Transferencia reactiva

8/10/2019 Transferencia reactiva

1/84


2/84

L

Q

L Q

L + Q

L1Q


3/84


4/84


5/84

L Q

L Q

L + Q L1Q


6/84


7/84


8/84


9/84

M[a, f](x)


10/84

Iw[a,f] Iw[a,a M]

R1az Iw[az ,fz] R1az

Iw[az,az Mz]

L1

[a, f

] Q

[a, f

]


11/84


12/84

f a

f fP={f(x), x P}

P

RaP[f

P] f

P

aP

aP

fP

P

P

fP

RaP[f

P] a

P

fP

a

P

aP

aP

fP

aP


13/84

aP

aP

fP

RaP

[f

P

]

P

(Pz)zR

Pz ={(y, z) R3, y R2} z R f R3 fz

(y

)=f

(y, z

), y R2 Pz

a

f R3

Pz

Raz[fz](s, )= R

fz(t + s)e 0 az(t++s)ddt s R, S1, S1 ={ R2, = 1} R2


14/84

P

=

0 1

1 0

(s,

) R S1

R2 Raz[fz] z

az fz

M[a, f](x) x R3

M[a, f](x)= S2

0f(x + t)e t0 a(x+)ddtd, S2 ={ R3,= 1}.

Pz

Raz[azMz[a, f]](s, )=

0 az(t+s

)Mz[a, f](t+s

)e

0 az(t++s)d

dt s R, S1

.

M[a, f] Pz

a f R3

Raz

[fz

](s,

) s, z R, S1

Raz

[az Mz

[a, f

]](s,

) s, z R, S1.


15/84

(a, f)


16/84


17/84

f Rn

f

(x

)dx<

Rn

f

(x

)2dx< f

Ff()= f()= Rn

eixf(x)dx. f

Ff, =f,F S(Rn), f f Rn Sm R Sm ={x Rm+1,x = 1}

Ff(, )= Rn eixf(x, )dx a.e. Sm.


18/84

f

f

F1f(x)= f(x)= 1

(2

)n

Rneixf(x)dx

f

x Rn

f nxif(x)

F( xj

f)()= ijf() j {1, 2,..n}.

f

g

f g(y)= Rnf(y x)g(x)dx F

(f g

)(

)= F

(f

) F

(g

)(

).

f

Rnf(x)2dx=(2)n

Rnf()2d.

Rn

C0(Rn)={f Rn R, f },C0()={f C0(Rn), (f)={x Rn, f(x) 0} },

C0( Sm)={f Rn Sm R, f (f) Sm} m N.

C(Rn)={f Rn R, f }C(Rn Sm)={f Rn Sm R, f }

C0(Rn)={f C(Rn), f }C0(Rn Sm)={f C(Rn Sm), f }


19/84

Rn

C()={f C(Rn), (f) }C( Sm)={f C(Rn Sm), (f) }

C0

(

)=

{f C

(

), f

}C0( Sm)={f C( Sm), f } Lp Lp(Rn) Lp(Rn) p[1, ]

fLp(Rn) =Rnf(x)pdx1p p[1, ), f = xRn f(x).

Rn, p[1, ] Lp

(

)=

{f Lp

(Rn

), f

(x

)= 0 x

}.

f

K K L2(K) L2(K) Lp Rn p[1, )

C0()Lp(Rn) Lp() p[1, ).

S

(Rn

)=

{f C

(Rn

), sup

xf

< , Nm, m N

}.

j 0 S(Rn) xj 0 Rn

S(Rn) S(Rn) Hs s R f S(Rn) f

f

Hs

(Rn

)=

f

s=

Rn f

(

)2

(1 +

2

)sd

12

.

Rn

Hs(Rn)={f S(Rn),fHs(Rn)


20/84

f(, ) Hs(Rn) a.e. Sm Hs(Rn Sm)

fHs(RnSm) =Smf(, )2sd12 ,

Rn

Hs(Rn Sm)={f S(Rn Sm),fHs(RnSm) 0

H

s(Rn) Hs(RSm) Hs(Rn) Hs(RSm) , Hs(R Sm) f, g= Smf(, ), g(, )d f(, ) Hs(Rn), g(, ) Hs(Rn) Sm f, g CfHs(Rn)gHs(Rn) f Hs(Rn), g Hs(Rn), f, g CfHs(RnSm)gHs(RnSm) f Hs(Rn Sm), g Hs(Rn Sm).

f, g

Sm

f

(,

), g

(,

)d

CSmf(, )Hs(Rn)g(, )Hs(Rn)d

C

Smf(, )2

Hs(Rn)d

Smg(, )2

Hs(Rn)d

=CfHs(RnSm)gHs(RnSm).


21/84

0 < 1 f Rn

f

C(Rn) = sup

x f

(x

)+ sup

xy f

(x

) f

(y

)x y ,

fC(Rn) = supxy

f(x) f(y)x y , Rn

C(Rn)={f C0(Rn),fC(Rn)


22/84

Hf(x)= f P.V. 1x

=1

lim0+

x

f

(t

)x tdt +

x+

f

(t

)x tdt

=

1

lim0+

f(x + t) f(x t)t

dt.

Lp(R) f Lp(R), p > 1

HfLp(R) CpfLp(R)

f L2(R)

F(Hf)()= isgn()f ,

sgn()=

1 >0

0 =0

1


23/84

F1x ()=

R

eix1

xdx

= R

cos

(x

)xdx i

R

sin

(x

)xdx

=0 iR

sin(x)(x)(dx)=i sgn().

Hs(R) f Hs(R) s R Hf Hs(R)

f

Hs(R) =

Hf

Hs(R).

Hs(R) HfHs(R) =

R

(1 + 2)s Hf()2d=

R

(1 + 2)s isgn()f()2d=

R

(1 + 2)sf()2d=fHs(R).

f H1(R) d

dxHf=H

d

dxf R.

L2

F( d

dxHf)()= i(i) sgn()

f()= isgn()if()=F(H d

dxf)().


24/84

f S(Rn) Rf[, ]=

x=f(x)dl(x), Sn1, R,

x = ni=1 xii Rn dl

(x

)

n 1 {x Rn, x = }

f S

(Rn

)

F(Rf)(, )= f(), Sn1, R.

F(Rf)(, )= R

eis[Rf](s, )ds=

R

eisx=

f(x)dl(x)ds=

R

x=

ei(x)f

(x

)dl

(x

)ds

= Rn

ei(x)f(x)dx =F(f)().


25/84

f S(Rn)

jRf(, )= R[xjf](, ) j {1, 2,...n}.

F(jRf(, )) = jiF(Rf(, ))=jif(), R, Sn1.

F(R[xjf](, )) =xjf()=i

(

)j

f

(

)=ijf(), R, S

n1

.

f S(Rn) 2Rf(, )= R[f](, ).

R R g(, ) C0(R Sn1) Rg(x)=

Sn1g(x , )d.

Rf,gL2(RSn1) =f, RgL2(Rn) . f C0(Rn) g C0(R Sn1)

Rf,g

L2(RSn1) =

Sn1

R

Rf

(,

)g

(,

)dd

= Sn1 R x= f(x)g(, )dl(x)dd= Sn1 Rnf(x)g(x , )dxd= Rnf(x) Sn1g(x , )ddx=f, RgL2(Rn) .


26/84

KRn f L2(K) Rf(, ) L2(R)

Rf(, )L2(R) C(K)fL2(Rn) Sn1, C(K) K g C0(Rn) (g) K

1K(x)= 1 x K0 x K ,

Rg

(s,

)2

L2(R

)

=

R x=s

g

(x

)dl

(x

)2

ds

= R

x=s

g(x)1K(x)dl(x)2 ds

R

x=s

(g(x))2dl(x)x=s

12K(x)dl(x)ds

(K)n1Rx=s

(g(x))2dl(x)ds= (K)n1

R2(g(x))2dx

=

(K

)n1

g

2L2(R2)

,

C0(K) L2(K) f L2(K) K Rn f Hk(K), k 0

Rf(, )Hk(R) C(K)fHk(Rn) Sn1, C(K) K

L2

(R

)

g C0(K) isRg(, )L2(R) =R[xig](, )L2(R)

C(K)xigL2(Rn)C(K)gH1(Rn).


27/84

sRg(, )2L2(R) = R

sRg(s, )2ds=

R

n

i=1

2i

sRg

(s,

)2ds

=ni=1RisRg(s, )2ds

=

n

i=1

isRg(, )2L2(R)

n

i=1

C(K)g2H1(Rn)nC(K)g2H1(Rn).

k =1

f Hk(K) g Hk+1(K) xig Hk(K) R[xig](, )Hk(R) C(K)xigHk(Rn) C(K)gHk+1(Rn) k N

f C0(Rn)

f(x)= cn RHn1 Rf

(x

) n

Rn1 Rf(x) n cn =

1

2(2)1n (1)(n2)2 (1)(n1)2 ,

Rf

f C0

f(x)= 1(2)nRn eixf()d =

=

f(x)= 1(2)n 0 Sn1 exf()n1dd

[0, ] R


28/84

n

n1

f(x)=1

2(2)nRSn1 eix

f()n1

dd

=1

2(2)nRSn1 eix 1in1F[n1 Rf](, )dd=

in1

2(2)n1Sn1 12R eixF[n1 Rf](, )=

in1

2(2)n1Sn1 n1 Rf( x, )d=cnR

n1 Rf(x). n

sgn(x)

f(x)= 12(2)nRSn1 eixf() sgn()n1dd

=1

2(2)nRSn1 eixiF[H(Rf)](, )n1dd=

1

2(2)nRSn1eix

i

in1

F

[n1 H

(Rf

)](,

)dd

=in2

2(2)n1 RHn1 Rf(x)

a C0(Rn) x Rn

Sn1

(Ba)(x, )= 0

a(x + t)dt x Rn, Sn1.


29/84

R> 0 x (a),x R xBa = a

Ba = 0 x > R

Ba = R0[a] x < R g(t) = a(x + t) d

dtg(t) = xa(x + t)

a

a(x + t)t= = 0 a

x > R

a

a C(Rn) K Ba(x, ) C(RnSn1) BaC C(K)aC C(K)

Ba(x, ) Ba(y, ) 0a(x + t) a(y + t)dt

0

a

(x + t

) a

(y + t

)(1K

(x + t

)+ 1K

(y + t

))dt

0 aC x +t y t(1K(x + t) + 1K(y + t))dt=aC x y

0(1K(x + t) + 1K(y + t))dt

2aC (K)x y. a C0(Rn) K

Ba(x, ) CKa x Rn, Sn1.

Ba(x, )= 0

a(x + t)dta

01K(x + t)dt

(K)a.


30/84

a H1(R2) C0(R2) K

Ba

(x,

) C

a

H1(R) x R

2, S1.

x= s +

Ba(x, )= 0

a(s + ( + t))dt

R

a(s + t)dt=R[ a ](s, ).

H1(R) L(R) Ba(x, ) CR[ a ](, )H1(R) x R2, S1

Ba(x, ) C(K) a H1(R2) x R2, S1 a H1(Rn) aH1(R2) Ba(x, ) C(K)aH1(R2), x R2, S1.

a, f C0(R2)

Raf(s, )= R

f(s + t)e(Ba)(s+t,)dt, s R, S1. f(x)

s e(Ba)(s+t,)

s + t

Raf(s, )= x=s

f(x)e(Ba)(x,)dl(x), s R, S1 Raf(s, )= Raf(s, ) Raf(s, )=Raf(s, )


31/84

a, f C1(R2)

f

(x

)=

1

4

S1eBa(x,

)

(ehHehRaf

)(x ,

)d,

h(s, )= 12(I+ iH)Ra(s, )

Raf

Raf Gg(s, )= g(s, )

f(x)= 14

S1

eBa(x,)(ehHehGRaf)(x , )d.

f L2(R2) w C0(R2 S1)

Iwf(s, )= Rw(s

+ t,)f(s

+ t)dt. f Iw(, ) L2(R)

K R2 f L2(K) w C0(Rn Sn1) KR2

Iwf(s, )= 1K(s)Iwf(s, ) s R, Sn1. R> 0 KB

(0, R

) f

(x

)= 1B(0,R)

(x

)f

(x

)

Iwf(s, )= Rw(s + t,)f(s + t)1B(0,R)(s + t)dt, s> R t R s + t B(0, R)

Iwf(s, )= 1[R,R](s)Iwf(s, ).


32/84

0 k < 12 k +12 < 1 f Hk(R2) w(x, ) C(R2 S1)

IwfHk+12(RS1) CwC(R2S1)fHk(R2).


33/84

u R3 S2 R

x R3 S2

xu(x, )= 0 x R3, S2 xu(x, )

u(x + s,)= u(x, ), s R, x R3, S2


34/84

xu(x, )= a(x)u(x, ), a

(x

) R3 R

u(x, )= Ce0 a(x+t)dt C R x R3 {, 1, 2} R3 x s, 1, 2 R x= s + 1

1 + 2

2

xu(x, )= su(s + 11 + 22 , )=s

Ce

0

a((s+t)+1

1+2

2)dt

=s Ces a(t+11+22)dt=Ces a(t+11+22)dt (a(s + 11 + 22))= Ce0 a(x+t)dta(x)= u(x, )a(x).

xu(x, ) + a(x)u(x, )= f(x) x R3, S2,limt

u(x t,)= 0 x R3, S2, f R3 R

f

a

R3 R f(x + t)dt< x R3, S2

u(x, )= 0

f(x + t)e0t a(x+s)dsdt.


35/84

x= + {w R3, w = 0}, R xu(x, )= u( + , )

xu

(x,

)=

0

f

(( + t

) +

)e

0

t a((+s)+)dsdt

=

f(t + )e 0ta((+s)+)dsdt=

f(t + )e t a(s+)dsdt

0g(t, )dt= g(, ) +

0g(t, )dt

xu(x, )= f( + )e a(s+)ds +

f(t + )e t a(s+)ds(a( + ))dt=f(x) a(x)u(x, ).

x= + ()= u( + , )

() + a( + )()= f( + )lim

()= 0,

s(x,,) x R3 S2

S2

x R3 S2

xu

(x,

)+ a

(x

)u

(x,

)+

S2u

(x,

)s

(x,,

)d =f

(x

)+

S2u

(x,

)s

(x, ,

)d

limt

u(x t,)= 0.

S2u(x, )s(x,,)d

(x, ) S2u(x, )s(x, , )d


36/84

x

x

x

x R3

(x, ) s(x,,) =s(x, )

xu(x, ) + a(x)u(x, ) +S2

u(x, )s(x, )d =f(x) +S2

u(x, )s(x, )d xu(x, ) + a(x)u(x, ) + u(x, )

S2s(x, )d =f(x) +

S2u(x, )s(x, )d

xu

(x,

)+ aT

(x

)u

(x,

)= f

(x

)+

S2u

(x,

)s

(x,

)d x R3, S2

aT(x)= a(x) +S2 s(x, )d. ui(x, ) i

u(x, )= i=0

ui(x, ),


37/84

xu0(x, ) + aT(x)u0(x, )= f(x) x R3, S2 xui

(x,

)+ aT

(x

)ui

(x,

)=

S2

s

(x,

)ui1

(x,

)d i 1, x R3, S2

limt

ui(x t,)= 0, i 0, x R3, S2

s(x, ) = s(x)k( ) s(x) C Cs(x)=a(x) k( )= 14 [0, 1]

aT(x)= a(x) +S2

s(x)k( )d =s(x)(C+ 1) C=

(4

(1 + C

))1

xu0(x, ) + aT(x)u0(x, )= f(x) x R3, S2 xui(x, ) + aT(x)ui(x, )= CaT(x)

S2ui1(x, )d i 1, x R3, S2

limt

ui(x t,)= 0. i 0, x R3, S2

f a D > 0, R f(x+ t)dt 0

yBaz(y, )= tBaz(t + s, )= az(t + s),


43/84

Baz(y, )= 0 y > R Baz(t + s, )= 0 t> R, Baz(y, )= R0[az](y , ) y < R

Baz

(t + s,

)= R0

[az

](s,

)= R az

( + s

)d t< R.

Raz+az fz(s, )= R

eB[az+az](t+s,)fz(t + s)dt,

d

d=0

Raz+az fz(s, )= R

eBaz(t+s,)fz(t + s) (B[a](t + s, ))dt.

eBaz(t+s)fz(t + s) = tB[eBaz(,)fz()](t + s, ),

dd =0Raz+az fz(s, )= R tB[eBaz(,)fz()](t + s, )B[az](t + s, )dt,

d

d=0

Raz+az fz(s, )= R

B[eBaz(,)fz()](t + s, )tB[az](t + s, )dt+ B[eBaz(,)fz()](t + s, )B[az](t + s, )t=

t=

d

d=0

Raz+az fz(s, )= R

B[eBaz(,)fz()](t + s, )az(t + s)dt

R

eBaz(+s,)fz( + s)d

R

az(t + s)dt=

R

B[eBaz(,)fz()](t + s, ) R

eBaz(+s,)fz( + s)d az(t + s)dt.

B[eBaz(,)fz()](t + s, ) R

eBaz(+s,)fz( + s)d

=

0eBaz(t++s

,)fz(t + + s)d R

eBaz(+s,)fz( + s)d

= 0

eBaz(t++s,)fz(t + + s)d,


44/84

d

d=0

Raz+az fz(s, )= R

0

eBaz(t++s,)fz(t + + s)d az(t + s)dt,

w

[az, fz

](y,

) = 0 eBaz(y+,)fz

(y +

)d,

d

d=0

Raz+az fz(s, )= Iw[az,fz]az(s, ).

a f (a,f) DRaz[fz](az, fz)= Iw[az ,fz][az] + Raz[fz].

Raz[az Mz[a, f]] a()a a

a(Raz(az Mz[a, f]))a = Iw[az ,az Mz[a,f]az + Raz(az Mz[a, f]) + Raz(az aMz[a, f]a). a f

(a,f

)

DRaz(az Mz[a, f])(a,f)=Iw[az ,az Mz[a,f]]az + Raz(az Mz[a, f])+ Raz(az aMz[a, f]a) + Raz(az M[a,f]).

aM[a, f]a

a, f

a

f

a = a a

f= f f

M=Mz[a, f] Az0 Raz(fz) Iw[az ,fz][az] + Raz[fz],

Az1 Raz(az M) Iw[az ,az Mz]az + Raz(az Mz) + Raz(az a Mza) + Raz(az Mz[a,f]).


45/84

Pz

R1az

R1azAz0 Raz(fz)R1az Iw[az,fz][az] + fz,R1azAz1 Raz(az M)R1az Iw[az,az Mz]az + az Mz + az a Mza + az Mz[a,f]. K R (f) (a) (a) (f) K3 =

K K K (fz) (az) (az) (fz)K2 =K K zR zKc, fz, az, az, fz =0

R1az

z K K K K

(y)= 1, y K2 1 Az0 = R

1az

Iw[az ,fz][az] + fz,Az1 = R

1az

Iw[az ,az Mz]az + az Mz + az a Mza + az Mz[a,f].

Az0 Az1 A

z0 A

z1

R1azAz0

fz Az0

R1azAz1 az

Mz Az1.

K2 () zRLz[a, f], Qz[a, f] L2(K3) L2(K3) L2(K2) L2(K2),

Lz[a, f](a,f)= R1az Iw[az,fz][az] + fzaz Mz

,Qz[a, f](a,f)= 0R1az Iw[az,az Mz][az] + (az a Mza) + (az Mz[a,f]) .

a,f L2

(K3

), K3 R3 az, fz

z R

az, fz

L

2

(K2

) Lz, Qz z R a,f L2(K3)

(Lz + Qz)[a, f](a,f)= Az0Az1

zR.


46/84

a,f

a,f L2(K3) L2(K) L2(K)

L

[a, f

], Q

[a, f

] L2

(K3

) L2

(K3

) L2

(K; L2

(K2

) L2

(K2

)),

L[a, f](a,f)=Lz[a, f](a,f)zR ,Q[a, f](a,f)=Qz[a, f](a,f)zR .

(L + Q)[a, f](a,f)= Az0Az1

zR

.

L Q

(L + Q)[a, f] a, f C0(K3) C>0 M[a, f](x) C x K3

L[a, f]

L1[a, f] L2(K; L2(K2) L2(K2)) L2(K; L2(K2) L2(K2)) L2(K3) L2(K3)

L1

[a, f

] g

h=

hz

Mz, gz R

1az

Iw[az,fz]

[hz

Mz

]zK

C>0 M[a, f](x) C x K3 f0 f0 a f M[a, f]

3.5

a, f C0(K3) Mz(y), az(y) fz(y) zK, y K2 L1z

[a, f

]

L1z[a, f] L2(K2) L2(K2) L2(K2) L2(K2)

L1z[a, f] gzhz =hz Mz, gz R1az Iw[az ,fz][hz Mz] gz, hz L2(K2) M[a, f](x) C>0 x K3 M z K az, fz az, fz L2(K2)


47/84

L1z[a, f]Lz[a, f](a,f)= L1z[a, f] R1az Iw[az ,fz][az] + fzaz Mz =

az

Mz

Mz, fz +

R1az Iw[aZ,fz]

[aZ

]

R1az Iw[az ,fz]

[az

Mz

Mz

]=(az, fz)zR=(a,f)

L1[a, f] gh=L1z gzhz

zK

,

L1[a, f]L[a, f](a,f)=(az, fz)zK=(a,f). z R

f 0 f0 f >0 A

f(x) 1A(x) f2

x K3

M[a, f](x)= S2

0f(x + t)et0 a(x+s)dsdtd

ediam(K3)a

S2

0f(x + t)dtd

ediam(K3)a

f

2 S2

0

1A

(x + t

)dtd

ediam(K3)a

f2

minxK3

S2

01A(x + t)dtd

>0.

x

a= 0 f C0

(K3

) C>0 M

(x

) C x K3 Q

[a, f

]= 0

(L + Q

)[a, f

]

L[a, f] Q[a, f]= 0 a= 0 az =0

az a Mza = 0,

az Mz[a,f]= 0,


48/84

w[az, az Mz]= w[0, 0]= 0, Iw[0,0]az(s, )=

R

w[0, 0](s + t,)az(s + t)dt= 0. Qz

[a, f

] Qz

[a, f

]= 0 Q

[a, f

]=

Qz[a, f]zK=0 K, K R K K a H52(K3) f C(K3) > 12 f0 f0 C0(K2) K L1 Q

L1[a, f] L2(K; L2(K2) L2(K2)) L2(K; L2(K2) L2(K2))Q[a, f] L2(K3) L2(K3) L2(K; L2(K2) L2(K2))

D >0

(L + Q

)[a, f

] L2

(K3

) L2

(K3

)

a H52

(K3

)

a

H52

(R3

)

< D

(L + Q)1[a, f]= L1[a, f] k=0

(QL1)[a, f]k .

a f K

H52

(K3

) H52

(K3

) C

(K3

)

C

(K

3

) I

w[a,f] Iw[a,a M]

Rn f C() z R fz(y) =f(y, z), y Rn1 z ={y Rn1,(y, z) } fz C(z) fzC(Rn1) fC(Rn),fzC(Rn1) fC(Rn).

fz f fz(y1) fz(y2) =f(y1, z) f(y2, z)fC (y1, z) (y2, z)fC y1 y2.


49/84

RN f1, f2 C() f1 f2 C() f1 f2C(Rn) 2f1C(Rn)f2C(Rn),

f1 f2

C(Rn) 3

f1

C(Rn)

f2

C(Rn).

f1f2(x) f1f2(y)=f1(x)(f2(x) f2(y)) f2(y)(f1(y) f1(x))f1f2C x y + f2f1C x y2f1C f2C x y,

f1f2 f1f2 f1C f2C,

f1f2C(Rn) 3f1C(Rn)f2C(Rn). K R a, f C(K3) M[a, f] C(R3)

M[a, f]C(R3) C(K)eC(K)a(1 + aC(R3))fC(R3), C

(K

)

K

a f

M[a, f](x)= S2

0f(x + t)e t0 a(x+s)dsdtd.

at,(x)= e t0 a(x+s)ds at,(x) at,(y) eC(K)a t

0a(x + s) + a(y + s)ds

eC(K)a

t

0 a

C

x y

1K3

(x + s

)+ 1K3

(y + s

)ds

C(K)eC(K)a aC x y, at,(x) eC(K)a

at,C(R3) C(K)eC(K)a(1 + aC(R)).


50/84

ft,(x) = f(x + t) f

M[a,f](x) M[a, f](y)

S2

0 ft,at,

(x

) ft,at,

(y

) (1K3

(x + t

)+ 1K3

(y + t

))dtd

2S2

0fC at,C(1K3(x + t) + 1K3(y + t))dtdx y

C(K)eC(K)a(1 + aC)fC S2

0(1K3(x + t) +1K3(y + t))dtdx y

C(K)eC(K)a(1 + aC)fC x y, M

M[a, f](x) C(K)eC(K)a f C(K)eC(K)a fC,

M[a, f]C(R3) C(K)eC(K)a(1 + aC)fC.

K R a, f C(K3) w[az, az Mz], w[az, fz] C(R2 S1) z K

w[az, fz]C(R2S1) C(K)eC(K)a(1 + aC(R3))fC(R3),

w

[az, az Mz

]C(R2S1) C

(K

)eC(

K)a

(1 +

a

C(R3)

)2

a

C(R3)

f

C(R3),

C(K) K w[az, fz]

w[az, fz](y, )= 0

e

0 az(y+t+)dfz(y + t)dt y R2, S1.

w[a, f]C(R2S1) 2C(K)eC(K)az (1 + azC)fz C,

3.6

w[a, f]C(R2S1) C(K)eC(K)a(1 + aC)fC. w[az, az Mz]= w[az, az Mz[a, f]]

w[a, a M]C(RS1) C(K)eC(K)a(1 + aC)a MC(R3),


51/84

w[a, a M]C(RS1) C(K)eC(K)a(1 + aC)aC MCC(K)eC(K)a(1 + aC)2aC fC.

K R a, f C(K3) > 12 z K

Iw[az ,fz], Iw[az ,az Mz[a,f]] L2(K2)H12(R S1),

az L2(K2)Iw[az ,fz][az]H12(RS1) C(K)eC(K)a 1 + aC(R3) fC(R3)az L2(R2),Iw[az ,az Mz[a,f]][az]H12(RS1) C(K)eC(K)a 1 + aC(R3)2 aC(R3)fC(R3)azL2(R2).

w[az,fz], w[az, az Mz[a,

f]] C

(R2

S1

) az

Iw[az ,fz][az]H12(RS1) w[az, fz]C(R2S1)azL2(R2)C(K)eC(K)a 1 + aC(R3) fC(R3)az L2(R2),

Iw[az ,az Mz[a,f]][az]H12(RS1) w[az, az Mz[a, f]]C(R2S1)azL2(R2)C(K)eC(K)a(1 + aC(R3))2aC(R3)fC(R3)az L2(R2).

R1

azIw[az,fz]

R1

azIw[az,az Mz]

KR a C(K3) z K kz(y, ) = eBaz(y,), y R2, S1 kzC(R2S1) C(K)eC(K)a 1 + aC(R3) z K C(K) K

kz(x, ) kz(y, ) C(K)eC(K)az azC(R2)x y, kz(, ) eC(K)az az a

kzC(R2S1) C(K)eC(K)a(1 + aC(R3)) z K.


52/84

f H12(R) g Hs(R), s> 1 f g H12(R) f gH12(R) CgHs(R)fH12(R).

f gL2(R) fL2(R)gL CfH12(R)gHs(R), C Hs(R) L(R)

12f g() L2(R)

12f g()2 = 12 [f g]()2=

Rf

( t

)g

(t

)dt

2

= R

f( t) 1(1 + ts)(1 + ts)g(t)dt2

R

f( t) 1(1 + ts)2

dtR

(1 + ts)g(t)2 dt=g2Hs(R)

R

f( t)2 (1 + ts)2 . L2

12f g2L(R)

= R

12f g()2 dg2Hs(R)

RR

f( t)2 (1 + ts)2 dtd( y + t)=g2Hs(R)

RR

f(y)2y + t(1 + ts)2 dtdyg2Hs(R)

R

f(y)2 R

y + t(

1 +

t

s

)2

dtdyg2Hs(R)CR f(y)2(y + 1)dyCg2Hs(R)

R

f(y)2(y 12 + 1)2dyCg2Hs(R)f2H12(R),

(1 + ts)2 t(1 + ts)2 s> 1


53/84

K R a H52(K3) hz(s, ) =12(I+ iH)Raz(s, ) K (s)W2, + (s)W2,2 C(K) C(K) K

(

)ehz(,) H2

(R

) z K

()ehz(,)H2(R) C(K)eC(K)a 1 + aH52(R3)2 S1, z K.

a H52(K3) az H2(K2) azH2(R2) CaH52(R3) Raz(, ) H2(R) Raz(, )H2(R) C(K)azH2(R2) C(K)aH52(R3) S1, z K,

Hk(Rn) hz(s, )= 12(I+ iH)Raz(s, )

hz(, )H2(R) C(K)aH52(R3) S1, z K. ehz(s,) (s)

()ehz(,) H2(R)

(s), s(s), 2s(s) C(K) s R,L2(R), sL2(R), 2sL2(R) C(K),

ehz(s,)

= e12Raz(s,) eC(

K)a s R, S1,

shz(s, )shz(, )H1(R) hz(, )H2(R) C(K)aH52(R3) s R, S1. L L2(R) (s)ehz(s,)

()ehz(,)L2(R) eC(K)a L2(R) C(K)eC(K)a,s ()ehz(,) L2(R) s()ehz(,)L2(R) + ()ehz(,)shz(, )L2(R)C(K)eC(K)a sL2(R) + shz(, )L2(R)C(K)eC(K)a 1 + aH52(R3) ,

2s

(

)ehz(,)

L2(R)

2s

(

)ehz(,)

L2(R) + 2

s

(

)ehz(,)shz

(,

)L2(R)

+ ()ehz(,) (shz(, ))2 L2(R) + ()ehz(,)2shz(, )L2(R)C(K)eC(K)a 2sL2 + shz(, )L2+(shz(, ))2L2 + 2shz(, )L2C(K)eC(K)a 1 + aH52(R3)2 .


54/84

f gL2 fL gL2 f2L2 fL fL2

()ehz(,)H2(R) C(K)eC(K) 1 + aH52(R3)2 S1, z K. h

K R a H52(K3) J H12(R S1) C0(R2), C0(R) K R1az [J]L2(R2) C(K)eC(K)a 1 + aH52(R3)5 JH12(RS1) z K.

J=J z K R1az

G

hz(s, ) = 12(I + iH)Raz(s, ) I H s

L2

(R2

) R1az

[J

] g C0

(R2

)

R1az [J], gL2(R2)=

1

4 1

Se(Baz)(y,

) ehzHehz J (y , )d, gL2(R2)

=1

4 1

Se(Baz)(y,

) ehzHehz J (y , )d,gL2(R2)

,

g =g gL2 gL2 (g) K g g(y)= B(0,R)(y)g(y) R> 0 K2 (g)B

(0, R

) B(0,R) C

0

(R2

) B(0,R)

(y

)= 1y B

(0, R

)

y(g(y)B(0,R)(y)) = yg(y)B(0,R)(y) +

0g(y)yB(0,R)(y).

R1az [J], gL2(R2) = 14 S1 B(0,R)(y)e(Baz)(y,) ehzHehz J (y , )d, g(y)L2(R2) , B(0,R) C

0

(R

)

B(0,R)(y)= (y ) y R2, S1, Fz(s, )= (s)ehzHehz J(s, )

R1az [J], gL2(R2) = 14 R2 S1 eBaz(y,)Fz(y , )d g(y)dy.


55/84

R2

R1az [J], gL2(R2) = 14 S1 Ry=s yg(y)e(Baz)(y,)dl(y)Fz(s, )dsd=

4y=s

(g

(y

)e(Baz)(y,

)

)dl

(y

)

y=sg

(y

) ye

(Baz)(y,)dl

(y

), Fz

L2(RS1),

L2(R2) Cy=s

y(g(y)e(Baz)(y,))dl(y), FzL2(RS1)

+ C

y=sg(y) ye(Baz)(y,)dl(y), Fz

L2(RS1)

.

yeBaz(y,) =eBaz(y,

)B

[ az

](y,

).

C(K) K

yeBaz(y,) C(K)eC(K)az B[ az](y, )C(K)eC(K)az azH1(R2)C(K)eC(K)az azH2(R2) y R2, S1.

S1 Ry=sg

(y

) ye

Baz(y,)dl

(y

)Fz

(s,

)dsd

S1Ry=s

g(y) yeBaz(y,)dl(y)Fz(s, )dsdC(K)eC(K)az azH2(R2)

S1Ry=s

g(y)dl(y)Fz(s, )dsdC(K)eC(K)az azH2(R2) Rg(s, ), Fz(s, )L2(RS1)C(K)eC(K)az azH2(R2)Rg(s, )L2(RS1)Fz(s, )L2(RS1)C(K)eC(K)az azH2(R2)gL2(R2)Fz(s, )L2(RS1),

y

y=s

(g(y)e(Baz)(y,))dl(y), FzL2(RS1)

=sy=s

g(y)e(Baz)(y,)dl(y), FzL2(RS1)

.


56/84

kz(y, )= e(Baz)(y,)

Hs

s

y=s

g

(y

)e(Baz)(y,

)dl

(y

), Fz

L2(RS1)=

sIkz g(

s,

), F

(s,

) L2(RS1)

sIkz gH12(RS1)FH12(RS1)Ikz gH12(RS1)FH12(RS1).

a H52(R) a C(K3) > 12 kz C(R2 S1)

Ikz gH12(RS1) C(K)kzC(R2S1)gL2(R2)C

(K

)eC(

K)a

1 +

a

C(R3)

g

L2(R2).

g

aC(R3) aH52(R3)

y=s (g(y)e(Baz)(y,))dl(y) C(K)eC(K)a 1 + aH52(R3) gL2(R2).

R1az J , g C(K)eC(K)a 1 + aH52(R2) gL2(R2)FzH12(RS1). Fz(s, )= (s)ehz(s,)Hehz(s,)(s)J(s, ) ()ehz(,)

H2(R) Fz(s, )2H12(RS1) =

S1()ehz(,)Hehz(,)()J(, )2

H12(R)d

S1()ehz(,)2H2(R)Hehz(,)()J(, )2H12(R)d

=

S1

(

)ehz(,)

2H2(R)

ehz(,)

(

)J

(,

)2

H

12

(R

)

d

C(K)eC(K)a 1 + aH52(R2)4S1ehz(,)(s)J(, )2

H12(R)d

C(K)eC(K)a 1 + aH52(R2)8S1J(, )2

H12(R)d

=C(K)eC(K)a 1 + aH52(R2)8 J2H12(RS1),


57/84

R1az J,gL2(R2) C(K)eC(K)a 1 + aH52(R3)5 JH12(RS1)gL2(R2)C

(K

)eC(

K)a

1 +

a

H52(R3)

5

J

H12(RS1)

g

L2(R2),

C(K) K z g g C0(R2) g L2(R2) z K z K

R1az JL2(R2) eC(K)a 1 + aH52(R3)5 JH12(RS1) z K. K R a H52(K3) aH52(R3) < D f C(K3) > 12 C0(K2) K

R1

azIw[az ,fz]L(L2(K2),L2(K2)) C(

K, D)fC(R3) z

K

R1az Iw[az ,az Mz[a,f]]L(L2(K2),L2(K2)) C(K, D)aH52(R3)fC(R3) z K, C(K, D) K D D

az L2(K2) Iw[az,fz], Iw[az ,az Mz[a,f]] H12(RS1) B R

Iw[az ,

fz][az

](s,

)= 1B

(s

)Iw[az,

fz][az

](s,

) s, zR, S1

, Iw[az ,az Mz[a,f]][az](s, )= 1B(s)Iw[az,az Mz[a,f]][az](s, ) s, z K, S1,

R1az Iw[az,fz][az]L2(R2) C(K)eC(K)a 1 + aH52(R3)5 Iw[az,fz][az]H12(RS1),C(K)eC(K)a 1 + aH52(R3)5 1 + aC(R3) fC(R3)azL2(R2),

R1az Iw[az,az Mz[a,f]][az]L2(R2) C(K)eC(K)a 1 + aH52(R3)5 Iw[az,az Mz][az]H12(RS1),C

(K

)eC(

K)a

1 +

a

H52(R3)

5

1 +

a

C(R3)

2

a

C(R3)

f

C(R3)

az

L2(R2).

aC(R3) CaH52(R3) >12 a aH52(R3) aH52(R3 D

R1az Iw[az ,fz][az]L2(R2) C(K, D)fC(R3)azL2(R2) z K,R1az Iw[az ,az Mz[a,f]][az]L2(R2) C(K, D)aH52(R3)azL2(R2) z K.


58/84

K2

() K2 C(K, D)

D

L1[a, f] Q[a, f]

KR a H52(K3) f C(K3) > 12 C0(K2) K aH52(R3)


59/84

az a Mz[]az a Mza(y) =az(y)

S2

0f((y, z) + t)et0 a((y,z)+)d t

0a((y, z) + s)dsdtd

a

1K3

(y, z

)S2

0 f

((y, z

)+ t

)e

t0 a((y,z)+)ddt

R a

((y, z

)+ s

)dsd.

K3 (y, z) ,t,s [ (K3), (K3)] = J(K) az a Mza(y) C(K)aeC(K)a f1K3(y, z)

S2J(K)

a((y, z) + s)dsd, a CaH52(R3) C D

f fC(R3)

az a Mza

(y

) C

(K, D

)a

H52(R3)

f

C(R3)1K3

(y, z

)S2J(

K)

a

((y, z

)+ s

)dsd,

L2

az a Mza2L2(R2) C(K, D)a252f2C1K(z)K2

S2J(K)

a((y, z) + s)dsd2 dyC(K, D)a252f2C1K(z)

K2S2J(K)

(a((y, z) + s))2 dsddy. z

R

az a Mza

2L2(R2) C

(K, D

)a

252

f

2C

K

K

2S2J(

K) (

a

((y, z

)+ s

))2

dsddydz

=C(K, D)a252f2C S2J(K)

KK2(a((y, z) + s))2 dydzdsd

=C(K, D)a252f2C S2J(K)

a2L2(R2)dsdC(K, D)a2

H52(R3)f2C(R3)a2L2(R2).

az Mz[a, ]

az Mz[a,f](y)=a(y, z)S2 0 f((y, z) + t)e 0 a((y,z)+)ddtdeC(

K)a a1K3(y, z)S2J(K)

f((y, z) + t)dtdC(K, D)aH52(R3)1K3(y, z)

S2J(K)

f((y, z) + t)dtd.


60/84

L2 y

az Mz[a,f]2L2(R2) C(K, D)a2521K(z)K2

S2J(K)

f((y, z) + t)dtd2 dyC

(K, D

)a

2

52

1 K(

z

)K2 S2 J(K) (f

((y, z

)+ t

))2

dtddy,

zR

R

az Mz[a,f]2L2(R2)dz C(K, D)a252KK2S2J(K)

(f((y, z) + t))2 dtddydz=C(K, D)a252

S2J(K)

KK2(f((y, z) + t))2 dydzdtd

C(K, D)a2H52(R3)

fL2(R3).

Q

Q[a, f](a,f)L2(R;L2(R2)L2(R2)) = R

Qz[a, f](a,f)2L2(R2)L2(R2)dz

R

R1az Iw[az,az Mz][az]2L2(R2) + az a Mza2L2(R2) + az Mz[a,f]2L2(R2)dzC(K, D)a2

H52(R3)f2C(R3)a2L2(R3) + f2L2(R3) a,f L2(K3),

Q

[a, f

] C

(K, D

)a

H52(R3)

(1 +

f

C(R3)

).

Q L2(K, L2(K2) L2(K2)), z K Qz[a, f](a,f) = 0 a,f L2(K2) Qz[a, f](a,f)L2(K2)2 z K R a C0(K3), f C(K3) f 0 f0

M

[a, f

](x

) C

(K

)eC(

K)a ff

C(R3)

3

f

x K

3.

f

x K3 f(x)=f > 0 A={x K3, f(x) f

2},


61/84

f

f(x) f(y)fC x y y Ac

f

2

f

C

x y

y Ac

R =f2fC

1

(x, Ac). fC 0 f f0 B(x, R) A

f(x) f2

x B(x, R)f(x) f

2 1B(x,R)(x) x K3.

M x K3

M[a, f](x)= S2

0f(x + t)et0 a(x+s)dsdtd

eC(K)a

S2

0f(x + t)dtd

eC(K)a

f2

S2

01B(x,R)(x + t)dtd,

t > (K3) (x + t) B(x, R)x K3, S2 B(x, R) K3

M[a, f](x) eC(K)a f2

S2

(K3)

01B(x,R)(x + t)dtd

=eC(K)a

f2

S2

(K3)

01B(x,R)(x + t)t2

t2dtd

eC(K)a

f2 (K3)2S2 (

K3)

01B(x,R)(x + t)t2dtd

=eC(K)a

f2

(K3

)2R31B(x,R)(x)dx

=eC(K)a

f

2 (K3)2 4R3C(K)eC(K)a ffC(R3)

3 f x K3.


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K R a H52(K3) f C(K3) > 12 f0, f0 C0(K2)

K aH52(R3) < D

L1

[a, f

] L2

(K2 K

) L2

(K2 K

) L2

(K; L2

(K2

) L2

(K2

)),

L1[a, f] gh=L1z[a, f] gh

zR

,

L1z

L1z[a, f] L2(K2 K) L2(K2 K) L2(K2) L2(K2)L1z

[a, f

] g

h

=

hz

Mz, gz R

1az

Iw[az ,fz]

[hz

Mz

],

L1[a, f] 1 + C(K, D)(1 + fC(R3)) f3C(R3)f3+1 , g, h L2(K3) gz, hz L2(K3) z R

L1z

[a, f

] g

h

2

(L2(R2))2

hz

Mz

2L2(R2) +

gz

2L2(R2) +

R1a Iw[a,f]

[hz

Mz

]2L2(R2).

R1az Iw[az ,fz][hz Mz]L2(R2) C(K, D)fC(R3)hz MzL2(R2) z K, z K zR

R1az Iw[az ,fz][hz Mz]L2(R2) C(K, D)fC(R3)hz MzL2(R2) z R.

hz MzL2(R2) C(K)eC(K)a fC(R3)f 3

1f hzL2(R2)C(K, D)fC(R3)f

3

1f hzL2(R2),


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z R

L1z[a, f] gh 2(L2(R2))2 gz 2L2(R2) + 1 + C(K, D)fC hz MzL2(R2)gz 2L2(R2) + C(K, D)(1 + fC)

f

3C

f3+1 hzL2(R2). R gz2L2(R2) =R R2g(y, z)2dydz =gL2(R3) hz L1[a, f] zL1[a, f] g

h L2(R;L2(R2))=

R

L1z[a, f] gh 2(L2(R2))2dz

R

gz2L2(R2) + C(K, D)(1 + fC)f3C

f

3+1

hzL2(R2)dzgL2(R3) + C(K, D)(1 + fC)f3Cf3+1 hL2(R3),

L1[a, f] 1 + C(K, D)(1 + fC(R3)) f3C(R3)f3+1 . L1

[a, f

] L2

(K; L2

(K2

)

L2

(K2

)) L2

(K3

) z K gz, hz, fz 0

L1z[a, f] z K gz, hz, L2(K2) L1z[a, f] K2 K2

K R f C(K3) > 12 f0, f0 C0(K2) K D> 0 (L+Q)[a, f] L2(K3)L2(K3) a H52(K3), aH52(R3) 0 aH52(R3) D (QL1)[a, f] < 1.

aH52(R3) D

Q

[a, f

] C

(K, D

)a

H52(R3)

(1 +

f

C(R3)

),

L1[a, f] 1 + C(K, D)(1 + fC(R3))

f3

C(R3

)f3+1C(K, D)(1 + fC(R3))1 +

f3C(R3)f3+1

, C(K, D)> 0 K D

(QL1)[a, f]Q[a, f]L1[a, f]C(

K, D)aH52(R3)(1 +

fC(R3))

2

1 + f

3)

C(R3)

f3+1 . C(K, D)> 0 D D > 0

D 12 K R a,f L2(K3) F[a + a, f + f] F

F H52(K3) C(K3) Y =(Y1 Y2)zR (Y1 Y2)zR

F

[a + a, f+ f

]= F

[a, f

]+ DF

[a, f

](a,f

)+ o

(a,f

).

K R K K C(K2) K

(Y1 Y2)zR R1a[(f, g)z]=R1az fz, R1az gzzR)

R1aF[a + a, f + f] F[a, f]= R1aDF[a, f](a,f) + R1a[o(a,f)]=(L + Q)[a, f](a,f) + R1a[o(a,f)],

L Q f 0, f

0

a

H52

(R3

)

=(L + Q)1[a, f] R1a[o(a,f)] (L + Q)1[a, f] R1aF[a + a, f+ f] F[a, f]=(az, fz)zR +

L1[a, f] k=0

(QL1)[a, f]k R1aF[a + a, f+ f] F[a, f]=(az, fz)zR + .


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(a,f)L2(R3)L2(R3) 0 R1a o(a+f)L2(R;L2(R2)L2(R2))

0 a,f L2(K3)

DF[a, f] DF[a, f]1 =L1[a, f]

k=0

(QL1)[a, f]k R1a .

a, f

F[a, f] F KR a, f H52(K3) GK L2(K3) H52(K3) a, f L2(R3) GK(a, f) =(GK(a), GK(f))

F L2(K3) L2(K3) Y =(Y1, Y2)zRF(a, f)= F[GK(a, f)] F[a, f].

a, f L2

(K3

) F

[GK

(a, f

)]=

F[a, f] F GK(a, f) =(a, f)

F

DF[GK(a, f)] (DF[GK(a, f)])1 =L1

k=0

(QL1)k[GK(a, f)]R1GK(a). F

DF(a, f)= DF[GK(a, f)] GKDF(a, f)1 =G1K (DF[GK(a, f)])1

=G1KL1

k=0

(QL1)k[GK(a, f)]R1GK(a),


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(an+1, fn+1

)=

(an, fn

) G1KL

1

k=0

(QL1

)k

[GK

(a, f

)]R1GK(a)F

(GK

(an, fn

)).

GK(an) GK(fn) n

GK GK(a) = a a

I+

(

)54

a= a x K3

a=

0

x

K

3

,

> 0 ()p

a f R3

zR

Az0 =Raz[fz], z R

Az1 =CRaz[az Mz[a, f]].

a0

R3

f0

a1 ai fi

i N


71/84

i N fi ai

A0

fi

(y, z

)= R1

aiz

[Az0

](y

) y R2, z R.

ai =a

A0

f

i N ai ai+1

A1 fi M[ai, fi] ai+1

ai+1(y, z)= R1aiz [ 1CAz1](y)M[ai, fi](y, z) y R2, zR.

ai =a fi =f A1 ai+1 =a

ai, fi 0 f > 0 M[ai, fi](y, z) C>0 y R2, zR ai fi ai ai+1

T[ai](y, z)= R1aiz [ 1CAz1](y)Mai,R1

aiz[Az0]

zR

(y, z) y R2, z R.

T

C

[1, 1

]

[1, 1

]

[1, 1

]

128 128 128 2128 2128 2128 2 2

128 ==H= 128 H = ={(2j 1)128 1, i {1, 2, ..128}}, ={2j128, j {0, 1, 2, ..127}}


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0 0.1


73/84

0 0.2


74/84

L2

L2

z


75/84


76/84

L2


77/84

L2

L2


78/84

az fz

= =H =128


79/84

C


80/84


81/84

F

[a, f

](s,,z

)=

Raz

[fz

](s,

)CRaz[

azMz[a, f

]](s,

) s, zR, S1

F


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