discrete green functions e. bendito, a. carmona and a.m. encinas depto. matemática aplicada iii,...
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DISCRETE GREEN FUNCTIONSDISCRETE GREEN FUNCTIONSE. BENDITO, A. CARMONA AND A.M. ENCINAS Depto. Matemática Aplicada III, UPC, Spain. e-mail: [email protected]
SETS AND FUNCTIONS
),( edge the of econductanc the is ),( network, anis),,( yxyxccEV
)(_
and )( of partition a, FFFFFF 21
boundary vertex some for ),(:)(, FyEyxcFxFVF
)( and )(),(),(),(22111
FCgFChFCgFCqFCf
OPERATORS
of Laplacian the is))()((),() ~ xy
yuxuyxcLu(x
)(at derivative normal the is))()((),()( ~
Fyuxuyxcxu
Fy
xy
BOUNDARY VALUE PROBLEM
),()(
),()()()(
),()()()(
22
11
Fxxgxu
Fxxgxuxhxu
FxxfxuxqxLu
F
2F
1F
NETWORKBOUNDARY PROBLEMS TYPE
otherwiseh
hF
F
FVF
:problem Robin-Dirichlet:problem Neumann-Dirichlet
, :problemNeumann:problem Robin
:problemDirichlet :equationPoisson
0
02
2
1
SEMIHOMOGENEOUS BOUNDARY VALUE PROBLEMS
u
xu
xuxhxu
xuxqxLu
)(
)()()(
)()()(
)(xf
0
0 0
0
00
1g
2g
3u
1u
2u u
GREEN FUNCTION
otherwise.,;and,when||
),(
),()(),(
)(1)(),()(),(
, all forthatsuch_
:
001
20
10
2
aqFF
a
FxyxG
FxyxGxhyxG
FxxF
axyyxGxqyxGL
FyRFFG
. on symmetric is
function. Green unique a exists There
FFG
)()()( , yfyxGxuFx
1
_
and_
where
),()(),(
_)()(
_)(
_
1
22
1
gffhqq
FxxgxuFFxxfxuxqxuL
EQUIVALENT BOUNDARY VALUE PROBLEM
_
1FF
2F
NETWORK FOR THE EQUIVALENT PROBLEMEXISTENCE OF SOLUTION
solution. unique a exists thereOtherwise,
constant. a to up unique is solution the and
iffsolution exists thereWhen
,
_
0
0
11
2
FxFx(x)gf(x)
q,F
)(),(_
)( ygyxGy
yFy x 22
Fy Fy
xgyxGyfyxGxu
11
)(),(_
)(),(_
)(
1u
2u
3u
Then,problem. equivalent the forfunction Green the)(_
:_
Let RFFFG 1
FUNDAMENTAL RESULTS: EXPRESSION OF THE SOLUTION
K
yxxqyxL
yxyxLyxK
RFFK
when
when
by given kernel the
),(_
),(_
),,(_
),(
__:
POTENTIAL THEORY
verifies the Equilibrium Principle
._
,_.,)(
_
)( )(
,_
formeasure
mequilibriuanexists also there
orwhen Moreover,
all forand
that such
unique aexists thereeach For
FA
qF
AxxL
AAC
FA
A
ASuppA
0
1
2
Otherwise, ),(_
),( yxGyxG
FwzwzGwyGzxGyxG
FyxG
,),(
_),(
_),(
_),(
_
||),( 2
112
0 FFq and,_
If
Poisson kernel:),()(),( yxG
yyxyxP
FUNDAMENTAL RESULTS: EXPRESSION OF THE GREEN FUNCTION
yFF
qFFFFF
G
qFc
G
FFy
FFy
FFy
FFy
FFy
FF
xyx
yzzzy
xxyx
11
21
11
1
21
11
01
01
set the of measure mequilibriu theiswhere
_andif)||||(||
||_
_orif
_
,)(),(
~)(),(
)()(),(
NEUMANN PROBLEM FOR A BALL OF RADIOUS r IN AN HOMOGENEOUS TREE OF DEGREE k+1>2
)(
)(||||
)(|)||(|
),(||||)(
xyr
ory
kkkk
k
xy
yxdxBx
112
1
1
11
R
kkkBk
kByxdyxd
kBkkyxG
yx
r
rr
or
r
;)(||
)(||),(),(
)(||)(),(
)( |||| 1121
111
|and ofconfluent |),(root, the and between distance||
yxyxd
xx
o
POISSON PROBLEM FOR A DISTANCE-REGULAR GRAPH
1 k kj kdk 1a1b1
aj adbjcj cd
d
j j
jjd
yxdj j
j
Bn
BnB
Bn
BnyxG
0
2 ||
|)|(||
||
||),(
),(
jjj
dd
BBjB
ccbbkdn
of
),...,;,...,,(
boundary edge,radius of ball
array onintersecti diameter, order,
111
DIRICHLET PROBLEM FOR THE END COMPACTIFICATION OF AN INFINITE DISTANCE-
REGULAR GRAPH WITH PARMETERS k AND l
),(),( ))((
)(||
),(yxd
yxdj jlklkk
lkB
yxG
11
REFERENCES
E. Bendito, A. Carmona and A.M. Encinas, Solving boundary value problems on networks using equilibrium measures,
Journal of Functional Analysis, 171, 155-176 (2000).