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Nonlinear problems with singular diffusivity and inhomogeneous terms (Mathematical Analysis of pattern dynamics and related topics)

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(1)

Nonlinear

problems

with

singular

diffusivity

and inhomogeneous

terms

Hirotoshi Kuroda (Hokkaido University)

In this talk

we

consider

a

singular diffusion equation associated with total variation with inhomogeneous terms

as

follows

$u:[0,1]\cross[0, T)arrow \mathbb{R}^{n}(n\geq 1)$ : unknown function

$(P)\{\begin{array}{ll}u_{t}-\frac{1}{b(x)}div(a(x)\frac{u_{x}}{|u_{x}|})=0, (x, t)\in(O, 1)\cross(0, T), (1)u(x, 0)=u_{0}(x), x\in(O, 1), (2)u(O, t)=g_{0}, u(1, t)=g_{1}, t\in(O, T), (3)\end{array}$

where $a(x),$$b(x)$

are

given positive, continuous functions on $[0,1]$ and $u_{0}$ is

an

initial data and

$g_{0},$ $g_{1}\in \mathbb{R}^{n}$

are

boundary condition. This equation (1) is written

as

the gradient system by

taking

energy

$E(u)= \int_{0}^{1}a(x)|u_{x}|dx$

with respect to the

norm

$\Vert f\Vert^{2}=\int_{0}^{1}b(x)|f(x)|^{2}dx$

.

The equation (1) describes the motion of

multi-grain problem studied in [3].

In the scalar valued

case

with boundary condition $u(O)=0,$$u(1)=1$, if $a(x)$ has a unique

ninimum point $x_{0}$, then

$E(u)= \int_{0}^{1}a(x)|u_{x}|dx\geq a(x_{0})\int_{0}^{1}u_{x}dx=a(xo)(u(1)-u(0))=a(x_{0})$

.

If $u$ is

a

step function and jumps only at $x_{0}$, then the equality holds. So global minimizer is

unique [2]. In generalcase,

a

global minimizerquite naturallyhas

a

discontinuity sinceit makes

the

energy

low by concentratingitsvariation at the point where $a(x)$ is minimal. It follows that

manyglobal minimizers may be piecewise constant functions.

We consider stationary problem of (P) in the vector valued

case.

Suppose that

inhomoge-neous term

$a(x),$$b(x)$ satisfy “concave condition” (cf [1]). We characterize stationary piecewise

constant solutions.

References

[1] M.-H. Giga, Y. Gigaand R. Kobayashi, Very singular diffusionequations, Proc. Taniguchi

Conf.

on

Math., Advanced Studies in Pure Mathematics, 31(2001), 93-125.

[2] R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. stat. Phis., 95(1999),

1187-1220.

[3] R. Kobayashi, J. A, Warren and W.

C.

Carter, A continuum model of grain boundaries,

Physica D, 140(2000), 141-150.

数理解析研究所講究録

参照

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