametrization of the surface.
We show that, If the initial boundary data is of small image in some sense, there exists a time-global weak solution The solution has the image of the same size as the datum, and its gradients are H"older continuous except some closed set in the domain. The size of the except set for regularity is estimated in the Hausdorff measure of some dimension.
To show the existence of a weak solution, we use the variational method called discrete Morse semi-flow, which is the minimization of the family of the functionals, of which the Euler-Lagrange equations are the time-discrete equations of the Rothe type.
To have the regularity of a weak solution, we use the fundamental regularity theorem for the evolution of p-Laplace operator with lower order term of the critical growth on the gradient, which was obtained by Masashi Misawa in 2002.
(2)Regularity and singularity for a singular perturbation problem
We study a singular perturbation problem in a phase transition., and in particular, we study the regularity of the interface which is the level set of the limit function, of the singular perturbation problem. To investigate the regularity and singularity of the interface of the limit function, we make device of the formula for the scaled energy, called monotonicity formula.
(3)Free boundary problem for minimal surfaces in high dimension
We study the free boundary problem for minimal surfaces in high dimension. The existence of a solution is proved by the variational method, in particular, the minimax method combined with some approximation., and the solution is nearly unstable. We also study the relation of the unstable solution with the singularity of the evolution of minimal surfaces in high dimension.. It is shown that there exists a time-global weak solution of the evolution of minimal surfaces with free boundaries in high dimension, and that the solution and its gradient is H"older continuous except finitely many times. Moreover, the singular time is characterized by the existence of a non-constant minimal surface with free boundaries.
We will try to study the free boundary problem for p-harmonic maps with values into smooth compact Riemannian manifold, the evolution, of p-harmonic maps, and moreover the wave equations and wave maps into smooth compact Riemannian manifold.

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Grant amount：\2900000 （ Direct Cost: \2900000 ）

本研究課題の研究目的である生データからタンパク質の3次元初期モデルを自動生成することは当初のアイデアでは、なかなか難しいことが分かった。それは部分的にはアイデアは有効であっても、その有効性の度合いがあまり顕著とはいえないこと、そしてそれを全体の中にどう組み込むかについてのソフトウエア作成技術上の問題のためである。そこで今年度は、関連する研究を行い具体的な成果をあげることにつとめた。具体的な研究成果は、次の二つである。
(1)Pyrococcus horikoshii由来のTenAホモログの構造解析を行った.分子表面には負の電荷を持った部位が存在しており,その部分に未知のリガンドが結合していた.このことからTenAはこれまで考えられていたような転写因子ではなく,おそらくは酵素として働いているものと考えられる.
(2)当該研究課題の関連研究として画像データの認識手段に有効な数値微分法を考案し、その数値実験を行った。数値実験結果は極めて良く、かつ誤差に対してもrobustな方法であることが分かった。この方法は、当該研究に関連する蛋白質の結晶生成過程に必要な溶液中の結晶成長の有無等を、写真撮影画像データから自動判別するのに有効な可能性が高い。またこのような応用ばかりでなく、MRE検査で得られるデータから初期乳がんの発見をするのにも有効と思われる。

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Grant amount：\48880000 （ Direct Cost: \37600000 、 Indirect Cost：\11280000 ）

Several mathematical foundation is established to investigate the mechanism of variation of shape of solutions due to anisotropy and diffusion effect for nonlinear diffusion equations. We just give two typical topics.
(i)Equation describing motion of a crystal surface. When anisotropy of surface energy is strong like snow crystal, there appears a plat surface called a facet in their equilibrium shape. The equations governing its evolution is formally a curvature flow equation. However, the diffusion is so strong that it cannot be viewed as a partial differential equation. We studied a Stefan type free boundary problem describing snow crystal growth assuming that its equilibrium shape is a cylinder. We established (a)Local solvability under the assumption that a facet stays as a facet ; (b)Berg's effect concerning behavior diffusion field near on a facet ; (c)a necessary and sufficient condition for facet bending ; (d)a sufficient condition for existence of a self-similar solution and size condition on facet bending ; (e)a rigorous proof that a facet preserved near equilibrium. A further problem is whether one can track evolution after facet bending.
(ii)Equations in fluid mechanics : An invicid Burgers equation is a typical coarse model to describe motion of compressible fluid. The solution develops jump discontinuity in finite time even if the initial data is smooth for the Burgers equation. Just before this proposal, the principal inverstigator introduced a notion of a proper viscosity solution to describe a solution with jumps which is also useful for equations with nondivergence type. In this project we established a notion of singular vertical diffusivity which is useful to calculate the solution numerically in the graph space. In particular we are successful to calculate a proper viscosity solution by a level set method numerically.

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Grant amount：\12100000 （ Direct Cost: \12100000 ）

We studied identifying the discontinuity of the medium such as inclusions, cavities, cracks and the physical property of the medium. For identifying the discontinuity for the medium, we improved and adopted the probe method and enclosure method. Especially, we studied the behavior of the reflected solution and the unique continuation property which are essential for the probe method, and we accomplished the probe method. As for the enclosure method, we enlarged its application by replacing the complex geometric optic solution which is difficult to construct and localize by introducing the osciallating-decaying solution. We also showed that the reconstruction methods for the inverse boundary value problem such as the probe method, singular source method, no response test are unified into the no response test, and the probe method and singular source method are the same methods. For the inverse scattering problem, we solved the difficulty of the linear sampling method by proposing two new reconstruction methods. Moreover, we succeeded in establishing the probe method for the one space dimensional parabolic equation and giving the theoretical frame work for Shirota's computational method for identifying the discontinuity of the coefficient for the wave equation.
As for identifying the physical property of the medium, we studied two inverse problems for identifying the residual stress and the damage of steel-concrete connected beam. We gave the dispersion formula of the speed of the Rayleigh wave and applied it for the former inverse problem. For the latter problem, we established identifying the damage from the frequency response function which is a practical measured data. We also studied identifying the coefficient for the nonlinear wave equation and succeeded in observing that we can identify the linear and the quadratic part of the coefficient by linearizing the Dirichlet to Neumann map.

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Grant amount：\8060000 （ Direct Cost: \6200000 、 Indirect Cost：\1860000 ）

平成16年度は(1)等方的な表面張力がその分離の主原因となっているvan del Waals 2相分離モデルにおいて、表面エネルギーの有界性と化学ポテンシャルに対応する量のソボレフノルムの有界性を仮定し、界面領域の厚みを0に近づけたときの極限界面について新しい結果を得た.同様な方向で2002年に得た結果では部分的なソボレフノルム有界下での結果が得られたが、今年度の結果で完全にシャープな結果を得た(論文準備中).この結果を示すにあたっては新しいregularizationの技術を用いたのであるが、他の現在懸案となっている関連問題への突破口となる結果であり、現在精査中である.(2)確率論の大偏差原理に動機付けられた、安定相間の遷移確率を決定する汎関数、Allen-Cahn Action(ACA)の特異極限問題についてReznikoffは各種のスケーリング則を得たが、特に空間的な非一様性が得られるスケーリングにおいてはACAの上からの評価を得ていた.一方、下からの評価についてのシャープな結果を共同研究で得ることが出来た(Kohn, Reznikoffとの共著論文を投稿中).この結果は空間1次元の結果であるが、(1)の結果を用いる事で各種の2、3次元の結果が得られることが期待される.(3)Penn State大学のLiu氏によって問題提起された非圧縮性流体移流効果のある場合の平均曲率流弱解構成への幾何学的測度論の応用に関して引き続き研究成果があった。

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Grant amount：\35360000 （ Direct Cost: \27200000 、 Indirect Cost：\8160000 ）

In this research project, various space-time behavior of solutions to nonlinear dispersive equations, such as nonlinear Schrodinger equations (NLS) and KdV type equations, nonlinear hyperbolic equations, such as nonlinear wave and Klein-Gordon equations, and coupled systems of those equations, such as nonlinear field equations. The main results are the following :
(1)Asymptotic completeness in the energy space H^1(R^3) for NLS with repulsive case has been proved.
(2)A unified treatment for small data scattering for nonlinear field equations has been given in terms of critical and subcritical setting.
(3)Existence and uniqueness of self-similar solutions for nonlinear wave equations have been proved in the framework of weak Lebesgue spaces.

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Grant amount：\10500000 （ Direct Cost: \10500000 ）

For the construction of Morse (variational) flows, we introduced the "Discrete Morse Flows" method, which has been adopted and named the Minimizing Movements method by De Giorgi. This approach can he used in the construction of Morse flows ; starting from initial data we look for minimizers of a series of variational functionals introduced inductively. By using this method, we construct Morse flows to variational problems of harmonic map type. We have noticed this method can still be used under weaker assumptions on the initial, and boundary data. We are trying to approach the construction problems of Morse flows to harmonic map variational problems between metric manifolds which have no smoothness of the coefficients of the second order differential operators. In the analysis of Discrete Morse Flows, we carry out local estimates of time-discrete partial differential equations of elliptic-parabolic type, and we have achieved De Giorgi-Nash type Holder estimates and Campanato estimates, and the higher integrability of the gradients of Gehring type. Such estimates are proved to hold independently of the approximation scheme, which enables us to construct Morse flows through Discrete Morse Flows. In conclusion, as a result of this research, we have recognized the characteristics of the Discrete Morse Flows method that can be made use of in the problem of constructing Morse flows for which Leray-Schauder theory with Schauder estimates cannot be directly applied. M.Misawa has treated the regularity and construction of p-harmonic map flows and S.Omata has made mathematical and numerical analysis of Morse flows to several kinds of variational problems. M.Kashiwagi has constructed an algorithm of nonlinear optimizations and composed its software available in http ://srg.pi.cuc.ac.jp/〜kasiwagi/numeric/20040221/

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Grant amount：\3300000 （ Direct Cost: \3300000 ）

(I) Non-trivial state solutions to the Ginzburg-Landau equation with magnetic effect are studied. Particularly, in a non-uniform thin 3-d domain, pattern is constructed (Jimbo with Morita). In 2-d convex domain, it is proved that no pattern formation exists (Jimbo with P. Sternberg).
(ii) Vortex motion in nonstationary Ginzburg-Landan equation (without magnetic effect) is studied. The reduced ODEs of vortex motion obtained by F.H. Lin and Jerrard-Soner are rewrittened in comprehensive form. The dynamics in the Neumann B.C. case is studied (Jimno and Morita).
(iii) The perturbation of eigenvalue problem of elliptic operator with discontinuous coefficients (or vaiable coefficients (or vaiable coefficients and perforated domain) is studied (Jimbo with Kosugi).
(iv) The phase transition boundary arising in the Allen-Cahn equation (with small diffusion coefficients) is studied. The regularity and the geometic property of the free boundaries are investigated (Tonegawa).
(v) The surface evolution equation driven by anisotropic effect of curvature (existence of solution and properties) is studied (Tonegawa).
(vi) Minimal surface problem with free boundary is studied. The hyperbolic evolution equation with free boundary is studied (Omata).
Numerical analysis are also done.
(vii) The vortex motion arising in the hyperbolic Ginzburg-Landau equation is studied by computational method (Omata).

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Grant amount：\2000000 （ Direct Cost: \2000000 ）

当該研究では物質が液体及び気体等の異なる2つの相を安定維持するような状態の重要な数理モデル化のひとつであるファンデルワールスのフェイズフィールドモデルの解析を主目標としている。物質の状態を表すために、表面張力効果を持つ特異摂動項を持つような非凸エネルギー汎関数をここでは考える。相分離面の厚みを表すパラメターによる停留点の漸近挙動が当初考えられた。予備的な結果としてJ.Hutchinsonとの共同研究により、相分離界面はパラメターが十分小さい時には定平均曲率曲面にハウスドルフ距離の意味で近いことが示された。これは以前知られていた結果がエネルギー最小解のみの知見であることと異なり、一般停留点に対しての結果であることが特記される。その後オイラー・ラグランジェ方程式におけるラグランジェ乗数が関数である場合について研究した。これは変数化学ポテンシャル項を考察した事にあたり、極限相分離界面の平均曲率値がこの化学ポテンシャルによって決定される事を示した。(研究発表1)研究集会における意見交換で特に応用数学者との情報交換を行い、当該研究に関連する問題としてリーマン多様体上での特異摂動、解析と、サイン-ゴードン方程式の不安定回転解との関係が指てきされ、現在リーマン多様体版の研究成果をあげつつある。

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Grant amount：\4200000 （ Direct Cost: \4200000 ）

For the construction of Morse (variational) flows, we introduced the Discrete Morse Flows method, which has been adopted and named Minimizing Movements method by De Giorgi. This approach has some merits in construction of Morse flows, since we can make the most of minimizers to a series of variational functionals inductively introduced starting with initial data. By using this method, we construct Morse flows to variational problems of harmonic map type, in the treatment of which we have noticed the method is available under the weaker assumptions on initial and boundary data. We are trying to approach the construction problems of Morse flows to harmonic map variational problems between metric manifolds, which have no smoothness of the coefficients of the second order differential operators. In the analysis of Discrete Morse Flows, we carry out local estimates of time-discrete partial differential equations of elliptic-parabolic type, so that we have achieved De Giorgi-Nash type Holder estimates and Campanato estimates, and the higher integrability to the gradients of Gehring type. Such kinds of estimates are proved to hold independently of the approximation schemes, which enables to construct Morse flows through Discrete Morse Flows. In conclusion of a series of these research work, we have recognized the characteristics of Discrete Morse Flows method that it can be made use of to the construction problems of Morse flows, to which Leray-Schauder with Schauder estimates cannot be directly applied.

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Grant amount：\1600000 （ Direct Cost: \1600000 ）

流体現象を中心とした非線形現象に現れる種々の非線形偏微分方程式の可解性及び解の定性的性質を調べてきた.1998年度の研究成果は以下の通りである.
渦糸方程式に関しては,軸流を伴う場合も伴わない場合も,その初期値問題及び周期境界条件を課した初期-境界値問題の時間大域解の一意存在が証明できた(Publ.RIMS Kyoto Univ.に掲載).二相問題と自由境界問題が物理学上も数学上も興味ある今後の重要な問題であるが,それに関しては次の結果が得られた.
(1) 圧縮性Euler方程式(渦あり)の二次元亭状領域での一相自由境界問題(深さが無限の時)の時間局所解の一意存在が証明できた(Adv.Math.Sci.Appl.に掲載予定).
(2) (1)と同じ問題が有限の深さの場合にも自由表面上で表面張力が作用する,しないに拘らず唯一つの時間局所解を持つこと及び表面張力係数が0に収束する時対応する解も収束することが証明できた.これらの結果は現在投稿準備中である.
(3) Boussinesq近似をした非圧縮性Navier-Stokes方程式に対する一相Stefan問題の古典解の存在が証明できた(SIAM J.Math.Anal.に掲載予定).
Steran問題とも関連するが,相転位を記述するモデル方程式であるChan-Hilliard方程式の特徴は空間4階微分を含むことにあり,そのためその数値解析はまだ十分にはなされていない.離散格子上でいくつかのパターンについて調べた結果が情報処理学会論文誌に掲載された.
相転位現象は自由エネルギーの形により考える方程式は変わってくるが,いずれにしてもエネルギー最小化関数の等高線集合を用いて議論が重要である.その基礎理論を構築する一環として,楕円-放物型差分微分方程式の解の正則性を変分法を用いて調べた(Z.Angew.Math.Phys.に掲載).さらに詳しい解の性質を調べるには何らかの仮定のもとで常微分方程式に帰着するのが常套手段である.その基礎理論として,2変数Bessel関数の満たす偏微分方程式系の不確定型特異集合のまわりで解の漸近挙動を調べた(Tohoku Math.J.に掲載).

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Grant amount：\2900000 （ Direct Cost: \2900000 ）

A smooth action of a discrete group GAMMA on a differentiable manifold M is, by definition, a homomorphism of GAMMA into Diff M, the diffeomorphism group of M.One of the main theme in the theory of group actions is to depict the whole space A (GAMMA, Diff M) of smooth actions of GAMMA on M.By rigidity (in a wide sense) is meant a claim which says that the space A(GAMMA, Diff M) is "small".
Invariant Geometric Structures. One of the approaches to the rigidity problem is frying to find a geometric structure that is invariant under a given group action. We found a new example for which this approach works.
Global Analysis on Foliated Manifolds. Rigidity for group actions often amounts to some global-analytic problem on a foliated manifold. We were able to describe the spectrum of the tangential laplacian on a certain foliated manifold.
Infinite-Dimensional Homogeneous Spaces of Diffeomorphism Groups. We studied geometry and topology of such spaces especially bearing an application to rigidity problem in mind.
Also there have been done researches on infinite-dimensional minimal submanifolds in infinite-dimensional spaces (by Maeda) , on the blow-up phenomenon of the Yang-Mills gradient flow (by Maeda) , on hydrodynamic limit of a spin system (by Suzuki) , and on a regularity theorem for a certain free boundary problem (by Tonegawa).

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