数理解析研究所講究録 1293 巻 2002 年 110-118

110

Asymptotics of Green functions
and Martin boundaries for
elliptic operators with periodic coefficients
Minoru Murata
Tokyo Institute of Technology
andTetsuo Tsuchida
Meijo University

村 田( 東 工 大 )土 田

哲 生

( 名 城大)July 9, 2002

The main purpose of this talk is to givethe asymptotics at infinity of aGreen function for an elliptic equation with.
periodic coefficients on
The second purpose is to completely determine the Martin compactification ofwith respect to an elliptic equation with periodic coefficients by using theexact asymptotics at infinity of the Green function.
$\mathrm{R}^{d}$$\mathrm{R}^{d}$

1. Asymptotics at infinity of Green functions
Let$L=- \sum_{j,k=1}^{d}\frac{\partial}{\partial x_{k}}(a_{jk}(x)\frac{\partial}{\partial x_{j}})-\mathrm{I}$ $b_{j}(x) \frac{\partial}{\partial x_{j}}+c(x)$$=-\nabla\cdot a(x)\nabla-b(x)\cdot\nabla+c(x)$be asecond order elliptic operator on
periodic Here
cients which are$d\geq 2$ ,$\mathrm{Z}^{d}$$\nabla=(\partial/\partial x_{1}, \cdots, \partial/\partial x_{d})$,$\mathrm{R}^{d}$with smooth real-valued coeffi-

111, and $b(x)=(b_{j}(x))_{j=1}^{d}$ .
, define an operator$a(x)=(a_{jk}(x))_{j,k=1}^{d}$For each (by$\in \mathrm{C}^{d}$$L(\zeta)$on the -dimensional torus$d$$\mathrm{T}^{d}=\mathrm{R}^{d}/\mathrm{Z}^{d}$$L(\zeta)=e^{-i\zeta\cdot x}Le^{i\zeta\cdot x}$$=-(\nabla+i\zeta)\cdot a(x)(\nabla+i\zeta)-b(x)\cdot(\nabla+i\zeta)+c(x|$where $i=\sqrt{-1}$ is the imaginary unit.
with the domain: aclosed operator in: the Sobolev space of order two: the formal adjointof,
For$H^{2}(\mathrm{T}^{d})$$L^{2}(\mathrm{T}^{d})$$L(\zeta)$,$H^{2}(\mathrm{T}^{d})$$L(\zeta)$$L(\zeta)^{*}$$\beta\in \mathrm{R}^{d}$$E(\beta)$: the principal eigenvalue of$L(i\beta)$

is areal eigenvalue of multiplicBy the Krein-Rutman theorem,
ity one such that the corresponding eigenspace is generated by apositive
function.
.
is also an eigenvalue of$E(\beta)$$L(i\beta)^{*}$$E(\beta)$$C_{L}=$$L$:subcriticalcase,:critical$C_{L}\neq\emptyset.${$u\in C^{2}(\mathrm{R}^{d});Lu=0$and $u>0$ inwhen apositive Green function for)when apositive Green function for

$L$

$L$

$L$

on

$\mathrm{R}^{d}$on$\mathrm{R}^{d}$}.

$\mathrm{R}^{d}$exists (In thisdoes not exist but$C_{L}\neq\emptyset$

For A$\in \mathrm{R}$, put, where is periodic}of the form {)and(Note: $L(i\beta)u=\lambda u$ onsuch that $(L-\lambda)\psi\geq 0$ and $\psi(x)=$$e^{-\beta\cdot x}u(x)>0$ , whereis periodic}
.
for
and
analogously toforandDefineFirst suppose that $\sup_{\beta}E(\beta)>0$ . Then
: subcritical.
$\Gamma_{\lambda}=${$\beta\in \mathrm{R}^{d};\exists\psi\in C_{L-\lambda}$$\mathrm{T}^{d}$$\mathrm{i}\mathrm{p}$$\mathrm{x})=e^{-\beta\cdot x}u(x)$$u$$E(\beta)=\lambda$$K_{\lambda}=\{\beta\in \mathrm{R}^{d};\exists\psi\in C^{2}(\mathrm{R}^{d})$$u$$K_{\lambda}^{*}$$L^{*}-\lambda$$\Gamma_{\lambda}^{*}$$K_{\lambda}$$\Gamma_{\lambda}$$L-\lambda$$L$$\forall S\in \mathrm{S}^{d-1}(\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e})\exists_{1}\beta_{s}\in\Gamma_{0}\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\sup_{S\{e_{s,1},\cdots,e_{s,d-1},s\}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}1\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{R}^{d}(\forall\in \mathrm{S})}\beta\in \mathrm{r}_{\theta_{-1}^{\beta\cdot s=\beta_{S}\cdot s}}$,For
: positive solution to $L(i\beta)u=E(\beta)u$: positive solution to $L(i\beta)^{*}v=E(\beta)v$, put
For functions and in$\beta\in \mathrm{R}^{d}$$u_{\beta}$$v_{\beta}$

$u$

$v$

$L^{2}(\mathrm{T}^{d})$$(u, v)= \int_{\mathrm{T}^{d}}u(x)\overline{v}(x)dx$.

112
Theorem 1Suppose that $\sup_{\beta}E(\beta)>0$ . Then the minimal Green function
has the following asymptotics as |x $-y|arrow\infty$ :G of L on$\mathrm{R}^{d}$$G(x, y)= \frac{e^{-(x-y)\cdot\beta_{s}}}{(2\pi|x-y|)^{(d-1)/2}}$$|\nabla E(\beta_{s})|^{(d-3)/2}$$\underline{u_{\beta_{*}}(x)v_{\beta_{s}}(y)}$$\cross\overline{(\det(-e_{s_{\dot{\beta}}}}\cdot$$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}E(\beta_{s})e_{s,k})_{jk})^{1/2}$$(u_{\beta_{s}}, v_{\beta_{*}})$,
where $s=(x-y)/|x-y|$ .
$\cross(1+O(|x-y|^{-1}))$

Here, let us recall some more facts.
: The generalized principal eigenvalue of$\lambda_{c}$$\lambda_{c}=\sup$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n}-\infty<\lambda_{c}<\infty$,$L-\lambda$

or critical.

{$\lambda\in \mathrm{R};L-\lambda$

$L$

on

$\mathrm{R}^{d}$, i.e.

is subcritical}

is subcritical for$\lambda<\lambda_{c}$, and$L-\lambda_{c}$is subcritical

The formal adjoint operator
of is subcritical (or critical) if and only
if is subcritical (or critical).
The generalized principal eigenvalue of andcoincide.
$L^{*}$

$L$

$L$

$L$

$L^{*}$, thenTheorem (Agmon&Pinsky) (i) If
strictry convex compact set with smooth boundary and$\lambda<\lambda_{c}$$\Gamma_{\lambda}=\partial K_{\lambda}$$K_{\lambda}$

is ad-dimensional.

, then.
for some, then(iii) If.
(iv) The function
is real analytic and strictly concave.
Its Hessian
is negative definite for any.
holds, and the supremum is attained uniquely
The equalityatin (ii).
.
if and only if(v) For any A,(ii) If A$=\lambda_{c}$$\Gamma_{\lambda}=K_{\lambda}=\{\beta_{0}\}$$\beta_{0}\in \mathrm{R}$$\Gamma_{\lambda}=K_{\lambda}=\emptyset$$\lambda>\lambda_{c}$$E(\beta)$$\beta\in \mathrm{R}^{d}$$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}E(\beta)$$\lambda_{c}=\sup_{\beta}E(\beta)$$\beta_{0}$$\nabla_{\beta}E(\beta)=0$$\beta=\beta_{0}$$\in \mathrm{R}$$\Gamma_{\lambda}=\{\beta\in \mathrm{R}^{d};E(\beta)=\lambda\}$$K_{\lambda}=\{\beta\in \mathrm{R}^{d};E(\beta)\geq\lambda\}$(vi)$K_{\lambda}^{*}=$.

-Kx, and $\beta_{0}=0$ if $L^{*}=L$ .

Now, let us look at the asymptotics of the Green function again. Notethat its main term is positive because of the assertion (iv)113
Theorem 1. Suppose that
. Then the minimal Green functionof L onhas the following asymptotics as$-y|arrow\infty$:|x$\lambda_{c}>0$$G$$\mathrm{R}^{d}$$G(x, y)= \frac{e^{-(x-y)\cdot\beta_{s}}}{(2\pi|x-y|)^{(d-1)/2}}$$|\nabla E(\beta_{s})|^{(d-3)/2}$$\underline{u_{\beta_{\epsilon}}(x)v_{\beta}.(y)}$$\cross\overline{(\det(-e_{s,j}}\cdot$$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}E(\beta_{s})e_{s,k})_{jk})^{1/2}$$(u_{\beta_{\theta},\beta_{*}}v)$,
where $s=(x-y)/|x-y|$ .
$\cross(1+O(|x-y|^{-1}))$This theorem is derived from the following theorem, where we regard
as aclosed operator in.
with the domain$L$$H^{2}(\mathrm{R}^{d})$$L^{2}(\mathrm{R}^{d})$Theorem 2Assume $E(0)>0$ . Then theexists, and the integral kernel
resolvent
totics as in Theorem 1.
$L^{-1}$$G$of$L^{-1}$

has the same asymp-

Actually, consider the operator$L_{1}=e^{\beta_{0}\cdot x}Le^{-\beta_{0}\cdot x}$.

Thensatisfies the assumption of Theorem 2, and the minimal Greenfunctionofsatisfies$L_{1}$$G_{1}$$L_{1}$$G_{1}(x, y)=e^{\beta_{0}\cdot x}G(x, y)e^{-\beta_{0}\cdot y}$.

Thus Theorem 1follows from Theorem 2.
Later, Iwill give an outline of the proof of Theorem 2.
Next, suppose that $\sup_{\beta}E(\beta)=0$ . Thenis critical if $d\leq 2$ , and subcritical if $d\geq 3$
Our second main theorem is the following$L$

Theorem 3Let $d\geq 3$ . Suppose that$\lambda_{c}=E(\beta_{0})=0$ . Put.
Then the minimal Green function of on
as $|x-y|arrow\infty$ :$H=-\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}E(\beta_{0})$

$G$

$L$

$\mathrm{R}^{d}$has the following asymptotics

114$G(x, y)= \frac{\Gamma(\frac{d-2}{2})}{2\pi^{d/2}(\det H)^{1/2}}\frac{e^{-(x-y)\cdot\beta_{0}}}{|H^{-1/2}(x-y)|^{d-2}}$$\cross\frac{u_{\beta_{0}}(x)v_{\beta_{0}}(y)}{(u_{\beta_{0}},v_{\beta_{0}})}(1+O(|x-y|^{-1}))$.

2. Martin boundaries

Now, let us determine explicitly the Martin compactification ofwithrespect to in the case.
Fix areference point in. Then the following proposition is adirectconsequence of Theorem 1.
Proposition 1. Suppose that
. Then for any sequenceinsuch thatand $y_{n}/|y_{n}|arrow\nu$ as $narrow\infty$ ,$\mathrm{R}^{d}$$L$$\lambda_{c}>0$$\mathrm{R}^{d}$$x_{0}$$\lambda_{c}>0$$\{y_{n}\}$$\mathrm{R}^{d}$$|y_{n}|arrow\infty$,$\lim_{narrow\infty}\frac{G(x,y_{n})}{G(x_{0},y_{n})}=e^{-(x-x_{0})\cdot\beta_{-\nu_{\frac{u_{\beta_{-\nu}}(x)}{u_{\beta_{-\nu}}(x_{0})}}}}$$x\in \mathrm{R}^{d}$.

Denote this right hand side by $K(x, \nu)$ . Then
, $K(x_{0}, \nu)=1$ ,if,is minimal in, i.e.,
If
satisfies $\psi(x)\leq K(x, \nu)$ on, then $\psi(x)=$Hence we can explicitly determine the Martin compactification of
as follows.
$K(\cdot, \nu)\in C_{L}$$K(\cdot, \nu)\neq K(\cdot, \mu)$$\forall\nu\in \mathrm{S}^{d-1}$$\nu\neq\mu$$K(\cdot, \nu)$$C_{L}$$\psi\in C_{L}$$L$$\mathrm{R}^{d}$$\mathrm{K}(\mathrm{x}, \nu)$$\mathrm{R}^{d}$for

Theorem 4Suppose that
. Then the Martin boundary and the minimal Martinboundary of
for are both equal to
the sphereat infinity which is homeomorphic to.
the Martin kernel at
is equal to
the Martin compactification offor is equal to$\lambda_{c}>0$$\mathrm{R}^{d}$$L$$\mathrm{S}^{d-1}$$\Gamma_{0i}$$\nu\in \mathrm{S}^{d-1}$$K(\cdot, \nu)_{f}$$\mathrm{R}^{d}$$L$$\{x\in \mathrm{R}^{d};|x|<1\}\cup[1, \infty]\cross \mathrm{S}^{d-1}$

equipped with the standard topology

115, we obtainIn the caseand ddirectly from Theorem 3the following theorem.
This result, however, is also asimple consequence of the known result
thatis one dimensional in this case.
$\geq 3$$\lambda_{c}=0$$C_{L}$

Theorem 5Suppose that $d\geq 3$ and$\lambda_{c}=E(\beta_{0})=0$ .
Then
in
with
for any sequence$\{y_{n}\}$$\mathrm{R}^{d}$$|y_{n}|arrow \mathrm{o}\mathrm{o}$as,$\lim_{narrow\infty}\frac{G(x,y_{n})}{G(x_{0},y_{n})}=e^{-(x-x\mathrm{o})\cdot\beta_{0_{\frac{u_{\beta_{0}}(x)}{u_{\beta_{0}}(x_{0})}}}}$$narrow\infty_{f}$$x\in \mathrm{R}^{d}$;the Martin boundary and the minimal Martin boundary are both equal to
one point
at infinity;
the Martin kernel at oo is equal to this right hand side;
the Martin compactification offor is equal to the one point compactificationof .
$\infty$$\mathrm{R}^{d}$$\mathrm{R}^{d}\cup\{\infty\}$$L$$\mathrm{R}^{d}$

3. Proof of Theorem 2

Finally, let us give an outline of the proof of Theorem 2.
Basic ingredients in establishing the asymptotics are an
integral representation of the Green function and the saddle point method
in complex integrations.
Let us give an integral expression of theresolvent of .
$L$

$2\pi \mathrm{T}^{d}=\mathrm{R}^{d}/(2\pi \mathrm{Z})^{d}$$\mathcal{H}=L^{2}(2\pi \mathrm{T}^{d}, \frac{d\zeta}{(2\pi)^{d}};L^{2}(\mathrm{T}^{d}))=\int_{2\pi \mathrm{T}^{d}}^{\oplus}L^{2}(\mathrm{T}^{d})\frac{d\zeta}{(2\pi)^{d}}$$\mathcal{F}$:$L^{2}(\mathrm{R}^{d})arrow H$$( \mathcal{F}f)(\zeta, x)=\sum_{l\in \mathrm{Z}^{d}}f(x-l)e^{-i(x-l)\cdot\zeta}$

Then is aunitary operator, andan isomorphism fromto.

$\mathcal{F}$$H^{2}(\mathrm{R}^{d})$$L^{2}(2\pi \mathrm{T}^{d}, (2\pi)^{-d}d\zeta;H^{2}(\mathrm{T}^{d}))$

.

.

116
The adjoint$\mathcal{F}^{*}$

is given by, for $g\in H$ ,$( \mathcal{F}^{*}g)(x-l)=\int_{2\pi \mathrm{T}^{d}}\frac{d\zeta}{(2\pi)^{d}}e^{i(x-l)\cdot\zeta}g((, x)$$x\in \mathrm{T}^{d}$,,$l\in \mathrm{Z}^{d}$,$L=\mathcal{F}^{*}\tilde{L}\mathcal{F}$$\tilde{L}=(2\pi)^{-d}\int_{2\pi \mathrm{T}^{d}}^{\oplus}L(\zeta)d\zeta$(since$(\nabla_{x}+i\zeta)\mathcal{F}f=\mathcal{F}(\nabla f)$)

Assume $E(0)>0$ . Thenis areal analytic function fromto the Banach space of bounded operators on$2\pi \mathrm{T}^{d}$$L(\zeta)^{-1}$$L^{2}(\mathrm{T}^{d})$.

$L^{-1}=\mathcal{F}^{*}M\mathcal{F}$$M=(2 \pi)^{-d}\int_{2\pi \mathrm{T}^{d}}^{\oplus}L(\zeta)^{-1}d\zeta$

That is,,$\forall x\in \mathrm{T}^{d}$$l\in \mathrm{Z}^{d}$, and$f\in L^{2}(\mathrm{R}^{d})$,$L^{-1}f(x-l)= \int_{2\pi \mathrm{T}^{d}}F(\zeta)\frac{d\zeta}{(2\pi)^{d}}$

,

where
$F( \zeta)=e^{:(x-l)\cdot\zeta}L(\zeta)^{-1}(\sum_{m\in \mathrm{Z}^{d}}f(\cdot-m)e^{-:(\cdot-m)\cdot\zeta})(x)$Meromorphic extension of, heFor each$s\in \mathrm{S}^{d-1}$$L(\zeta)^{-1}$$\beta_{s}\in\Gamma_{0}$$\mathrm{t}$.

such that$\sup_{\beta\in\Gamma_{0}}\beta\cdot s=\beta_{s}$.

$s$$\eta_{s}=\beta_{s}/|\beta_{s}|$$\{e_{s,1}, \ldots, e_{s,d-1}, s\}$: orthonormal basis of$\mathrm{R}^{d}$$e_{s}=(e_{s,1}, \ldots, e_{s,d-1})$

Introduce new coordinates $(w, z)$ near$i\beta_{s}$such that$\zeta=w\eta_{\theta}+z\cdot e_{s}=w\eta_{s}+\sum_{j=1}^{d-1}z_{j}e_{s_{\dot{\theta}}}$$w\in \mathrm{C}$,$z=(z_{1}, \ldots, z_{d-1})\in \mathrm{R}^{d-1}$,.

Proposition 2.
such thatandwith $|z|<r$ ,the inverse $L(w\eta_{s}+z\cdot e_{s})^{-1}$ has asimple pole $w_{s}(z)$ as afunction ofand has the following asymptotics at the pole$\exists r>0$$\forall s\in \mathrm{S}^{d-1}$$z\in \mathrm{R}^{d-1}$$L(w \eta_{s}+z\cdot e_{s})^{-1}=\frac{A_{s,z}}{w-w_{s}(z)}+O(1)$

.

$w$

117
Here

$A_{s,z}$is arank one operator-valued function with
$A_{s,0}= \frac{i}{\eta_{s}\cdot\nabla E(\beta_{s})}\frac{(\cdot,v_{\beta_{s}})u_{\beta_{s}}}{(u_{\beta_{s}},v_{\beta_{s}})}$

and$w_{s}(z)$,is asmooth function having the following properties:$w_{s}(0)=i|\beta_{s}|$$\partial w\vec{\partial z_{j}}(0)=0$$(1\leq j\leq d-1)$$\frac{\partial^{2}w_{s}}{\partial z_{j}\partial z_{k}}(0)=\dot{\iota}\frac{\partial^{2}{\rm Im} w_{s}}{\partial z_{j}\partial z_{k}}(0)=i.\frac{e_{s},{}_{j}\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}E(\beta_{s})e_{s,k}}{\eta_{s}\cdot\nabla E(\beta_{s})}$$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}{\rm Im} w_{s}(0)=(\frac{\partial^{2}{\rm Im} w_{s}}{\partial z_{j}\partial z_{k}}(0))_{1\leq j,k\leq d-1}>0$

Now, regard the integral expression of the resolvent of via $L(w\eta_{s}+z$ .
as acomplex integration for .
Deform the contour of the integral in .
Finally, apply the residue theorem and the following saddle point
method to get the asymptotics at infinity of the Green function.
Proposition 3. Let $n=d-1$ . Let be an open neighborhood of the.
origin in-functions on aneighborhood of satisfying
and $a(x)$ beLetfor some constants and .
andAssume thatand it is positive definite.
Further suppose that $\exists p>0$ such thatand ${\rm Re}(\varphi(x)-\varphi(0))\geq p|x|^{2}/4$ for $x\in U$ .
for
Then the asymptotics
$L$

$w$

$e_{s})^{-1}$

$w$

$U$

$\mathrm{R}^{n}$$\overline{U}$$C^{\infty}$$\varphi(x)$$b_{1}$$||a||_{C^{6}(U)}\leq b_{2}$$||\varphi||_{C^{9}(U)}\leq b_{1}$$b_{2}$$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}\varphi(0)=\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}{\rm Re}\varphi(0)$$p|x|^{2}\leq x\cdot \mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{p}(0)\mathrm{x}$$x\in \mathrm{R}^{n}$$\int_{U}e^{-\lambda\varphi(x)}a(x)dx=(\frac{2\pi}{\lambda})^{n/2}\frac{e^{-\lambda\varphi(0)}}{(\det \mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}\varphi(0))^{1/2}}$$\mathrm{x}(\mathrm{a}(0)+O(\lambda^{-1}))$as A$arrow \mathrm{o}\mathrm{o}$holds.

4. References[A] S. Agmon, On positive solutions of elliptic equations with periodic, spectral results and extensions to elliptic operators on
coefficients in$R^{d}$

118
Riemannian manifolds,
Proc. Internat. Conf. on Differential Equations. I.W.Knowles and R.T.Lewised., North-Holland, 1984, pp.7-17[KS] M. Kotani and T. Sunada, Albanese maps and off diagonal long time
asymptotics for the heat kernel,
Comm. Math. Phys. 209(2000), 633-670[P] R.G. Pinsky, Second order elliptic operators with periodic coefficients:
Criticality theory, perturbations, and positive harmonic functions,
J. Funct. Anal., 129(1995), 80-107[Sch] C. Schroeder, Green functions for the Schrodinger operator withperiodic potential,
J. Funct. Anal., 77(1988), 60-8