IAOe?IIAEUeIA AEAAeAI?ss IAOE OE?A?IE

?inoeooo i?eeeaaeii? iaoaiaoeee ? iaoai?ee

EA??I Aeieo?i A?eoi?iae/

OAeE 517.946

ONA?AAeIAIIss AE?IAeAEAIEO IAE?I?EIEO CAAeA*

A IA?OI?IAAIEO IAEANOssO

01.01.02 – aeeoa?aioe?aeuei? ??aiyiiy

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

Aeiiaoeuee – 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia o a?aeae?e? iae?i?eiiai aiae?co ?inoeoooo i?eeeaaeii?
iaoaiaoeee ? iaoai?ee Iaoe?iiaeueii? aeaaeai?? iaoe Oe?a?ie.

Iaoeiaee ea??aiee: aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?,

aeaaeai?e IAI Oe?a?ie

Ne?eiiee ?ai? Aieiaeeie?iae/,

?inoeooo i?eeeaaeii? iaoaiaoeee ? iaoai?ee IAI Oe?a?ie, aee?aeoi?.

Io?oe?ei? iiiiaioe: aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?

Iaieia Ieaenaiae? Aiae??eiae/,

A?iieoeueeee aea?aeaaiee iaaeaaia?/iee oi?aa?neoao,

eaoaae?a iaoaiaoeee, i?ioani?;

eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe, aeioeaio

Ai?iae?i Ieoaeei Ieaen?eiae/,

Aeiiaoeueeee aea?aeaaiee oi?aa?neoao,

eaoaae?a iaoaiaoe/ii? o?ceee, aeioeaio.

I?ia?aeia onoaiiaa:

?inoeooo iaoaiaoeee IAI Oe?a?ie, i.Ee?a, a?aeae?e aeeoa?aioe?aeueieo
??aiyiue c /anoeiieie iio?aeieie.

Cao?no a?aeaoaeaoueny “ 28 ” ea?oiy 1999 ?ieo i 15 aiae. ia can?aeaii?
niaoe?ae?ciaaii? a/aii? ?aaee Ae 11.193.01 i?e ?inoeooo?
i?eeeaaeii? iaoaiaoeee ? iaoai?ee IAI Oe?a?ie ca aae?anith:

340114, i.Aeiiaoeuee, aoe.?.Ethenaiao?a, 74.

C aeena?oaoe??th iiaeia iciaeiieoenue o a?ae?ioaoe? ?inoeoooo
i?eeeaaeii? iaoaiaoeee ? iaoai?ee

IAI Oe?a?ie ca aae?anith:

340114, i.Aeiiaoeuee, aoe.?.Ethenaiao?a, 74.

Aaoi?aoa?ao ?ic?neaiee “ 25 ” aa?aciy 1999 ?ieo.

A/aiee nae?aoa? niaoe?ae?ciaaii? a/aii? ?aaee Eiaaeaanueeee I.A.
CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. O aaaaoueio ?icae?eao o?ceee oa iaoai?ee /anoi
aeine?aeaeothoueny i?ioeane a neeueii iaiaeii??aeieo na?aaeiaeuao, ye?
iienothoueny aeeoa?aioe?aeueieie ??aiyiiyie, ui iathoue oaeaeeieieeai?
eiao?oe??ioe, aai ae ?icaeyaeathoueny a iaeanoyo neeaaeii? no?oeoo?e.
Oea i?eaiaeeoue aei iaiao?aeiino? iiaoaeiae ona?aaeiaieo iiaeaeae aeey
oaeeo na?aaeiaeu. Nooue ieoaiiy ona?aaeiaiiy iieyaa? a iiaeeeaino?
iiaoaeiae e?aeiai? caaea/? aeey ??aiyiiy c i?inoeie eiao?oe??ioaie, aai
a i?ino?e iaeano?, ?ica’ycee yei?, a iaaiiio ?icoi?ii?, iaei
a?ae??ciythoueny a?ae ?ica’yce?a ii/aoeiaeo caaea/.

sse i?eeeaae iaeanoae neeaaeii? no?oeoo?e, ui aeieeathoue a oai???
ona?aaeiaiiy, iiaeia ?icaeyaeaoe ia?oi?iaai? iaeano?, ye? iaea?aeai? ?c
o?eniaaii? iaeano? oeyoii aeeeaeaiiy aaeeei? e?eueeino? ae??aieo
iaia?aoeiieo eiiiiiaio. Aia?oa ieoaiiy ona?aaeiaiiy e?aeiaeo caaea/ o
oaeeo iaeanoyo aoee ?icaeyioo? a 60-o? ?iee o ?iaioao A.I.Ia?/aiei oa
?.ss.O?oneiaa. ?acoeueoaoe ?o iiiia?ao?? (Ia?/aiei A.A., O?oneia E.ss.
E?aaaua caaea/e a iaeanoyo n iaeeica?ienoie a?aieoeae. – E.:Iaoeiaa
aeoiea, 1974. – 278 n.), i?enay/aiii? oe?i ieoaiiyi, io?eiaee
iiaeaeueoee nooo?aee ?icaeoie a ?iaioao ?.ss.O?oneiaa. ?i aoee
?ic?iaeai? aa??aoe?ei? iaoiaee aeine?aeaeaiiy aneiioioe/ii? iiaaae?iee
?ica’yce?a caaea/ Ae???oea oa Iaeiaia aeey e?i?eieo ??aiyiue a ci?iieo
iaeanoyo, acaaae? eaaeo/?, iaia??iaee/ii? no?oeoo?e. O oa?i?iao
ca?aeiino? niaoe?aeueieo /eneiaeo oa?aeoa?enoee oe?o iaeanoae
?.ss.O?oneia io?eiaa aeinoaoi?, a o aeayeeo aeiaaeeao ? iaiao?aei?,
oiiae ca?aeiino? ?ica’yce?a ?icaeyaea?ieo caaea/ aei ?ica’yce?a
ona?aaeiaieo caaea/, o yeeo aeiaeaoeiaee /eai aecia/a?oueny ii a?aieoeyi
oe?o oa?aeoa?enoee.

Iaoiaeeea ?icaeyaeo ieoaiiue ona?aaeiaiiy, cai?iiiiiaaia ?.ss.O?oneiaei,
o iiaeaeueoiio ?icaeaaeanue a ?iaioao E.A.A??eyiaea, ?.TH.*oae?iiae/a,
I.A.Aii/a?aiei, ?.A.Na?uaai?, E.N.Iaie?aoiaa oa ?i.

Nooo?a? ?acoeueoaoe a aeine?aeaeaii? iae?i?eieo caaea/ Ae???oea a
iaeanoyo neeaaeii? iaia??iaee/ii? no?oeoo?e aoee io?eiai?
?.A.Ne?eiieeii. ?i aoee ?ic?iaeai? iaoiaee ona?aaeiaiiy caaea/ Ae???oea
aeey iae?i?eieo ae?ioe/ieo ??aiyiue ae?oaiai ii?yaeeo a ci?iieo
iaeanoyo, ca aeiiiiiaith yeeo aoee anoaiiaeai? aeinoaoi? oiiae
ca?aeiino? iine?aeiaiino? ?ica’yce?a iae?i?eieo caaea/ a ia?oi?iaaieo
iaeanoyo, aeienai? o oa?i?iao i?noeino? oe?o iaeanoae ? iiaoaeiaaia
a?aie/ia caaea/a c aeiaeaoeiaei /eaiii, ui ia? i?noe?niee oa?aeoa?.
Aeei?enoiaoaath/e cai?iiiiiaaiee iaoiae aneiioioe/iiai ?iceeaaeo,
?.A.Ne?eiiee aeieeaaeii aea/ea ieoaiiy ona?aaeiaiiy iae?i?eieo caaea/
Ae???oea a iaeanoyo c ae??aiica?ienoith iaaeath, a iaeanoyo c eaiaeaie,
a oaeiae a ia?oi?iaaieo iaeanoyo caaaeueii? no?oeoo?e, oiaoi aac
aoaeue-yeeo aaiiao?e/ieo i?eiouaiue a?aeiinii no?oeoo?e ci?iieo
iaeanoae.

Aiaeia?/iee iaoiae ona?aaeiaiiy aoa ?ic?iaeaiee ?.A.Ne?eiieeii aeey
iae?i?eieo ia?aaie?/ieo caaea/ a iine?aeiaiino? ia?oi?iaaieo iaeanoae.

Iiaeaeueoee ?icaeoie iaoiaee, cai?iiiiiaai? ?.A.Ne?eiieeii, ciaeoee a
?iaioao A.?.I?ieiiaiei, ?.?.Ne?eiieea, I.A.Iaoiiai? oa ?i.

A?aecia/eii oaeiae ?iaioe G. Dal Maso, F.Murat, A.Garroni,
J.Casado-Diaz, D.Cioranescu, I.A.Eiaaeaanueeiai, i?enay/ai? ieoaiiyi
ona?aaeiaiiy caaea/ Ae???oea aeey e?i?eieo oa eaac?e?i?eieo ??aiyiue o
ci?iieo iaeanoyo caaaeueii? no?oeoo?e.

Ia?o? ?iaioe ii ona?aaeiaiith ??aiyiue c /anoeiieie iio?aeieie c
ia??iaee/ieie oaeaeeieieeaieie eiao?oe??ioaie c’yaeeeny ia ii/aoeo 70-o
?ie?a. Oea aoee, ianaiia?aae, ?iaioe E.Sanchez-Palencia, I.N.Aaoaaeiaa,
A.Bensoussan, J.-L. Lions, G.Papanicolaou. Iiaeaeueoa nenoaiaoe/ia
aea/aiiy oe?o ieoaiiue iia’ycaia, ia?o ca ana, c ?iaioaie I.A.Ie?eiee,
A.A.AEeeiaa, N.I.Eiceiaa, I.N.Aaoaaeiaa, A.A.?in?oueyia. C i?iaeaiaie
ona?aaeiaiiy e?aeiaeo caaea/ aeey aeeoa?aioe?aeueieo ??aiyiue c
/anoeiieie iio?aeieie o?nii iia’ycaia oai??y G-ca?aeiino? iia?aoi??a oa
A-ca?aeiino? ooieoe?iiae?a. Ieoaiiy G-ca?aeiino? oa A-ca?aeiino?
aeine?aeaeoaaeeny a ?aaioao oaeeo iaoaiaoee?a, ye E.De Giorgi,
S.Spagnolo, I.A.Ie?eiee, A.A.AEeeia, N.I.Eiceia, I.A.Iaieia,
I.A.Eiaaeaanueeee, G.Dal Maso oa ?i.

Aea ne?ae a?aecia/eoe, ui ia caaaeath/e ia aaeeeo e?euee?noue ?ia?o c
oai??? ona?aaeiaiiy, aaaaoi ieoaiiue caeeoothoueny a?aee?eoeie. Oae,
iai?eeeaae, oe?eaaeie aeey aeine?aeaeaiiy ? ieoaiiy ona?aaeiaiiy
ae?iaeaeaieo iae?i?eieo caaea/ a ia?oi?iaaieo iaeanoyo, aea ae?iaeaeaiiy
?icoi??oueny ii i?inoi?iaeo ci?iieo. Cie?aia, noaiiaeyoue ?ioa?an
caaea/? ona?aaeiaiiy aeey ae?iaeaeaieo iae?i?eieo ae?ioe/ieo ??aiyiue
ae?oaiai ii?yaeeo, oeio aaaiaiai ?-Eaieana

iaeaaeeoue iaaiiio eeano Iaeaioaoioa. Oae? eeane ooieoe?e aoee
ai?iaaaeaeai? B.Munkenhoupt ia ii/aoeo 70-o ?ie?a (Munkenhoupt A.
Weighted norm inequalities for the Iardy maximal function // Trans.
Amer. Math. Soc. – 1972. – V.165. – P.207-226). Iiaeaeueoa aea/aiiy
aeanoeainoae aaa Iaeaioaoioa iia’ycaia, ianaiia?aae, c ?iaioaie
R.L.Wheeden, C.Fefferman, C.Segovia, R.R.Coifman, N.Miller. O na?aaeei?
80-o ?ie?a E.Fabes, C.Kenig, R.Serapioni ? S.Chanillo oa R.L.Wheeden
io?eiaee aaaia? ia??aiino? Niaie?aa oa Ioaiea?a, ? ii/aeiny ?ioaineaia
aea/aiiy nii/aoeo ae?iaeaeaieo ae?ioe/ieo ??aiyiue oeio aaaiaiai
?-Eaieana (E.Fabes, D.Jerison, C.Kenig, R.Serapioni, B.Franchi,
J.Heinonen, T.Kilpelainen, O.Martio, ?.A.Ne?eiiee, F.Nicolosi,
N.E.Aiaeiiueyiia, oa ?i.), a caiaeii ? ae?iaeaeaieo ia?aaie?/ieo
??aiyiue (F.Chiarenza, R.Serapioni, C.Gutierrez, G.Nelson, R.L.Wheeden,
?.A.Ne?eiiee, F.Nicolosi, ?.?.Ne?eiiee oa ?i.). O oaia?aoi?e /an
aea/aiiy ae?iaeaeaieo iae?i?eieo ??aiyiue ? iaeiei c iai?yie?a ?icaeoeo
oai??? ae?oa?aioe?aeueieo ??aiyiue c /anoeiieie iio?aeieie.

Oaeei /eiii, i?aaeiaoii aeine?aeaeaiiy aeaii? aeena?oaoe?eii? ?iaioe ?
ieoaiiy ona?aaeiaiiy n?i’? caaea/ Ae???oea aeey ae?iaeaeaieo iae?i?eieo
ae?ioe/ieo ??aiyiue ae?oaiai ii?yaeeo a iine?aeiaiino? ia?oi?iaaieo
iaeanoae ??cii? no?oeoo?e, i?e oiia?, ui aaaiaa ooieoe?y iaeaaeeoue
iaaiiio eeano Iaeaioaoioa.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. Oaiaoeea
aeena?oaoe?? iia’ycaia c iaoeiaeie aeine?aeaeaiiyie a?aeae?eo
iae?i?eiiai aiae?co ?inoeoooo i?eeeaaeii? iaoaiaoeee ? iaoai?ee IAI
Oe?a?ie, aiia aoea aeeiiaia a ?aieao aea?aeaaii? oaie ?0196U002838
“Aneiioioe/ia iiaaae?iea ?ica’yce?a iae?i?eieo ae?ioe/ieo ? ia?aaie?/ieo
??aiyiue”.

?acoeueoaoe aeena?oaoe?eii? ?iaioe aoee aeei?enoai? i?e aeeiiaii?
i?iaeoo 1.4/148 “Aeine?aeaeaiiy aneiioioe/ii? iiaaae?iee ?ica’yce?a
iae?i?eieo ae?ioe/ieo ? ia?aaie?/ieo a?aie/ieo caaea/ a ia?oi?iaaieo
iaeanoyo caaaeueii? no?oeoo?e” Aea?aeaaiiai oiiaeo ooiaeaiaioaeueieo
aeine?aeaeaiue Oe?a?ie oa a?aioo INTAS ?96-1061 “Homogenization of
problems of mathematical physics”.

Iaoa ? caaea/? aeine?aeaeaiiy. ?icaeyiooe ieoaiiy ona?aaeiaiiy n?i’?
caaea/ Ae???oea aeey ae?iaeaeaieo iae?i?eieo ae?ioe/ieo ??aiyiue
ae?oaiai ii?yaeeo a ia?oi?iaaieo iaeanoyo ??cii? no?oeoo?e, i?e oiia?,
ui aaaiaa ooieoe?y iaeaaeeoue iaaiiio eeano Iaeaioaoioa. Aeey oeueiai
io?eiaoe ??ci? ?ioaa?aeuei? oa iioi/eia? ioe?iee ?ica’yce?a
niaoe?aeueieo iiaeaeueieo ae?iaeaeaieo iae?i?eieo caaea/ Ae???oea, ia
iniia? yeeo aea/eoe aneiioioe/io iiaaae?ieo iine?aeiaiino? ?ica’yce?a
?icaeyaeoaaieo caaea/, ? iieacaoe, ui ?ica’ycee ae?iaeaeaieo iae?i?eieo
caaea/ Ae???oea a n?i’? ia?oi?iaaieo iaeanoae aeecuee? aei ?ica’yceo
iaaii? ona?aaeiaii? ae?iaeaeaii? iae?i?eii? caaea/? a iaia?oi?iaai?e
iaeano?. C’ynoaaoe oiiae, i?e yeeo ?nio? ona?aaeiaia e?aeiaa caaea/a, ?
aeaoe eiie?aoiee niin?a ?? iiaoaeiae.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a. Aia?oa ?icaeyiooi i?iaeaio
ona?aaeiaiiy n?i’? caaea/ Ae???oea aeey ae?iaeaeaieo iae?i?eieo
ae?ioe/ieo ??aiyiue ae?oaiai ii?yaeeo a iine?aeiaiino? ia?oi?iaaieo
iaeanoae, i?e oiia?, ui aaaiaa ooieoe?y iaeaaeeoue iaaiiio eeano
Iaeaioaoioa. Aeey iine?aeiaiino? iaeanoae c ae??aiica?ienoith iaaeath (o
aeiaaeeo ia’?iiiai oa iiaa?oiaaiai ?iciiae?eo “ca?ai”), a oaeiae aeey
ia?oi?iaaieo iaeanoae c “ca?iaie”, ye? aonoi oiaeiaai? (aea, cie?aia, ia
i?eionea?oueny iae?noue ae?aiao??a ii?iaeiei a?aeiinii a?aeaeaeae i?ae
ieie), aea/aii aneiioioe/io iiaaae?ieo iine?aeiaiinoae ?ica’yce?a
?icaeyaeoaaieo caaea/; oa io?eiaii aeinoaoi? oiiae ca?aeiino? ?ica’yce?a
?icaeyaeoaaieo ae?iaeaeaieo iae?i?eieo caaea/ Ae???oea, iiaoaeiaaii
ei?aeoi?e aeey iaaeeaeaiiy oaeeo iine?aeiaiinoae ?ica’yce?a ? e?aeia?
caaea/? aeey a?aie/ieo ooieoe?e. O aeiaaeeo iaeanoae c “ca?iaie”, ye?
aonoi oiaeiaai?, aeeo/aii n?i’th iiiaeei, ui ia aieeaathoue ia iiaoaeiao
ona?aaeiaii? caaea/?.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao?a. Aeena?oaoe?y ia?
oai?aoe/iee oa?aeoa?, ?? ?acoeueoaoe iiaeooue aooe aeei?enoai? aeey
iiaeaeueoiai aea/aiiy ieoaiiue ona?aaeiaiiy ae?iaeaeaieo iae?i?eieo
ae?ioe/ieo ? ia?aaie?/ieo e?aeiaeo caaea/ a iaeanoyo neeaaeii?
no?oeoo?e.

Iniaenoee aianie aeena?oaioa. ?iaioa [5] iaienaia aeena?oaioii o
ni?aaaoi?noa?. ?.A.Ne?eiieeo iaeaaeeoue aea?? iai?yieo aeine?aeaeaiue,
iinoaiiaea caaea/? oa iaaiai?aiiy io?eiaieo ?acoeueoao?a, aeiaaaeaiiy ae
iniiaieo ?acoeueoao?a iaeaaeeoue aeena?oaioo.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. ?acoeueoaoe aeena?oaoe??
aeiiia?aeaeenue oa iaaiai?thaaeenue ia ia’?aeiaiiio nai?ia?? a?aeae?eo
iae?i?eiiai aiae?co oa a?aeae?eo ??aiyiiue iaoaiaoe/ii? o?ceee ?inoeoooo
i?eeeaaeii? iaoaiaoeee ? iaoai?ee IAI Oe?a?ie (ea??aieee: aeieoi?
o?c.-iao. iaoe, i?ioani?, aeaaeai?e IAI Oe?a?ie ?.A.Ne?eiiee, aeieoi?
o?c.-iao. iaoe, i?ioani? A.A.Aacae?e); I?aeia?iaei?e eiioa?aioe??
“Nonlinear Differential Equations” (i.Ee?a, 1995 ?.); I?aeia?iaei?e
eiioa?aioe?? “Nonlinear Partial Differential Equations” (i.Ee?a, 1997
?.).

Ioae?eaoe??. Iniiai? ?acoeueoaoe aeena?oaoe?? iioae?eiaaii a ?iaioao
[1-7], c yeeo [1-3,5] iaae?oeiaaii o aeaeaiiyo c ia?ae?eo ?1,
caoaa?aeaeaiiai AAE Oe?a?ie, a [4] — o aeaeaii? c ia?ae?eo ?5,
caoaa?aeaeaiiai AAE Oe?a?ie.

No?oeoo?a ? ia’?i ?iaioe. Aeena?oaoe?y neeaaea?oueny ?c anooio, o?ueio
?icae?e?a, aeniiae?a ? nieneo aeei?enoaieo aeaea?ae oa aeeeaaeaia ia 150
noi??ieao iaoeiiieniiai oaenoo. Nienie e?oa?aoo?e i?noeoue 92
iaeiaioaaiiy.

CI?NO ?IAIOE

O anooi? aea?oueny ei?ioeee aiae?c no/aniiai noaio i?iaeaie ?
iaa?oioiao?oueny aeooaeuei?noue oaie, iiaeathoueny iaoa oa caaea/?
aeine?aeaeaiiy, iaoeiaa iiaecia, i?aeoe/ia cia/aiiy, ai?iaaoe?y oa
no?oeoo?a ?iaioe.

O ia?oiio ?icae?e? ?icaeyiooa niaoe?aeueia iiaeaeueia ae?iaeaeaia
iae?i?eia caaea/a Ae???oea, aeey ?ica’yce?a yei? io?eiai? ??ci?
?ioaa?aeuei? oa iioi/eia? ioe?iee.

O i?ae?icae?e? 1.1 aeathoueny iniiai? icia/aiiy oa aeiaiaeyoueny ??ci?
aeiiii?aei? oaa?aeaeaiiy.

caaeiaieueiy? oiiao

(1)

.

ii?iaeaeo? aaaiao i??o ?aaeiia-I?eiaeeia

.

, oaeeo, ui

c

.

iieeaaeaii

.

.

.

iicia/eii iiiaeeio

.

/enei

.

A i?ae?icae?eao 1.2-1.3 ?icaeyaea?oueny niaoe?aeueia iiaeaeueia
ae?iaeaeaia iae?i?eia caaea/a Ae???oea.

.

?icaeyaea?oueny ianooiia caaea/a Ae???oea

(2)

(3)

.

, ? caaeiaieueiythoue ianooii? oiiae:

;

aeeiiai? ia??aiino?

,

.

aeeiiaia ?ioaa?aeueia oioiaei?noue

A oiiaao 1), 2) iaeiicia/ia ?ica’yci?noue caaea/? (2), (3) aeoiaeeoue ?c
caaaeueii? oai??? iiiioiiieo iia?aoi??a.

Aeei?enoiaoth/? iaaio iiae?o?eaoe?th iaoiaea Iica?a, o i?ae?icae?e? 1.3
aeiaaaeaia ianooiia oai?aia, yea ? iniiaiei ?acoeueoaoii ?icae?eo 1.

a??ia ianooiia ioe?iea

, (4)

.

O ae?oaiio ?icae?e? ?icaeyaea?oueny ae?iaeaeaia iae?i?eia caaea/a
Ae???oea a iaeanoyo c ae??aiica?ienoith iaaeath.

A i?ae?icae?e? 2.1 aeine?aeaeothoueny ieoaiiy ona?aaeiaiiy n?i’?
ae?iaeaeaieo iae?i?eieo caaea/ Ae???oea a iine?aeiaiino? iaeanoae c
ae??aiica?ienoith iaaeath o aeiaaeeo ia’?iiiai ?iciiae?eo “ca?ai”.

.

?icaeyaea?oueny eaac?e?i?eia ae?ioe/ia caaea/a

(5)

, (6)

— iaaia a?aeiia ooieoe?y.

oa caaeiaieueiythoue ianooii? oiiae:

aeeiiai? ia??aiino?

,

.

a i?eiouaii? A2) caaeiaieueiythoue oiiae

(7)

.

caaeiaieueiythoue ianooii? oiiae:

, oaea, ui

, oae? ui

, ui aecia/athoueny ye ?ica’ycee iiaeaeueieo caaea/.

? caaeiaieueiy? ?ioaa?aeueio oioiaeiinoue

(8)

.

.

I?eionea?oueny aeeiiaiiy ianooiii? oiiae:

aeeiiaia ??ai?noue

, (9)

.

aeeiiaia ?ioaa?aeueia oioiaei?noue

.

.

Aeei?enoiaoth/e iaoiaee caaaeueii? oai??? iiiioiiieo iia?aoi??a, iiaeia
aeiaanoe ianooiio oai?aio

aeeiiaia ioe?iea

.

? caaeiaieueiythoue ioe?ieo

(10)

.

caaeiaieueiy? oiiao

, (11)

oaea naia, ui ? a oiia? (7).

Iniiaiei ?acoeueoaoii i?ae?icae?eo 2.1 ? ianooiia oai?aia

? ocaaaeueiaiei ?ica’yceii caaea/?

(12)

(13)

Aeiaaaeaiith oai?aie 3 i?enay/ai? ioieoe 2.1.2-2.1.4.

caaea/o (12), (13).

A i?ae?icae?e? 2.2 aeine?aeaeothoueny ieoaiiy ona?aaeiaiiy n?i’?
ae?iaeaeaieo iae?i?eieo caaea/ Ae???oea a iine?aeiaiino? iaeanoae c
ae??aiica?ienoith iaaeath o aeiaaeeo iiaa?oiaaiai ?iciiae?eo “ca?ai”.

.

.

.

caaeiaieueiythoue ianooii? oiiae:

;

aeeiiai? ia??aiino?

,

;

;

aeeiiaia ia??ai?noue

;

aeeiiaia ??ai?noue

(14)

iathoue oie aea ci?no, ui ? a (9).

.

).

Iniiaiei ?acoeueoaoii i?ae?icae?eo 2.2 ? ianooiia oai?aia

? ocaaaeueiaiei ?ica’yceii ianooiii? caaea/? ni?yaeaiiy

(15)

.

Aeiaaaeaiith oai?aie 4 i?enay/ai? ioieoe 2.2.2-2.2.4.

, ca aeiiiiiaith aneiioioe/iiai ?iceeaaeo, iiaoaeiaaii caaea/o
ni?yaeaiiy (15).

O o?aoueiio ?icae?e? aeine?aeaeothoueny ieoaiiy ona?aaeiaiiy n?i’?
ae?iaeaeaieo iae?i?eieo caaea/ Ae???oea a iine?aeiaiino? ia?oi?iaaieo
iaeanoae c “ca?iaie”, ye? aonoi oiaeiaai? (aea, cie?aia, ia
i?eionea?oueny iae?noue ae?aiao??a ii?iaeiei a?aeiinii a?aeaeaeae i?ae
ieie).

.

.

iathoue oie aea ci?no, ui ? a i?ae?icae?e? 2.1, ? aaaaeaii aeiaeaoeia?
iicia/aiiy:

.

caaeiaieueiythoue ianooii? oiiae:

A1) a??ia ??ai?noue

, oaea, ui

oae?, ui

,

? aeeiiaii oiiae:

, aeey yei? a??ia ??ai?noue

, oae? ui

;

aeeiiaia ??ai?noue:

, (16)

ia? oie aea ci?no, ui ? a (9).

.

Iniiaiei ?acoeueoaoii ?icae?eo 3 ? ianooiia oai?aia

? ocaaaeueiaiei ?ica’yceii caaea/?

(17)

, (18)

Aeiaaaeaiith oai?aie 5 i?enay/ai? i?ae?icae?ee 3.2-3.3.

iiaoaeiaaii e?aeiao caaea/o (17), (18).

AENIIAEE

O aeena?oaoe?ei?e ?iaio? aia?oa aeine?aeaeaii ieoaiiy ona?aaeiaiiy n?i’?
ae?iaeaeaieo iae?i?eieo caaea/ Ae???oea a ia?oi?iaaieo iaeanoyo ??cii?
no?oeoo?e oa io?eiaii ianooii? iniiai? ?acoeueoaoe:

Aeiaaaeaii iioi/eiao ioe?ieo ?ica’yceo niaoe?aeueii? iiaeaeueii?
ae?iaeaeaii? iae?i?eii? caaea/? Ae???oea.

Aea/aii aneiioioe/io iiaaae?ieo iine?aeiaiino? ?ica’yce?a caaea/
Ae???oea aeey ae?iaeaeaieo iae?i?eieo ae?ioe/ieo ??aiyiue ae?oaiai
ii?yaeeo a n?i’? iaeanoae c ae??aiica?ienoith iaaeath o aeiaaeeo
ia’?iiiai ?iciiae?eo “ca?ai”, i?e oiia?, ui aaaiaa ooieoe?y iaeaaeeoue
iaaiiio eeano Iaeaioaoioa. Io?eiaii aeinoaoi? oiiae ca?aeiino?
iine?aeiaiino? ?ica’yce?a ?icaeyaeoaaieo caaea/, iiaoaeiaaii ei?aeoi?
aeey iaaeeaeaiiy oaei? iine?aeiaiino? ?ica’yce?a oa e?aeiao caaea/o aeey
a?aie/ii? ooieoe??.

Aiaeia?/i? ?acoeueoaoe io?eiaii aeey n?i’? iaeanoae c ae??aiica?ienoith
iaaeath o aeiaaeeo iiaa?oiaaiai ?iciiae?eo “ca?ai”.

Aiaeia?/i? ?acoeueoaoe io?eiaii aeey iine?aeiaiino? ia?oi?iaaieo
iaeanoae c “ca?iaie”, ye? aonoi oiaeiaai? (aea, cie?aia, ia
i?eionea?oueny iae?noue ae?aiao??a ii?iaeiei a?aeiinii a?aeaeaeae i?ae
ieie). Aeey oaeeo iaeanoae aeeo/aii n?i’th iiiaeei, ui ia aieeaathoue ia
iiaoaeiao aeiaeaoeiaiai /eaio a a?aie/i?e caaea/?.

Nienie iioae?eiaaieo AAOI?II i?aoeue ca oaiith aeEna?oaoe??

Larin D.V. Pointwise estimate of solution of a model degenerate
nonlinear elliptic problem // Nonlinear Boundary Value Problems. – 1997.
– V.7. – P.132-137.

Ea?ei Ae.A. Au?iaeaeathuayny eaaceeeiaeiay caaea/a Aee?eoea aeey
iaeanoae n iaeeica?ienoie a?aieoeae // Aeiiia?ae? IAI Oe?a?ie. – 1997. —
?10. – N.39-43.

Ea?ei Ae.A. I noiaeeiinoe ?aoaiee au?iaeaeathuaeny eaaceeeiaeiie caaea/e
Aee?eoea i?e eciaeue/aiee a?aieoeu iaeanoe // Aeiiia?ae? IAI Oe?a?ie. –
1998. — ?8. – N.37-41.

Ea?ei Ae.A. Au?iaeaeathuayny eaaceeeiaeiay caaea/a Aee?eoea aeey
iaeanoae n iaeeica?ienoie a?aieoeae. Neo/ae iiaa?oiinoiiai
?ani?aaeaeaiey “ca?ai” // O?oaeu EIII IAI Oe?aeiu. – 1998. – O.2. –
N.104-115.

Ne?uiiee E.A., Ea?ei Ae.A. I?eioeei aaeaeeoeaiinoe a on?aaeiaiee
au?iaeaeathueony iaeeiaeiuo caaea/ Aee?eoea // Oe?. iaoai. aeo?i. –
1998. – O.50, ?1. – N.118-135.

Larin D.V. A pointwise estimate of the solution of a model degenerate
nonlinear elliptic problem // Book of abstracts of International Conf.
“Nonlinear Differential Equations”. — Kiev, 1995. — P.95.

Larin D.V. Homogenization of degenerate nonlinear elliptic boundary
value problems in domains with finely granulated boundary // Book of
abstracts of International Conf. “Nonlinear Partial Differential
Equations”. — Kiev, 1997 — P.113.

AIIOAOe??

Ea??i Ae.A. Ona?aaeiaiiy ae?iaeaeaieo iae?i?eieo caaea/ a ia?oi?iaaieo
iaeanoyo. – ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.01.02 —
aeeoa?aioe?aeuei? ??aiyiiy. – ?inoeooo i?eeeaaeii? iaoaiaoeee ? iaoai?ee
IAI Oe?a?ie, Aeiiaoeuee, 1999.

Aeena?oaoe?th i?enay/aii ieoaiiyi ona?aaeiaiiy n?i’? caaea/ Aee??oea
aeey ae?iaeaeaieo iae?i?eieo ae?ioe/ieo ??aiyiue ae?oaiai ii?yaeeo a
ia?oi?iaaieo iaeanoyo ??cii? no?oeoo?e, i?e oiia?, ui aaaiaa ooieoe?y
iaeaaeeoue iaaiiio eeano Iaeaioaoioa. A ?iaio? aea/aii aneiioioe/io
iiaaae?ieo iine?aeiaiino? ?ica’yce?a ?icaeyaeoaaieo caaea/ aeey n?i’?
iaeanoae c ae??aiica?ienoith iaaeath (o aeiaaeeo ia’?iiiai oa
iiaa?oiaaiai ?iciiae?eo “ca?ai”), a oaeiae aeey ia?oi?iaaieo iaeanoae c
“ca?iaie”, ye? aonoi oiaeiaai?, oa io?eiaii aeinoaoi? oiiae ca?aeiino?
?ica’yce?a ?icaeyaeoaaieo ae?iaeaeaieo iae?i?eieo caaea/ Ae???oea,
iiaoaeiaaii ei?aeoi?e aeey iaaeeaeaiiy oaeeo iine?aeiaiinoae ?ica’yce?a
? e?aeia? caaea/? aeey a?aie/ieo ooieoe?e.

?acoeueoaoe iniiaaii ia ??cieo iioi/eiaeo ioe?ieao ?ica’yceo
niaoe?aeueii? iiaeaeueii? ae?iaeaeaii? iae?i?eii? caaea/? Ae???oea.

Eeth/ia? neiaa: ona?aaeiaiiy, ae?iaeaeaiiy, iae?i?eia ae?ioe/ia
??aiyiiy, eeane Iaeaioaoioa, ia?oi?iaai? iaeano?, aneiioioe/ia
iiaaae?iea, iioi/eiaa ioe?iea.

Ea?ei Ae.A. On?aaeiaiea au?iaeaeathueony iaeeiaeiuo caaea/ a
ia?oi?e?iaaiiuo iaeanoyo. – ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.02 –
aeeooa?aioeeaeueiua o?aaiaiey. – Einoeooo i?eeeaaeiie iaoaiaoeee e
iaoaieee IAI Oe?aeiu, Aeiiaoee, 1999.

Aeenna?oaoeey iinayuaia aii?inai on?aaeiaiey naiaenoaa caaea/ Aee?eoea
aeey au?iaeaeathueony iaeeiaeiuo yeeeioe/aneeo o?aaiaiee aoi?iai
ii?yaeea a ia?oi?e?iaaiiuo iaeanoyo ?acee/iie no?oeoo?u, i?e oneiaee,
/oi aaniaay ooieoeey i?eiaaeeaaeeo iaeioi?iio

eeanno Iaeaioaoioa.

p-Eaieanna, aaea aan i?eiaaeeaaeeo iaeioi?iio eeanno Iaeaioaoioa,
eniieuecoy iaeioi?oth iiaeeoeeaoeeth iaoiaea Iica?a, iieo/aiu ?acee/iua
eioaa?aeueiua e iioi/a/iua ioeaiee, e iieacaia oi/iinoue oaeeo ioeaiie.
Ia iniiaa iaoiaea Iica?a aeieacaia ?aaiiia?iay ia?aie/aiiinoue
iineaaeiaaoaeueiinoe ?aoaiee ?anniao?eaaaiuo au?iaeaeathueony iaeeiaeiuo
caaea/ Aee?eoea a ia?oi?e?iaaiiuo iaeanoyo. Aeey naiaenoaa iaeanoae n
iaeeica?ienoie a?aieoeae (a neo/aa iauaiiiai e iiaa?oiinoiiai
?ani?aaeaeaiey “ca?ai”), a oaeaea aeey ia?oi?e?iaaiiuo iaeanoae n aonoi
oiaeiaaiiuie “ca?iaie” (aaea, a /anoiinoe, ia i?aaeiieaaaaony iaeinoue
aeeaiao?ia iieinoae ioiineoaeueii ?annoiyiee iaaeaeo ieie), iino?iaiu
ei??aeoi?u aeey i?eaeeaeaiey iineaaeiaaoaeueiinoae ?aoaiee
?anniao?eaaaiuo caaea/, e, n iiiiuueth iieo/aiiuo ioeaiie ?aoaiee
niaoeeaeueiuo iiaeaeueiuo caaea/, eniieuecoy aeieacaiioth ?aaiiia?ioth
ia?aie/aiiinoue, eco/aii aneiioioe/aneia iiaaaeaiea oaeeo
iineaaeiaaoaeueiinoae ?aoaiee. Iieo/aiu aeinoaoi/iua oneiaey noiaeeiinoe
?aoaiee ?anniao?eaaaiuo au?iaeaeathueony iaeeiaeiuo caaea/ Aee?eoea aeey
oeacaiiuo auoa iaeanoae, auienaiiua a oa?ieiao aaniauo aieinoae yoeo
iaeanoae e niaiaaeathuea n ecaanoiuie ?acoeueoaoaie aeey aacaaniaiai
neo/ay. Aeey i?aaeaeueiuo ooieoeee iino?iaiu a?aie/iua caaea/e a
iaia?oi?e?iaaiiuo iaeanoyo n aeiaaai/iuie /eaiaie, eiathueie aieinoiie
oa?aeoa?.

A neo/aa ia?oi?e?iaaiiuo iaeanoae n aonoi oiaeiaaiiuie “ca?iaie”
auaeaeaii naiaenoai iiiaeanoa, ia aeeythueo ia iino?iaiea on?aaeiaiiie
a?aie/iie caaea/e.

Eeth/aaua neiaa: on?aaeiaiea, au?iaeaeaiea, iaeeiaeiia yeeeioe/aneia
o?aaiaiea, eeannu Iaeaioaoioa, ia?oi?e?iaaiiua iaeanoe, aneiioioe/aneia
iiaaaeaiea, iioi/a/iay ioeaiea.

Larin D.V. Homogenization of degenerate nonlinear problems in perforated
domains. — Manusnript.

Thesis for a candidate’s degree by speciality 01.01.02 — differential
equations. — The Institute of Applied Mathematics and Mechanics of
National Academy of Science of Ukraine, Donetsk, 1999.

The dissertation is devoted to the questions of homogenization of a
family of Dirichlet problems for degenerate nonlinear elliptic second
order equations in perforated domains of various structure provided that
the weight function belongs to a certain Muckenhoupt class. The
asymptotic behavior of a sequence of solutions of problems under
consideration is studied for a family of domains with a finely
granulated boundary (in the case of the volume and surface distribution
of “grains”) and for perforated domains with densely packing “grains”.
Sufficient conditions of convergence of solutions of considered
degenerate nonlinear Dirichlet problems are obtained. Correctors for
approximation of such sequences of solutions and a boundary value
problems for the limit functions are constructed.

The results are based on a various pointwise estimates of a solution of
a special model degenerate nonlinear Dirichlet problem.

Key words: homogenization, degeneration, nonlinear elliptic equation,
Muckenhoupt classes, perforated domains, asymptotic behavior, pointwise
estimate.

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