Îñíîâíûì óñëîâèåì óáûâàíèÿ ìëàäøåãî êîýèöèåíòà, ãàðàíòèðóþùåãî òðèõîòîìèþ ðåøåíèé, ÿâëÿåòñÿ êîíå÷íîñòü èíòåãðàëà R Q q(x) ln |x| dx

(1)

ÓÄÊ517.956

DOI10.23671/VNC.2019.1.27733

ÒÈÕÎÒÎÌÈß ÅØÅÍÈÉÝËËÈÏÒÈ×ÅÑÊÈÕ ÓÀÂÍÅÍÈÉ ÂÒÎÎÎ

ÏÎßÄÊÀ ÑÓÁÛÂÀÞÙÈÌ ÏÎÒÅÍÖÈÀËÎÌ ÍÀÏËÎÑÊÎÑÒÈ

À. B. Íåêëþäîâ

1

Ìîñêîâñêèéãîñóäàðñòâåííûéòåõíè÷åñêèéóíèâåðñèòåòèì.Í.Ý.Áàóìàíà,

îññèÿ,105005,Ìîñêâà,óáöîâñêàÿíàá.,2/18

E-mail:nekl5yandex.ru

Àííîòàöèÿ.Âäâóìåðíîéîáëàñòè

Q

^,^{âíåøíåé}^ïî^{îòíîøåíèþ}^ê^êðóãó,ðàññìàòðèâàåòñÿðàâíîìåðíî ýëëèïòè÷åñêîåóðàâíåíèåâòîðîãî ïîðÿäêàâäèâåðãåíòíîéîðìå ñèçìåðèìûìèêîýèöèåíòàìè,

ñîäåðæàùåå ìëàäøèé íåîòðèöàòåëüíûé êîýèöèåíò

q(x) = q(x 1 , x 2 )

^òèïà ^{ïîòåíöèàëà} ^â ^ñòàöè-

îíàðíîì óðàâíåíèè Øð¼äèíãåðà. Èçó÷àþòñÿîáîáùåííûå ðåøåíèÿ, ïðèíàäëåæàùèå ïðîñòðàíñòâó

Ñ.Ë.Ñîáîëåâà

W 2 ¹

^â^ëþáîéîãðàíè÷åííîéïîäîáëàñòè.àññìàòðèâàåòñÿâîïðîñîâîçìîæíîìðîñòå ðåøåíèé íà áåñêîíå÷íîñòè. Äîêàçàíî, ÷òî ïðè äîñòàòî÷íî áûñòðîì óáûâàíèè ìëàäøåãî êîýè-

öèåíòà

q(x)

^íàáåñêîíå÷íîñòèñóùåñòâóåòïîëîæèòåëüíîå ðåøåíèå,ðàñòóùååêàêëîãàðèììîäóëÿ ðàäèóñ-âåêòîðàòî÷êè,ò.å.òàêæå,êàêóíäàìåíòàëüíîåðåøåíèåñîîòâåòñòâóþùåãîýëëèïòè÷åñêîãî

îïåðàòîðàáåçìëàäøåãî÷ëåíà.Ïîñòðîåííîåðåøåíèåîáëàäàåòðàâíîìåðíîîãðàíè÷åííûì¾ïîòîêîì

òåïëà¿÷åðåçîêðóæíîñòèïðîèçâîëüíîãîðàäèóñà

R

^,êîíöåíòðè÷åñêèåñãðàíèöåéîáëàñòè

Q

^.^Äàëåå

óñòàíàâëèâàåòñÿ,÷òîäëÿëþáîãîðåøåíèÿ,óäîâëåòâîðÿþùåãîíåêîòîðîéñòåïåííîéîöåíêåðîñòàíà

áåñêîíå÷íîñòè,âûïîëíåíà îöåíêàèíòåãðàëàÄèðèõëåòèïàïðèíöèïàÑåí-Âåíàíàâòåîðèèóïðóãî-

ñòè.àíååïîäîáíàÿîöåíêàøèðîêîèñïîëüçîâàëàñüâðàáîòàõäëÿýëëèïòè÷åñêèõóðàâíåíèéâòîðîãî

ïîðÿäêàáåçìëàäøèõ÷ëåíîââíåîãðàíè÷åííûõîáëàñòÿõ.ÎöåíêàòèïàÑåí-Âåíàíàïîçâîëÿåòïîëó-

÷èòüîöåíêóäëÿèíòåãðàëàÄèðèõëåðåøåíèÿâêîëüöåâîéîáëàñòè÷åðåçñðåäíååçíà÷åíèåðåøåíèÿ

íà îäíîé èç îêðóæíîñòåéýòîéêîëüöåâîéîáëàñòè. Èç ýòîãî ñëåäóåò,÷òîðåøåíèå íàîêðóæíîñòè

ðàäèóñà

R

^èìååò^òîò^æå^{ïîðÿäîê}^ðîñòà^ïî

R

^,^÷òî^è^{ñðåäíåå}^{çíà÷åíèå}^íà^ýòîéîêðóæíîñòè.Èñïîëü- çîâàíèå ïðèíöèïà ìàêñèìóìà ïîçâîëÿåò ïîêàçàòü, ÷òîëþáîå ðàñòóùåå íàáåñêîíå÷íîñòèðåøåíèå

èìååò ëîãàðèìè÷åñêèé ðîñò.Îñíîâíîéðåçóëüòàòñòàòüè ñîñòîèòâ òîì,÷òî äëÿäàííîãî óðàâíå-

íèÿèìååòìåñòîòðèõîòîìèÿðåøåíèé,êàêèäëÿóðàâíåíèÿáåçìëàäøåãî÷ëåíà:ðåøåíèåÿâëÿåòñÿ

ëèáî îãðàíè÷åííûì,ëèáî ðàñòåòñ ëîãàðèìè÷åñêîéñêîðîñòüþ,ñîõðàíÿÿçíàê, ëèáî îñöèëëèðóåò

è ðàñòåò ïî ìàêñèìóìó ìîäóëÿ êàêìèíèìóì ñòåïåííûì îáðàçîì. Îñíîâíûì óñëîâèåì óáûâàíèÿ

ìëàäøåãî êîýèöèåíòà, ãàðàíòèðóþùåãî òðèõîòîìèþ ðåøåíèé, ÿâëÿåòñÿ êîíå÷íîñòü èíòåãðàëà

R

Q q(x) ln |x| dx

^.

Êëþ÷åâûå ñëîâà: ýëëèïòè÷åñêîå óðàâíåíèå, íåîãðàíè÷åííàÿ îáëàñòü, ìëàäøèé êîýèöèåíò,

àñèìïòîòè÷åñêîåïîâåäåíèåðåøåíèé,òðèõîòîìèÿðåøåíèé.

MathematialSubjet Classiation(2000): 35J15.

Îáðàçåö öèòèðîâàíèÿ:ÍåêëþäîâÀ. B.Òðèõîòîìèÿðåøåíèéýëëèïòè÷åñêèõóðàâíåíèéâòîðîãî

ïîðÿäêàñóáûâàþùèì ïîòåíöèàëîìíàïëîñêîñòè//Âëàäèêàâê.ìàò.æóðí.2019.Ò.21, âûï.1.

Ñ.3750.DOI:10.23671/VNC.2019.1.2 773 3.

1. Ââåäåíèå

Ïîâåäåíèå ðåøåíèé ýëëèïòè÷åñêèõ óðàâíåíèé âòîðîãî ïîðÿäêà â íåîãðàíè÷åííûõ

îáëàñòÿõèçó÷àëîñüðàçëè÷íûìèàâòîðàìè[14℄.Õîðîøîèçâåñòíî[3 ℄,÷òîïðèíåêîòîðûõ

ïðåäïîëîæåíèÿõ îòíîñèòåëüíî íåîãðàíè÷åííîé îáëàñòè, â ñëó÷àå óðàâíåíèé âòîðîãî

ïîðÿäêà áåç ìëàäøèõ ÷ëåíîâ, äëÿ ðåøåíèé, óäîâëåòâîðÿþùèõ íà ãðàíèöå íåîãðàíè-

÷åííîéîáëàñòèîäíîðîäíîìóóñëîâèþÍåéìàíà,òèïè÷íîéÿâëÿåòñÿòðèõîòîìèÿðåøåíèé.

²⁰¹⁹^{Íåêëþäîâ}^À.^B.

(2)

Òðèõîòîìèÿ îçíà÷àåò, ÷òî ëþáîå ðåøåíèå ïðèíàäëåæèò ê îäíîìó èç òðåõ êëàññîâ: 1)

îãðàíè÷åííûå ðåøåíèÿ, 2) çíàêîïîñòîÿííûå ðåøåíèÿ, ðàñòóùèå ïî ìîäóëþ íà áåñêî-

íå÷íîñòèñî ñêîðîñòüþ, îïðåäåëÿåìîé ãåîìåòðèåé îáëàñòè, 3) îñöèëëèðóþùèåðåøåíèÿ,

ðàñòóùèå ïî ìàêñèìóìó ìîäóëÿ áûñòðåé, ÷åì ðåøåíèÿ èç êëàññà 2). Íàïðèìåð, â ñëó-

÷àå öèëèíäðè÷åñêèõ îáëàñòåé êëàññ 2) îáðàçóþò ðåøåíèÿ ëèíåéíîãî ðîñòà, êëàññ 3)

ðåøåíèÿ, ðàñòóùèå ïî ìàêñèìóìó ìîäóëÿ íà ñå÷åíèè öèëèíäðà ýêñïîíåíöèàëüíî. Äëÿ

äâóìåðíûõ óãëîâûõ îáëàñòåé, à òàêæå âî âíåøíîñòè êðóãà ðåøåíèÿ èç êëàññîâ 2) è 3)

îáëàäàþò ñîîòâåòñòâåííî ëîãàðèìè÷åñêèì è ñòåïåííûì ðîñòîì. Â äàííîé ðàáîòå âî-

ïðîñ î òðèõîòîìèè ðàññìàòðèâàåòñÿ âî âíåøíîñòè êðóãà äëÿ óðàâíåíèÿ, ñîäåðæàùåãî

ìëàäøèé÷ëåíâèäà

q(x)u(x)

^ñ^{äîñòàòî÷íî} ^áûñòðî^{óáûâàþùèì}^âáåñêîíå÷íîñòèïîòåíöè- àëîì

q(x)

^. ^àíåå ^âîïðîñ ^î ^{òðèõîòîìèè} ^{ðåøåíèé} ^{óðàâíåíèé} ^ñ ^{óáûâàþùèì} ïîòåíöèàëîì áûëðàññìîòðåí[5℄äëÿöèëèíäðè÷åñêèõîáëàñòåéñóñëîâèåìÍåéìàíàíàáîêîâîéïîâåðõ-

íîñòè. Áëèçêîé ê ýòîìóñëó÷àþ îêàçàëàñü [6℄ òàêæå ñèòóàöèÿ ñ òðèõîòîìèåé ðåøåíèéâ

öèëèíäðè÷åñêèõ îáëàñòÿõ äëÿ óðàâíåíèé áåç ìëàäøåãî ÷ëåíà ïðè òðåòüåì ãðàíè÷íîì

óñëîâèè (óñëîâèè îáåíà).

2. Îñíîâíûå îáîçíà÷åíèÿ è îïðåäåëåíèÿ.

Âñïîìîãàòåëüíûå óòâåðæäåíèÿ

Âäâóìåðíîéîáëàñòè

Q = {x : |x| > R ₀ }

^(áóäåì^{ñ÷èòàòü,}^÷òî

R ₀ > 1

⁾ðàññìàòðèâàåòñÿ óðàâíåíèå ýëëèïòè÷åñêîãî òèïà

Lu ≡ L ₀ u − q(x)u ≡

2

X

i,j=1

∂

∂x _j

a _ij (x) ∂u

∂x _i

− q(x)u = 0,

⁽¹⁾

ãäå

x = (x ₁ , x ₂ ) ∈ R ² _x

,

a _ij (x)

^{èçìåðèìûå} ^óíêöèè ^â

Q

^,

a _ij = a _ji

^,

λ ₁ |ξ| ² 6 P ₂

i,j=1 a _ij (x)ξ _i ξ _j 6 λ ₂ |ξ| ²

^,

ξ ∈ R ²

,

λ ₁ , λ ₂ = const > 0

^,

q(x) > 0

îãðàíè÷åííàÿ èçìå- ðèìàÿóíêöèÿ.

Ââåäåì ñëåäóþùèåîáîçíà÷åíèÿ:

Q(a, b) := Q ∩ {x : a < |x| < b}, Q _R = Q(R, R + 1), S R := {x : |x| = R}, ∇u := grad u, u(R) := (2πR) ⁻¹

Z

S R

u ds.

Ïîä ðåøåíèÿìè (1) â

Q

^áóäåì ^{ïîíèìàòü} ^{îáîáùåííûå} ^{ðåøåíèÿ,} ^ò. ^å. ^{óíêöèè,} ^ïðè-

íàäëåæàùèå ïðîñòðàíñòâóÑîáîëåâà

W ₂ ¹ (Q(R 0 , R))

^äëÿ^âñåõ

R > R 0

^èóäîâëåòâîðÿþùèå èíòåãðàëüíîìó òîæäåñòâó

Z

Q(R 0 ,R) 2

X

i,j=1

a ij ∂u

∂x _i

∂v

∂x _j dx + Z

Q(R 0 ,R)

quv dx = 0

⁽²⁾

äëÿâñåõ óíêöèé

v ∈ W ₂ ¹ (Q(R ₀ , R))

^òàêèõ, ^÷òî

v

S R0 ∪S R = 0

^.

Äëÿ ðåøåíèÿ

u(x)

^{óðàâíåíèÿ}⁽¹⁾ ñòàíäàðòíûì îáðàçîì ââåäåì ïîíÿòèå ¾ïîòîêà òåï- ëà¿ ÷åðåçîêðóæíîñòü

S _R

^:

P (R, u) = lim

h→0+





 h ⁻¹ Z

Q(R,R+h) 2

X

i,j=1

a _ij ∂u

∂x _i x _j

|x| dx





 = Z

S R

2

X

i,j=1

a _ij ∂u

∂x _i x _j

|x| ds,

(3)

ïîñëåäíåå ðàâåíñòâî ñïðàâåäëèâî äëÿ ïî÷òè âñåõ

R > R ₀

^, ^åãî ^òàêæå ^ìîæíî ^{çàïèñàòü}

ââèäå

P (R, u) = Z

S R

∂u

∂ν ds,

ãäå

∂u

∂ν = P ₂

i,j=1 a _ij _∂x ^∂u

i

x j

|x|

ïðîèçâîäíàÿ ïîêîíîðìàëè êîêðóæíîñòè

S _R

^.

Ïóñòü

R ₀ 6 r < R

^,

h ₁ > 0

^,

h ₂ > 0

^.^{Ïîëîæèì}^â⁽²⁾

v = Φ

^,^ãäå

Φ = Φ(|x|)

íåïðåðûâíàÿ óíêöèÿ,

Φ = 1

^ïðè

r + h ₁ < |x| < R

^,

Φ(r) = Φ(R + h ₂ ) = 0

^,

Φ

^{ëèíåéíàÿ} ^ïðè

r < x ₁ < r + h ₁

^è ^ïðè

R < x ₁ < R + h ₂

^:

h ⁻¹ ₁ Z

Q(r,r+h 1 ) 2

X

i,j=1

a _ij ∂u

∂x _i x _j

|x| dx − h ⁻¹ ₂ Z

Q(R,R+h 2 ) 2

X

i,j=1

a _ij ∂u

∂x _i x _j

|x| dx + Z

Q(r,R+h 2 )

quΦ dx = 0.

Óñòðåìëÿÿê íóëþ

h ₁

^, ^à^çàòåì

h ₂

^,^{ïîëó÷àåì} ñîîòíîøåíèå

P(R, u) − P (r, u) = Z

Q(r,R)

qu dx.

⁽³⁾

Ëåãêîâèäåòü,÷òîïðè

R > R ₀

^âîïðåäåëåíèèïîòîêàîáëàñòüèíòåãðèðîâàíèÿ

Q(R, R +h)

ìîæíîçàìåíèòü íà

Q(R − h, R)

^.

Äàëåå áóäåìèñïîëüçîâàòü íåðàâåíñòâî Ïóàíêàðå ñëåäóþùåãî âèäà:

Z

S R

v ² ds 6 cR ² Z

S R

|∇v| ² ds

(

c > 0

^íå^{çàâèñèò} ^îò^óíêöèè

v

^è

R

⁾ ^äëÿ

v ∈ W ₂ ¹ (S _R )

v(R) = 0

^; ^è

Z

Q(aR,bR)

v ² dx 6 c(a, b)R ²





 Z

Q(aR,bR)

|∇v| ² dx + v ² (R)







äëÿ

v ∈ W ₂ ¹ (Q(aR, bR))

^,

0 < a 6 1

^,

b > 1

^.

Áóäåì èñïîëüçîâàòü òàêæå îöåíêó [7, ëåììà1℄

|v(R) − v(R ₁ )| 6 (2π) ^−1/2 ln ^1/2 R

R ₁





 Z

Q(R 1 ,R)

|∇v| ² dx







1/2

,

⁽⁴⁾

R > R ₁ > 0

^,

v ∈ W ₂ ¹ (Q(R ₁ , R))

^.

3. Ñóùåñòâîâàíèå ïîëîæèòåëüíîãî ðåøåíèÿ ñ ëîãàðèììè÷åêèì ðîñòîì

àññìîòðèì ñîîòâåòñòâóþùååóðàâíåíèþ (1)óðàâíåíèå áåç ìëàäøåãî÷ëåíà

L ₀ V ≡

2

X

i,j=1

∂

∂x j

a _ij (x) ∂V

∂x i

= 0.

⁽⁵⁾

(4)

Õîðîøî èçâåñòíî, íàïðèìåð [8 , îðìóëà (7.5)℄, ÷òî â

Q

^{ñóùåñòâóåò} óíäàìåíòàëüíîå ðåøåíèå

V (x)

^{óðàâíåíèÿ} ^(5),óäîâëåòâîðÿþùååïðè

|x| > R 0

^îöåíêå

C ₁ ln |x| 6 V (x) 6 C ₂ ln |x|, C ₁

^,

C ₂

íåîòðèöàòåëüíûå êîíñòàíòû.

Åñòåñòâåííî îæèäàòü, ÷òî ïðèäîñòàòî÷íî áûñòðîìóáûâàíèè êîýèöèåíòà

q(x)

^íà

áåñêîíå÷íîñòèðåøåíèåñëîãàðèìè÷åñêèìïîâåäåíèåìñóùåñòâóåò èäëÿóðàâíåíèÿ(1) .

Òåîðåìà 1.Ïóñòü

q(x) > 0

^â

Q

^,

R

Q q(x) ln |x| dx < ∞

^,

0 6 q(x) 6 c|x| ⁻² ln |x|

^ïðè

|x| >

R ₁ = const

^;

c > 0

^{íåêîòîðàÿ} ïîñòîÿííàÿ, çàâèñÿùàÿ îò

λ ₁

^,

λ ₂

^. ^Ò^îãäà ^â

Q

^{ñóùåñòâóåò}

ïîëîæèòåëüíîå ðåøåíèå

U (x)

^{óðàâíåíèÿ} ^{(1) ,} óäîâëåòâîðÿþùååóñëîâèÿì

U S R 0

= 0, A ₁ ln |x| 6 U (x) 6 A ₂ ln |x| (0 < A _i = const, i = 1, 2), P (R, U ) → p ₀

^ïðè

R → ∞ (0 < p ₀ = const).

⊳

^Äëÿïðîèçâîëüíîãî

N ∈ N

,

N > R ₀

^,^â^{îáëàñòè}

Q(R ₀ , N )

^{ðàññìîòðèì}^{ðåøåíèå}

U _N (x)

çàäà÷è

LU N = 0, U N

S _R0 = 0, ∂U N

∂ν S N

= (2πN ) ⁻¹ .

Ôóíêöèÿ

U _N

óäîâëåòâîðÿåò èíòåãðàëüíîìó òîæäåñòâó

Z

Q(R 0 ,N) 2

X

i,j=1

a _ij ∂U _N

∂x _i

∂v

∂x _j dx + Z

Q(R 0 ,N)

qU _N v dx = (2πN) ⁻¹ Z

S N

v ds

⁽⁶⁾

äëÿâñåõ óíêöèé

v ∈ W ₂ ¹ (Q(R ₀ , N ))

v| _S _R0 = 0

^.

Ïîëàãàÿ â èíòåãðàëüíîì òîæäåñòâå (6) äëÿ ðåøåíèÿ

U _N

^{ïðîáíóþ} ^óíêöèþ

v = U _N

è èñïîëüçóÿ îöåíêóâèäà (4)äëÿ

U _N (N )

^, ^{ïîëó÷àåì}

Z

Q(R 0 ,N) 2

X

i,j=1

a _ij ∂U _N

∂x i

∂U _N

∂x j

dx + Z

Q(R 0 ,N)

qU _N ² dx

= (2πN ) ⁻¹ Z

S N

U _N ds = U _N (N ) 6 (2π) ^−1/2 ln ^1/2 N

R 0





 Z

Q(R 0 ,N)

|∇U _N | ² dx







1/2

.

Îòñþäàñ ó÷åòîìýëëèïòè÷íîñòè óðàâíåíèÿ(1) ïîëó÷àåì, ÷òî

Z

Q(R 0 ,N)

|∇U _N | ² dx + Z

Q(R 0 ,N)

qU _N ² dx 6 c ₁ ln N,

⁽⁷⁾

çäåñüè äàëåå âäîêàçàòåëüñòâå íåîòðèöàòåëüíûå êîíñòàíòû

c _i

^{çàâèñÿò} ^òîëüêî ^îò

λ ₁

^,

λ ₂

^.

Î÷åâèäíî, ÷òî

P (N, U _N ) = 1

^.

Â ñèëó ïðèíöèïà ýêñòðåìóìà

U N

^íå ^ìîæåò ^èìåòü îòðèöàòåëüíûé ìèíèìóì â

Q(R ₀ , N )

^, ^à ^â ^ñèëó ^{ãðàíè÷íîãî} ^{óñëîâèÿ} ^íà

S _N

^íå ^ìîæåò ^èìåòü ^{ìèíèìóì} ^íà

S _N

^. ^Îò-

ñþäà

U _N > 0

^â

Q(R ₀ , N )

^. ^Ò^àê ^êàê^{ñîãëàñíî} ⁽³⁾ ^ïðè

R ₀ 6 r < N P(r, U _N ) = P (N, U _N ) −

Z

Q(r,N)

qU _N dx,

(5)

òî èç ïîëîæèòåëüíîñòè

U _N

^è ^{ðàâåíñòâà}

P(N, U _N ) = 1

^{ïîëó÷àåì,} ^÷òî

P (r, U _N ) 6 1

^ïðè

R 0 6 r < N

^.

Ïîëó÷èì îöåíêó èíòåãðàëà Äèðèõëå äëÿ

U _N

^ïî ^{îáëàñòè}

Q(R ₀ , r)

^,

R ₀ < r < N

^. ^Äëÿ

ïî÷òè âñåõ

r

^èìååì

Z

Q(R 0 ,r) 2

X

i,j=1

a _ij ∂U N

∂x _i

∂U N

∂x _j dx + Z

Q(R 0 ,r)

qU _N ² dx = Z

S r

U _N ∂U N

∂ν ds.

Îòñþäà ñ ó÷åòîì ýëëèïòè÷íîñòè óðàâíåíèÿ, èñïîëüçóÿ íåðàâåíñòâà Êîøè Áóíÿêîâ-

ñêîãî è Ïóàíêàðå, ïîëó÷àåì

Z

Q(R 0 ,r)

|∇U _N | ² + qU _N ²

dx 6 c ₂ Z

S r

U _N ∂U _N

∂ν ds = c ₂ Z

S r

U _N − U _N (r) ∂U _N

∂ν ds + c ₂ P (r, U _N )U _N (r) 6 c ₃ r

Z

S r

|∇U _N | ² ds + c ₂ P (r, U _N )U _N (r).

(8)

Îòñþäà ñ ó÷åòîìîöåíêè âèäà (4)äëÿ

U _N (r)

^èíåðàâåíñòâà

P (r, U _N ) 6 1

^{ïîëó÷èì}

I(r) ≡

Z

Q(R 0 ,r)

|∇U _N | ² + qU _N ²

dx 6 c ₃ r Z

S r

|∇U _N | ² ds + c ₄ ln ^1/2 r





 Z

Q(R 0 ,r)

|∇U _N | ² dx







1/2

6 c ₃ r Z

S r

|∇U _N | ² ds + 1

2 c ² ₄ ln r + 1 2

Z

Q(R 0 ,r)

|∇U _N | ² dx.

Îòñþäà

I (r) 6 2c ₃ r Z

S r

|∇U _N | ² ds + c ² ₄ ln r 6 2c ₃ rI ^′ (r) + c ² ₄ ln r.

Çàïèøåìýòî íåðàâåíñòâî â âèäå

I(r)r ^−δ ′

> −c ₅ r ^−δ−1 ln r, δ = (2c ₃ ) ⁻¹ > 0, c ₅ = c ² ₄ /(2c ₃ ).

Èíòåãðèðóÿ è ó÷èòûâàÿ ëîãàðèìè÷åñêóþ îöåíêó (7) èíòåãðàëà Äèðèõëå äëÿ

U _N

ïîîáëàñòè

Q(R ₀ , N )

^, ^{ïîëó÷àåì}^îöåíêó

I (r) 6 I(N )

r N

δ

+ c ₅ r ^δ Z N

r

ρ ^−δ−1 ln ρ dρ 6 c ₆ ln r,

⁽⁹⁾

åñëè

r ² 6 N

^.

Òàêèì îáðàçîì, ïîñëåäîâàòåëüíîñòü

U _N

⁽

N > r ²

⁾ ^{îãðàíè÷åíà} ^â

W ₂ ¹ (Q(R ₀ , r))

^äëÿ

ëþáîãî

r > R ₀

^. ^{Îòñþäà,} ^{ïðèìåíÿÿ} äèàãîíàëüíûé ïðîöåññ, ïîëó÷àåì ïîñëåäîâàòåëü- íîñòü

U _N _k

^, ^ñëàáî ^{ñõîäÿùóþñÿ} ^â

W ₂ ¹ (Q(R ₀ , r))

^è ^ñèëüíî ^{ñõîäÿùóþñÿ} ^â

L ₂ (Q(R ₀ , r))

^äëÿ

ëþáîãî

r > R ₀

^ê^{íåêîòîðîé}^óíêöèè

U

^,^{ÿâëÿþùåéñÿ}^{ðåøåíèåì}^{óðàâíåíèÿ}^(1).^{Î÷åâèäíî,}

÷òî

U > 0

^â

Q

^, ^è ñïðàâåäëèâàîöåíêà

Z

Q(R 0 ,r)

|∇U | ² + qU ²

dx 6 c ₆ ln r.

⁽¹⁰⁾

(6)

Òîãäà âñèëó (4)

|U (r)| 6 c 4 ln ^1/2 r





 Z

Q(R 0 ,r)

|∇U | ² dx







1/2

6 c 7 ln r.

Èçîöåíêè Äå Äæîðäæè [9, òåîðåìà 8.17℄ èíåðàâåíñòâà Ïóàíêàðå ïîëó÷àåì,÷òî

sup _S _r |U _N (x)| 6 c ₈ r ⁻¹





 Z

Q(r/2,2r)

U _N ² dx







1/2

6 c ₉











 Z

Q(r/2,2r)

|∇U _N ² | dx







1/2

+ U _N (r)





 .

Îòñþäà, ó÷èòûâàÿ îöåíêè (4) äëÿ

U _N (r)

^è ⁽⁹⁾ ^äëÿ ^{èíòåãðàëà} ^{Äèðèõëå} ^óíêöèè

U _N

^,

ïîëó÷àåì

sup _S _r |U _N (x)| 6 c ₁₀ ln r,

îòêóäà

sup _S _r |U (x)| 6 c 10 ln r.

⁽¹¹⁾

Èç ýòîé îöåíêè, ñ ó÷åòîì òîãî, ÷òî

P(N, U _N ) = 1

^è

R

Q q(x) ln |x| dx < ∞

^, ^{ïîëó÷àåì,} ^÷òî

ïðè

R > R 1 = const

^è

N > R

ñïðàâåäëèâî íåðàâåíñòâî

P (R, U N ) = P (N, U N ) − Z

Q(R,N)

qU N dx > 1 2 .

Èç(3) ñëåäóåò, ÷òî

P (R, U _N ) =

R 0 +1

Z

R 0





 P (r, U _N ) + Z

Q(r,R)

qU _N dx





 dr,

îòêóäà âûòåêàåò, ÷òî

P (R, U ) = lim _N _→∞ P(R, U _N )

^è, ñëåäîâàòåëüíî,

P (R, U ) > 1/2

^äëÿ

äîñòàòî÷íî áîëüøèõ

R

^. ^Èç ⁽³⁾^è ^îöåíêè ^ñâåðõó ⁽¹¹⁾^äëÿ ^óíêöèè

U

^òàêæå ^{ñëåäóåò,} ^÷òî

P (R, U ) → p ₀ = const

^, ^ïðè÷åì

p ₀ > 1/2

^.

Î÷åâèäíî òàêæå,÷òî

Z

S r

|∇U | ² ds > c ₁₁ r ⁻¹ P ² (r, U ) > c ₁₂ r ⁻¹ ,

Z

Q(R 0 ,R)

|∇U| ² dx > c ₁₃ ln R.

⁽¹²⁾

Ïîëó÷èì ëîãàðèìè÷åñêóþ îöåíêó ñíèçóäëÿ

U (x)

^.

ÄëÿèíòåãðàëàÄèðèõëåóíêöèè

U

ñïðàâåäëèâîäèåðåíöèàëüíîåíåðàâåíñòâîâè- äà (8):

J (r) ≡ Z

Q(R 0 ,r)

|∇U | ² dx 6 c ₃ r Z

S r

|∇U | ² ds + c ₂ P (r, U )U (r)

6 c 3 r Z

S r

|∇U | ² ds + c 2 U (r) = c 3 rJ ^′ (r) + c 2 U (r),

(7)

J (r)r ^−ε > −c ₁₄ r ^−ε−1 U (r), ε = c ⁻¹ ₃ , c ₁₄ = c ₂ /c ₃ .

Èíòåãðèðóÿ ýòî íåðàâåíñòâîîò

R

^äî

mR

^ïðè

m > 1

^, ^{ïîëó÷àåì}

Z mR

R

U (r)r ^−ε−1 dr > c ⁻¹ ₁₄ J(R)R ^−ε − J(mR)(mR) ^−ε .

Îòñþäà,èñïîëüçóÿäâóñòîðîííþþëîãàðèìè÷åñêóþ îöåíêó (10) , (12)äëÿ

J (R)

^, ^ïî-

ëó÷èì

Z mR

R

U (r)r ^−ε−1 dr > c ₁₅ R ^−ε ln R − c ₁₆ (mR) ^−ε ln(mR).

Ïóñòü

m > 1

^{òàêîâî,} ^÷òî

c ₁₅ − 2m ^−ε c ₁₆ > c ₁₅ /2

^. ^{Âîçüìåì}

R > m

^. ^{Ïîëó÷èì}^îöåíêó

Z mR

R

U (r)r ^−ε−1 dr > 1

2 c ₁₅ R ^−ε ln R.

Òîãäà äëÿíåêîòîðîãî

ξ ∈ (R, mR)

^èìååì

(m − 1)RU (ξ)ξ ^−ε−1 > 1

2 c ₁₅ R ^−ε ln R,

îòñþäà

U (ξ) > 1

2 (m − 1) ⁻¹ c ₁₅ ln R.

Òàê êàê â ñèëó(4)

U (ξ) − U (R)

6 (2π) ^−1/2 ln ξ

R





 Z

Q(R,ξ)

|∇U | ² dx







1/2

6 c ₁₇ ln ^1/2 R,

òîèç ïðåäûäóùåé îöåíêè äëÿ

U (ξ)

^{ïîëó÷àåì}^äëÿ ^{äîñòàòî÷íî} ^{áîëüøèõ}

R

^îöåíêó

U (R) > c ₁₈ ln R.

⁽¹³⁾

Òàê êàê óíêöèÿ

w = U − U (R)

óäîâëåòâîðÿåò óðàâíåíèþ

L ₀ w = qU

^, ^òî ^èç ^îöåíêè

ÄåÄæîðäæè [9 ,òåîðåìà 8.17℄ èìååì

sup _S _R |U (x) − U (R| ² 6 c ₈





 R ⁻² Z

Q(R/2,2R)

U − U (R)

2 dx + R ² Z

Q(R/2,2R)

q ² U ² dx





 .

ÈñïîëüçóÿíåðàâåíñòâîÏóàíêàðå,óñëîâèÿíàóíêöèþ

q(x)

^è^îöåíêó^{(10) ,}^{ïîëó÷àåì,}^÷òî

sup _S _R |U (x) − U (R| ² 6 c ₁₉





 Z

Q(R/2,2R)

|∇U | ² dx + c ln R Z

Q(R/2,2R)

qU ² dx





 6 1

4 c ² ₁₈ ln ² R,

åñëè êîíñòàíòà

c

^{äîñòàòî÷íî} ^ìàëà. ^C ^ó÷åòîì ^íèæíåé ëîãàðèìè÷åñêîé îöåíêè (13) äëÿ

U (R)

^{ïîëó÷èì}^îöåíêó

U (x) > A ₁ ln |x|

^,

A ₁ = const

^,

A ₁ > 0

^.

⊲

Çàìåòèì òàêæå, ÷òîòî÷å÷íóþ îöåíêó

q(x) 6 c|x| ⁻² ln |x|

^äëÿíåîòðèöàòåëüíîé óíê- öèè

q(x)

^â ^{óñëîâèè} ^{òåîðåìû} ^ìîæíî^{çàìåíèòü} ^íà èíòåãðàëüíóþîöåíêó

R

Q(R/2,2R) q ² dx 6

cR ⁻²

^, ^åñëè ^{ïîñòîÿííàÿ}

c > 0

^{äîñòàòî÷íî} ^ìàëà.

(8)

4. Òðèõîòîìèÿ ðåøåíèé

Ëåììà 1.Ïóñòü

u(x)

^{ðåøåíèå}⁽¹⁾^â

Q

^,

q(x) > 0

^. ^Ò^îãäà^ïðè

R > R ₀ + 1

ñïðàâåäëèâà îöåíêà

Z

Q(R 0 +1,R)

|∇u| ² dx 6 c 1 R ⁻² Z

Q(R,2R)

u ² dx,

c ₁ > 0

^{çàâèñèò}^òîëüêî ^îò

λ ₁

^,

λ ₂

^.

⊳

^Ïóñòü

Φ = Φ(|x|) = 1

^ïðè

R 0 + 1 < |x| < R

^,

Φ(|x|) = |x| − R 0

^ïðè

R 0 < |x| <

R ₀ + 1

^,

Φ(|x|) = ϕ ² (|x|)

^ïðè

R < |x| < 2R

^, ^ãäå

ϕ(|x|) = ^2R−|x| _R

^. ^{Ïîëîæèì} ^â èíòåãðàëüíîì òîæäåñòâå (2)

v = uΦ

^:

Z

Q(R 0 ,2R)

Φ

2

X

i,j=1

a _ij ∂u

∂x i

∂u

∂x j

dx + Z

Q(R 0 ,2R)

qu ² Φ dx = 2R ⁻¹ Z

Q(R,2R)

ϕu

2

X

i,j=1

a _ij ∂u

∂x i

x _j

|x| dx.

Èñïîëüçóÿýëëèïòè÷íîñòüóðàâíåíèÿ(1)èíåðàâåíñòâîÊîøèÁóíÿêîâñêîãî,ïîëó÷àåì

Z

Q(R 0 ,2R)

Φ|∇u| ² dx 6 Z

Q(R,2R)

ϕ ² |∇u| ² dx + c ₁ R ⁻² Z

Q(R,2R)

u ² dx.

Ó÷èòûâàÿ,÷òî

Φ = ϕ ²

^â ^{îáëàñòè}

Q(R, 2R)

^, ^òî ^îòñþäà^ñðàçó ^{ïîëó÷àåì}óòâåðæäåíèå ëåì- ìû.

⊲

Ëåììà 2. Ïóñòü

u(x)

^{ðåøåíèå} ⁽¹⁾^â

Q

^,

q(x) > 0

^; ^â

Q

^{âûïîëíåíî} ^{óñëîâèå}

|u(x)| 6 c ₀ |x| ^γ

äëÿ íåêîòîðûõ ïîñòîÿííûõ

γ > 0

^,

c 0 > 0

^. ^Ò^îãäà, ^åñëè

4 ^γ /(1 + δ) < 1

^,

δ > 0

^, ^òî ^äëÿ

íåêîòîðîé ïîñëåäîâàòåëüíîñòè

R _k → ∞

^,

k → ∞

^, ñïðàâåäëèâà îöåíêà

Z

Q(R _k ,2R _k )

|∇u| ² dx 6 δ Z

Q(R 0 ,R _k )

|∇u| ² dx.

⊳

Ïðåäïîëîæèì ïðîòèâíîå. Òîãäà äëÿâñåõ

R > R ^′ ₀ = const

^èìååì

Z

Q(R 0 ,2R)

|∇u| ² dx − Z

Q(R 0 ,R)

|∇u| ² dx = Z

Q(R,2R)

|∇u| ² dx > δ Z

Q(R 0 ,R)

|∇u| ² dx,

ò. å.

Z

Q(R 0 ,R)

|∇u| ² dx < (1 + δ) ⁻¹ Z

Q(R 0 ,2R)

|∇u| ² dx < (1 + δ) ⁻² Z

Q(R 0 ,4R)

|∇u| ² dx

< · · · < (1 + δ) ^−k Z

Q(R 0 ,2 ^k R)

|∇u| ² dx.

(9)

Èñïîëüçóÿ ëåììó1, ïîëó÷àåì

Z

Q(R 0 ,R)

|∇u| ² dx 6 (1 + δ) ^−k





 c ₁ 2 ^k R −2 Z

Q(2 ^k R,2 ^k+1 R)

u ² dx + I ₀







6 (1 + δ) ^−k

c 1 2 ^k R −2

π 2 ^k+1 R 2

c ² ₀ (2 ^k+1 R) ^2γ + I 0

→ 0, k → ∞,

åñëè

4 ^γ /(1+δ) < 1

^(çäåñü

I ₀

^íå^{çàâèñèò}^îò

k

^).^Ò^àêèì^{îáðàçîì,}

∇u ≡ 0

^,^÷òîíåâîçìîæíî.

⊲

Ëåììà 3. Ïóñòü

u(x)

^{ðåøåíèå} ⁽¹⁾ ^â

Q

^,

|u(x)| 6 c ₀ |x| ^γ

^,

0 < c ₀ = const

^,

0 6 q(x) 6 c|x| ⁻²

^,

γ > 0

^è

c > 0

^{íåêîòîðûå}ïîñòîÿííûå, çàâèñÿùèå îò

λ ₁

^,

λ ₂

^.^Ò^îãäà^äëÿ^{íåêîòîðîé}

ïîñëåäîâàòåëüíîñòè

R ^′ _k → ∞

^,

k → ∞

^, ^äëÿ

x ∈ S _R ^′

k

u(R ^′ _k ) − 1

2 |u(R _k ^′ )| − I 1 6 u(x) 6 u(R ^′ _k ) + 1

2 |u(R ^′ _k )| + I 1 ,

ãäå

I 1 > 0

^íå ^{çàâèñèò}^îò

k

^.

⊳

^Ïóñòü ^äëÿ

δ > 0

^{âûïîëíåíî} ^{óñëîâèå}

4 ^γ /(1 + δ) < 1

^. ^Êàê ^è ^ðàíåå, ^÷åðåç

c ₁ , c ₂ , . . .

áóäåìîáîçíà÷àòü ïîëîæèòåëüíûå ïîñòîÿííûå, çàâèñÿùèå òîëüêîîò

λ ₁

^,

λ ₂

^.

Èñïîëüçóÿ ëåììû 2 è 1è íåðàâåíñòâî Ïóàíêàðå, ïîëó÷èìäëÿíåêîòîðîé ïîñëåäîâà-

òåëüíîñòè

R _k → ∞

^îöåíêó

Z

Q(R k ,2R k )

|∇u| ² dx 6 δ Z

Q(R 0 ,R k )

|∇u| ² dx 6 c 1 δ





 R ⁻² _k Z

Q(R k ,2R k )

u ² dx + I 0







6 c ₂ δ





 Z

Q(R _k ,2R _k )

|∇u| ² dx + u ² (3R _k /2) + I ₀





 ,

ãäå

I 0

k

^. ^Åñëè

c 2 δ 6 1/2

^, ^òî

Z

Q(R _k ,2R _k )

|∇u| ² dx 6 2c ₂ δ u ² (3R _k /2) + I ₀

.

⁽¹⁴⁾

Ââåäåìîáîçíà÷åíèå

q ˜ _k := sup _Q(R _k _,2R _k ₎ q(x)

^.^Ò^àê ^êàê ^óíêöèÿ

w = u − u(3R _k /2)

^{ÿâëÿåòñÿ}

ðåøåíèåì óðàâíåíèÿ

L ₀ w = qu

^, ^òî, ^{èñïîëüçóÿ} ^îöåíêó ^Äå ^{Äæîðäæè} ^[9, ^{òåîðåìà} ^8.17℄

(10)

è äàëåå äâàæäû íåðàâåíñòâî Ïóàíêàðå,ïîëó÷àåì

sup _S ₃

Rk/ 2

u − u(3R _k /2)

2 6 c ₃





 R ⁻² _k Z

Q(R k ,2R k )

u − u(3R _k /2) ²

dx + R ² _k Z

Q(R k ,2R k )

q ² u ² dx







6 c 4





 Z

Q(R k ,2R k )

|∇u| ² dx + ˜ q ² _k R ² _k Z

Q(R k ,2R k )

u ² dx







6 c ₅





 Z

Q(R k ,2R k )

|∇u| ² dx + ˜ q ² _k R ⁴ _k





 Z

Q(R k ,2R k )

|∇u| ² dx + u ² (3R _k /2)













= c ₅ 1+˜ q ² _k R ⁴ _k Z

Q(R k ,2R k )

|∇u| ² dx+c ₅ q ˜ _k ² R ⁴ _k u ² (3R _k /2) 6 c ₆ Z

Q(R k ,2R k )

|∇u| ² dx+c ₅ c ² u ² (3R _k /2).

Èñïîëüçóÿ îöåíêó (14)ïîëó÷àåì, ÷òî ïðè

k > k ₀ = const sup _S ₃

Rk/ 2

u − u(3R _k /2)

2 6 c ₇ δ u ² (3R _k /2) + I ₀

+ c ₅ c ² u ² (R _k ) 6 2c ₇ δu ² (3R _k /2) + I ₀ ^′ ,

åñëè

c 5 c ² 6 c 7 δ

^. ^Çäåñü

I ₀ ^′

k

^.

Ïóñòü

δ > 0

^,

γ > 0

^è

c > 0

^{òàêîâû,}^÷òî

c ₂ δ 6 1/2

^,

2c ₇ δ 6 1/4

^,

4 ^γ /(1+ δ) < 1

^,

c ₅ c ² 6 c ₇ δ

^.

Òîãäà ïðè

k > k ₀ = const

sup

S 3 Rk/ 2

u − u(3R _k /2)

2 6 1

4 u ² (3R _k /2) + I ₀ ^′ .

Îòñþäàñðàçó âûòåêàåò óòâåðæäåíèå ëåììû äëÿïîñëåäîâàòåëüíîñòè

R ^′ _k = 3R _k /2

^.

⊲

Ëåììà 4. Ïóñòü äëÿ

u(x)

^{âûïîëíåíû} ^{óñëîâèÿ} ^{òåîðåìû}

1

^è ^ëåììû

3

^. ^Ò^îãäà ^ïðè

|x| > R ₁ = const

|u(x)| 6 c ₀ ln |x|, 0 < c ₀ = const

^íå^{çàâèñèò} ^îò

x

^.

⊳

Ïðåäïîëîæèì ïðîòèâíîå, òîãäà äëÿ íåêîòîðîé ïîñëåäîâàòåëüíîñòè

R _k → ∞

^èìå-

åì

sup _R _k |u|/ ln R _k → ∞

^,

k → ∞

^. ^Ïóñòü

U (x)

ïîëîæèòåëüíîå ðåøåíèå óðàâíåíèÿ (1) , óäîâëåòâîðÿþùåå ëîãàðèìè÷åñêîé îöåíêå, ñóùåñòâîâàíèå êîòîðîãî äîêàçàíî â òåîðå-

ìå 1. Ïðèìåíÿÿ ê óíêöèÿì

u ± c ₁ U

^ïðè ^{äîñòàòî÷íî} ^{áîëüøîì}

c ₁

^{ïðèíöèï} ^{ìàêñèìóìà,}

ëåãêî ïîëó÷èòü, ÷òî

sup _S _R |u|/ ln R → ∞

^,

R → ∞

^. ^Â ^{ïðîòèâíîì} ^ñëó÷àå ^óíêöèÿ

u(x)

óäîâëåòâîðÿëà áûëîãàðèìè÷åñêîé îöåíêå â

Q

^,^÷òî ïðîòèâîðå÷èò ïðåäïîëîæåíèþ.

Ïóñòü

R _k ^′

ïîñëåäîâàòåëüíîñòü, äëÿêîòîðîéñïðàâåäëèâîóòâåðæäåíèåëåììû3.Áåç îãðàíè÷åíèÿ îáùíîñòè ìîæíî ñ÷èòàòü, ÷òî

sup _S

R ′ k

u > 0

^. ^Ò^îãäà ^èç ^ëåììû ³ ^{ïîëó÷àåì,}

÷òî

inf _S _R _′

k

u/ ln R ^′ _k → +∞

^,

k → ∞

^.^{Ïðèìåíÿÿ} ^{ïðèíöèï}^{ìàêñèìóìà}^ê^óíêöèè

U − c ₂ − εu

ïðèäîñòàòî÷íî áîëüøîì

c ₂

^è^{óñòðåìëÿÿ}

ε > 0

^ê^íó^ëþ,^{ïîëó÷àåì,} ^÷òî

U 6 c ₂

^â

Q(R ₁ ^′ , ∞)

^,

÷òî íåâîçìîæíî.

⊲

u(x)

^{âûïîëíåíû}^{óñëîâèÿ} ^{òåîðåìû}

1

^è ^ëåììû

3

^. ^Ò^îãäà

Z

Q(R 0 ,r)

|∇u| ² dx 6 c ₀ ln r,

(11)

0 < c ₀ = const

^íå ^{çàâèñèò} ^îò

r > R ₀

^.

⊳

^{Èñïîëüçóåì}^äëÿ ^óíêöèè

u(x)

äèåðåíöèàëüíîå íåðàâåíñòâî âèäà(8) :

I(r) ≡ Z

Q(R 0 ,r)

|∇u| ² dx 6 c 1 rI ^′ (r) + c 2 P (r, u)u(r),

⁽¹⁵⁾

çäåñüèäàëåå âäîêàçàòåëüñòâå

c i > 0

^{çàâèñÿò} ^òîëüêî ^îò

λ 1

^,

λ 2

^. ^Â^ñèëó^ëåììû ⁴

|u(x)| 6 c ₃ ln |x|

^, ^{ïîýòîìó}

R

Q q|u| dx < ∞

^. ^Òîãäà ^èç ⁽³⁾^{ñëåäóåò,} ^÷òî

|P (r, u)| 6 c ₄

^. ^À ^èç ⁽¹⁵⁾ ^òîãäà

âûòåêàåò, ÷òî

I(r) 6 c 1 rI ^′ (r) + c 5 ln r.

Èíòåãðèðóÿ êàê èâ äîêàçàòåëüñòâå òåîðåìû 1,ïðè

r ² < R

^{ïîëó÷àåì}^îöåíêó

I(r) 6 I(R)

r R

δ

+ c ₆ ln r 6 c ₀ ln r,

ïîñëåäíåå íåðàâåíñòâî ñëåäóåò èç òîãî, ÷òî â ñèëó ëåììû 1 âûïîëíåíà îöåíêà

I(R) 6 c ₇ ln ² R

^, ^çäåñü

0 < δ = const

^.

⊲

u(x)

^{âûïîëíåíû} ^{óñëîâèÿ} ^{òåîðåìû}

1

^è ^ëåììû

3

^, ^ïðè÷åì ^äëÿ

íåêîòîðîé ïîñëåäîâàòåëüíîñòè

R _k , R _k → ∞

^,

k → ∞

^, ñïðàâåäëèâà îöåíêà

inf _S _Rk |u| = o(ln R _k )

^,

k → ∞

^. ^Ò^îãäà^{ðåøåíèå}

u(x)

^{îãðàíè÷åíî}^â

Q

^.

⊳

^{Ñîãëàñíî} ^ëåììå ⁴

|u(x)| 6 c ₀ ln |x|

^. ^Â ^ñèëó ^îöåíêè ^Äå ^{Äæîðäæè} ^è íåðàâåíñòâà Ïóàíêàðå

sup

S _Rk

|u(x) − u(R _k )| ² 6 c ₁





 R ⁻² _k Z

Q(R _k /2,2R _k )

|u − u(R _k )| ² dx + R _k ² Z

Q(R _k /2,2R _k )

q ² u ² dx







6 c ₂





 Z

Q(R k /2,2R k )

|∇u| ² dx + sup _Q(R _k _/2,2R _k ₎ (qu ² )R ² _k ln ⁻¹ R _k Z

Q(R k /2,2R k )

q ln |x| dx







6 c 3





 Z

Q(R k /2,2R k )

|∇u| ² dx + ln R _k Z

Q(R k /2,2R k )

q ln |x| dx





 .

Îòñþäà, ó÷èòûâàÿ ëîãàðèìè÷åñêóþ îöåíêó èíòåãðàëà Äèðèõëå â ñèëó ëåììû 5,

ïîëó÷àåì, ÷òî ïðè

x ∈ S _R _k

^{âûïîëíåíà} ^îöåíêà

u(x) − u(R _k ) = o(ln R _k )

^, ^îòêóäà ^{ñëåäóåò,}

÷òî

u(x) = o(ln R _k )

^íà

S _R _k

^. ^{Ïðèìåíÿÿ} ^{ïðèíöèï}^{ìàêñèìóìà}^ê ^{óíêöèÿì}

u ± c ₀ ± εU

^(ãäå,

êàê èâûøå,

U (x)

^{ðåøåíèå}^{óðàâíåíèÿ}⁽¹⁾^ñ ëîãàðèìè÷åñêèì ðîñòîì)ïðèäîñòàòî÷íî áîëüøîì

c ₀

^, ^{ïîëó÷èì,} ^÷òî

|u| 6 c ₀ + εU

^â

Q(R ₁ , R _k )

^äëÿ

k > k ₀ (ε)

^. ^{Óñòðåìèâ}

ε

^ê ^íóëþ,

ïîëó÷èìóòâåðæäåíèå ëåììû.

⊲

Òåîðåìà 2. Ïóñòü âûïîëíåíû óñëîâèÿ

R

Q q(x) ln |x| dx < ∞

^,

0 6 q(x) 6 c|x| ⁻²

^äëÿ

çàâèñÿùåé îò

λ ₁

^,

λ ₂

^{ïîñòîÿííîé}

c > 0

^, óäîâëåòâîðÿþùåé óñëîâèÿì íàêîíñòàíòó

c

^â ^òåî-

ðåìå

1

^è ^ëåììå

3

^. ^Ò^îãäà ^ëþáîå ^{ðåøåíèå} ^{óðàâíåíèÿ} ⁽¹⁾ ^â

Q

^âåäåò ^ñåáÿ ^îäíèì ^èç ^òðåõ

âîçìîæíûõñïîñîáîâ:

1) sup _S

Rk |u| > c ₀ R _k ^γ

^äëÿ^{íåêîòîðîé}ïîñëåäîâàòåëüíîñòè

R _k → ∞

^,

k → ∞

^,^ïðè÷åì

u(x)

ìåíÿåò çíàê íà ëþáîé îêðóæíîñòè

S _R

^ïðè

R > R ^′ ₀ = const

^; ^{ïîñòîÿííàÿ}

γ > 0

^{çàâèñèò}

òîëüêîîò

λ ₁

^,

λ ₂

^;

0 < c ₀ = const

^;

(12)

2) C ₁ ln |x| 6 u(x) 6 C ₂ ln |x|

^,

C ₁ > 0, C ₂ > 0

^;

3) u(x)

^{îãðàíè÷åíî} ^â

Q

^.

⊳

^Åñëè^{ðåøåíèå}

u(x)

^íåóäîâëåòâîðÿåòóñëîâèþ1),òî,ñîãëàñíîëåììå4,ñïðàâåäëèâà îöåíêà

|u(x)| 6 c ln |x|

^,

c = const

^. ^Åñëè^ïðè ^ýòîì ^íå^{âûïîëíåíî} ^{óñëîâèå} ^2),^òî ^ïî^ëåììå ⁶

ðåøåíèå

u(x)

^{îãðàíè÷åíî} ^â

Q

^.

Ïîêàæåì, ÷òî ëþáîå ðåøåíèå, óäîâëåòâîðÿþùåå 1), ÿâëÿåòñÿ çíàêîïåðåìåííûì.

Ïðåäïîëîæèì ïðîòèâíîå. Ïóñòü ñóùåñòâóåò ðåøåíèå

u(x)

^èç ^êëàññà ^1), íåîòðèöàòåëü- íîå ïðè

|x| > R ₁ = const

^.

Ëåãêî âèäåòü, ÷òî äëÿ íåîòðèöàòåëüíûõ ðåøåíèé óðàâíåíèÿ (1) â îáëàñòè

Q(R/2, 3R/2)

^{âûïîëíåíî} íåðàâåíñòâî Õàðíàêà ñ êîíñòàíòîé

K

^, ^íå ^{çàâèñÿùåé} ^îò

R

^:

u(A)/u(B) 6 K

^äëÿ ^âñåõ

A, B ∈ Q(R/2, 3R/2)

^. Äåéñòâèòåëüíî, îòîáðàçèì

Q(R/2, 3R/2)

íà îáëàñòü

Q(1/2, 3/2)

ïðåîáðàçîâàíèåì

x → y = x/R

^. ^{Óðàâíåíèå} ⁽¹⁾ ^{ïåðåéäåò} ^â ^óðàâ-

íåíèå

L _R u − q _R (y)u = 0

^, ^ãäå

L _R

^{ðàâíîìåðíî} ýëëèïòè÷åñêèé äèâåðãåíòíûé îïåðàòîð ïîïåðåìåííûì

y

^ñïîñòîÿííûìè ýëëèïòè÷íîñòè, íå çàâèñÿùèìèîò

R

^;

q R (y) = R ² q(x) 6 c ₂ = const

^, ^{ïîýòîìó}^{êîíñòàíòà} ^{Õàðíàêà} ^äëÿ ^{ðåøåíèé}

u(y)

^â

Q(1/2, 3/2)

R

^.

Ñîîòâåòñòâåííî íå çàâèñèòîò

R

^{êîíñòàíòà} ^{Õàðíàêà} ^äëÿ^{ðåøåíèé}

u(x)

^â

Q(R/2, 3R/2)

^.

Òàê êàê

sup _S

Rk u/ ln R _k → ∞

^,

k → ∞

^, ^òî ^â ^ñèëó íåðàâåíñòâà Õàðíàêà ïîëó÷àåì

inf _S _Rk u/ ln R _k → ∞

^. ^{Ïðèìåíÿÿ} ^{ïðèíöèï} ^{ìàêñèìóìà,} ^êàê ^è ^â äîêàçàòåëüñòâå ëåììû 4, ïîëó÷èì,÷òîñóùåñòâîâàíèå ðåøåíèÿ

u(x)

^, ^{ðàñòóùåãî}^íà^{íåêîòîðîé} ïîñëåäîâàòåëüíîñòè îêðóæíîñòåé

S _R _k

^{áûñòðåå,}^÷åì

ln |x|

^,ïðîòèâîðå÷èòñóùåñòâîâàíèþðåøåíèÿ

U (x)

^ñ^ëîãà-

ðèìè÷åñêèìðîñòîì.Òàêèìîáðàçîì,ðåøåíèåèçêëàññà1)ìîæåò áûòüòîëüêîçíàêîïå-

ðåìåííûìâ ëþáîé îáëàñòèâèäà

|x| > R ₁

^. ^Îòñþäà^{ñëåäóåò}^, ^÷òî^îíî ^äîëæíî^ìåíÿòü ^çíàê

íà ëþáîé îêðóæíîñòè

S _R

^äëÿ ^{äîñòàòî÷íî} ^{áîëüøèõ}

R

^, ^{ïîñêîëüêó,} ^åñëè ^áû ñóùåñòâîâàëà ïîñëåäîâàòåëüíîñòü

R ^′ _k → ∞

^òàêàÿ, ^÷òî

u > 0

^íà

S _R ′

k

, òî â ñèëó ïðèíöèïà ìàêñèìóìà

u > 0

^â

Q(R ^′ ₁ , ∞)

^, ^÷òî íåâîçìîæíî. Òàêèì îáðàçîì,òåîðåìà ïîëíîñòüþäîêàçàíà.

⊲

Â çàêëþ÷åíèå îòìåòèì, ÷òî èíòåãðàëüíîå óñëîâèå óáûâàíèÿ ïîòåíöèàëà

R

Q q(x) ln |x| dx < ∞

^{ÿâëÿåòñÿ} ^{àíàëîãîì} ^{óñëîâèÿ} ^{òðèõîòîìèè}

R

x ₁ q(x) dx < ∞

^äëÿ

ðåøåíèé çàäà÷è Íåéìàíà â áåñêîíå÷íîì öèëèíäðå [5 ℄ (çäåñü

x ₁

ïåðåìåííàÿ, ñîîòâåò- ñòâóþùàÿ îñè öèëèíäðà).

Ëèòåðàòóðà

1. ËàíäèñÅ.Ì.,Ïàíàñåíêî.Ï.ÎáîäíîìâàðèàíòåòåîðåìûòèïàÔðàãìåíàËèíäåëåàäëÿýë-

ëèïòè÷åñêèõóðàâíåíèéñêîýèöèåíòàìè,ïåðèîäè÷åñêèìèïîâñåìïåðåìåííûì,êðîìåîäíîé//

Òð.ñåìèíàðàèì.È..Ïåòðîâñêîãî.1979.Ò.5.Ñ.105136.

2. ÎëåéíèêÎ.À.,Èîñèüÿí.À.Îïîâåäåíèèíàáåñêîíå÷íîñòèðåøåíèéýëëèïòè÷åñêèõóðàâíåíèé

âòîðîãîïîðÿäêàâîáëàñòÿõñíåêîìïàêòíîéãðàíèöåé//Ìàò.ñá.1980.4.Ñ.588610.

3. Ëàíäèñ Å. Ì., Èáðàãèìîâ À. È.Çàäà÷è Íåéìàíà â íåîãðàíè÷åííûõ îáëàñòÿõ // Äîêë. ÀÍ.

1995.Ò.343,4.Ñ.1718.

4. Êîíäðàòüåâ Â. À., Îëåéíèê Î. À. Îá àñèìïòîòèêå â îêðåñòíîñòè áåñêîíå÷íîñòè ðåøåíèé

ñ êîíå÷íûì èíòåãðàëîì Äèðèõëå ýëëèïòè÷åñêèõ óðàâíåíèé âòîðîãî ïîðÿäêà // Òð. ñåìèíàðà

èì.È..Ïåòðîâñêîãî.1987.Ò.2.C.149163.

5. ÍåêëþäîâÀ.Â.Îðåøåíèÿõýëëèïòè÷åñêèõóðàâíåíèéâòîðîãîïîðÿäêàâöèëèíäðè÷åñêèõîáëà-

ñòÿõ//Óèìñê.ìàò.æóðí.2016.Ò.8,âûï.4.Ñ.135146.

6. ÍåêëþäîâÀ.Â.Îçàäà÷åîáåíàäëÿýëëèïòè÷åñêèõóðàâíåíèéâòîðîãîïîðÿäêàâöèëèíäðè÷åñêèõ

îáëàñòÿõ//Ìàò.çàìåòêè.2018.Ò.103,âûï.3.Ñ.417436.

7. ÍåêëþäîâÀ.Â.ÀñèìïòîòèêàðåøåíèéäâóìåðíîãîóðàâíåíèÿàóññàÁèáåðáàõààäåìàõåðà

ñïåðåìåííûìèêîýèöèåíòàìèâîâíåøíåé îáëàñòè//Ñèá. ýëåêòðîí.ìàò.èçâ.2018.Ò.15.

Ñ.338354.

(13)

8. Littman W., Stampahia G., Weinberger H. F.Regular Pointsfor Ellipti Equations with Dison-

tinuousCoeients//Ann.SuolaNorm.Sup.Pisa.Ser.3.1963.Vol.17,3.P.4377.

9. èëáàðãÄ.,ÒðóäèãåðÍ.Ýëëèïòè÷åñêèåäèåðåíöèàëüíûåóðàâíåíèÿñ÷àñòíûìèïðîèçâîäíû-

ìèâòîðîãîïîðÿäêà.Ì.:Íàóêà,1989.464ñ.

Ñòàòüÿïîñòóïèëà 16ìàÿ2018ã.

Íåêëþäîâ Àëåêñåé Âëàäèìèðîâè÷

Ìîñêîâñêèéãîñóäàðñòâåííûéòåõíè÷åñêèé

óíèâåðñèòåòèì.Í.Ý.Áàóìàíà,

äîöåíòêàåäðûâûñøåéìàòåìàòèêè

ÎÑÑÈß,105005,Ìîñêâà,óáöîâñêàÿíàá.,2/18

Vladikavkaz MathematialJournal

2019,Volume 21,Issue1,P. 3750

TRICHOTOMYOF SOLUTIONS OFSECOND-ORDER ELLIPTIC EQUATIONS

WITHA DECREASING POTENTIAL INTHE PLANE

Neklyudov, A.V.

1

BaumanMosowStateTehnialUniversity,

2/18Rubtsovskayanab.,Mosow105005,Russia

Abstrat.Weonsiderauniformlyelliptiseond-orderdivergentequationwithmeasurableoeients

intwo-dimensional domain

Q

êxternal^to ^the îrle. Ânêquation ôntains ^the^lowernonnegative oeient

q(x) = q(x 1 , x 2 )

ôf ^potential ^type ⁱⁿ ^the ^stationary ^Shrôdinger êquation. ^Weak ^solutions ⁱⁿ ^the ^Sobolev

spae

W 2 ¹

ⁱⁿ âny ^bounded ^subdomain âre ^studied. ^The ^possible ^rate ôf ^solutions ât înnity îs ônsidered.

Itis establishedthat ifthe loweroeientdereases withasuient ratethenthe positivesolutionexists

andhasthesamerateatinnityasthefundamentalsolutionofrespetiveelliptiequationwithoutlowerterm.

Therateislogarithmi.Thissolutionhasuniformlyboundedheatowonirlesofradius

R

^.^It^isestablished Sen-Venan typeinequality forDirihletintegral ofsolutionofpowerrate.Sen-Venaninequalityleadsto the

evaluationof Dirihlet integral in a ringdomain viaaverage value of solution onthe irle. It means that

thesolutionhasthesamerateontheirleasitsaveragevalue.Maximumprinipleimpliesthatanytending

to innity solution has the logarithmi rate. The main result of paper is the trihotomy of solutions: The

solutionis eitherbounded,ortends toinnitywithalogarithmi rate,preservingthe sign,or osillates and

has a power-law rate of the maximum of the modulus. The basi ondition for the derease of the lower

oeientisformulatedinintegral form

R

Q q(x) ln |x| dx < ∞

^.

Key words:ellipti equation,unboundeddomain, lower oeient, asymptotibehaviourof solutions,

trihotomyofsolutions.

MathematialSubjet Classiation(2000): 35J15.

For itation: Neklyudov, A. V. Trihotomy of Solutions of Seond-Order Ellipti Equations with

a Dereasing Potential in the Plane, Vladikavkaz Math. J., 2013, vol. 21, no. 4, pp. 3750 (in Russian).

DOI:10.23671/VNC.2019.1.27 733 .

Referenes

1. Landis, Å. Ì., Panasenko, G. P. A Variant of a Theorem of Phragmen-Lindelof Type for Ellipti

Equations with Coeients That Are Periodi in All Variables But One, Trudy Seminara imeni

I.G.Petrovskogo [ProeedingsofthePetrovskiySeminar℄,1979,vol.5,pp.105136(inRussian).

2. Oleinik, O. A., Iosif'yan, G. A. On the Behavior at Innity of Solutions of Seond Order Ellipti

EquationsinDomainswithNonompatBoundary,Mathematis of theUSSR-Sbornik, 1981,vol. 40,

no.4,pp.527548.DOI:10.1070/SM1981v040n04ABEH00 184 9.

(14)

3. Landis,E. M.Ibragimov,A. I.NeumannProblems inUnboundedDomains,Doklady AkademiiNauk

[ReportsofAkademyofSiene℄,1995,vol.343,no.1,pp.1718(inRussian).

4. Kondrat'ev, V. A. and Oleinik, O. A. Asymptotis in a Neighborhood of Innity of Solutions with

Finite Dirihlet Integral of Seond-Order Ellipti Equations, Journal of Soviet Mathematis, 1989,

vol.47,no.4,pp.25962607. DOI:10.1007/BF01105913 .

5. Nekludov, A. V. On Solutions of Seond Order Ellipti Equations in Cylindrial Domains, Ufa

MathematialJournal,2016,vol.8,no.4,pp.131143.DOI:10.13108/2016-8-4-131.

6. Nekludov, A.V.OntheRobinProblemfor Seond-Order ElliptiEquations inCylindrialDomains,

MathematialNotes,2018,vol.103,no.34, pp.430446.DOI:10.1134/S00014346180 300 94.

7. Nekludov, A. V. Asymptoti of Solutions of Two-Dimesional GaussBierbahRademaher Equation

withVariableCoeientsinExternalArea,SibirskieEletronnie MatematiheskieIzvestiya[Syberian

EletroniMathematialReports℄,2018,vol.15, pp.338354(inRussian).

8. Littman, W., Stampahia, G. and Weinberger, H. F. Regular Points for Ellipti Equations with

DisontinuousCoeients,Ann.SuolaNorm.Sup.Pisa.Ser.3,1963,vol.17, no.12,pp.4377.

9. Gilbarg, D. and Trudinger,N. Ellipti PartialDierential Equations of Seond Order, Berlin, N.Y.,

SpringerVerlag,1977,401p.

ReeivedMay16,2018

AlekseyV.Neklyudov

BaumanMosowStateTehnialUniversity,

2/18Rubtsovskayanab.,Mosow105005,Russia,

AssosiateProfessoroftheDepartmentofHigherMathematis

Îñíîâíûì óñëîâèåì óáûâàíèÿ ìëàäøåãî êîýèöèåíòà, ãàðàíòèðóþùåãî òðèõîòîìèþ ðåøåíèé, ÿâëÿåòñÿ êîíå÷íîñòü èíòåãðàëà R Q q(x) ln |x| dx

1

1

Q

q(x) = q(x 1 , x 2 )

W 2 1

q(x)

R

Q

R

R

R

Q q(x) ln |x| dx

q(x)u(x)

q(x)

Q = {x : |x| > R 0 }

R 0 > 1

Lu ≡ L 0 u − q(x)u ≡

2

X

i,j=1

∂

∂x j

a ij (x) ∂u

∂x i

− q(x)u = 0,

x = (x 1 , x 2 ) ∈ R 2 x

a ij (x)

Q

a ij = a ji

λ 1 |ξ| 2 6 P 2

i,j=1 a ij (x)ξ i ξ j 6 λ 2 |ξ| 2

ξ ∈ R 2

λ 1 , λ 2 = const > 0

q(x) > 0

Q(a, b) := Q ∩ {x : a < |x| < b}, Q R = Q(R, R + 1), S R := {x : |x| = R}, ∇u := grad u, u(R) := (2πR) −1

Z

S R

u ds.

Q

W 2 1 (Q(R 0 , R))

R > R 0

Z

Q(R 0 ,R) 2

X

i,j=1

a ij ∂u

∂x i

∂v

∂x j dx + Z

Q(R 0 ,R)

quv dx = 0

v ∈ W 2 1 (Q(R 0 , R))

v

S R0 ∪S R = 0

u(x)

S R

P (R, u) = lim

h→0+





 h −1 Z

Q(R,R+h) 2

X

i,j=1

a ij ∂u

∂x i x j

|x| dx





 = Z

S R

2

X

i,j=1

a ij ∂u

∂x i x j

|x| ds,

R > R 0

P (R, u) = Z

W 2 ¹

Q = {x : |x| > R ₀ }

R ₀ > 1

Lu ≡ L ₀ u − q(x)u ≡

∂x _j

a _ij (x) ∂u

∂x _i

x = (x ₁ , x ₂ ) ∈ R ² _x

a _ij (x)

a _ij = a _ji

λ ₁ |ξ| ² 6 P ₂

i,j=1 a _ij (x)ξ _i ξ _j 6 λ ₂ |ξ| ²

ξ ∈ R ²

λ ₁ , λ ₂ = const > 0

Q(a, b) := Q ∩ {x : a < |x| < b}, Q _R = Q(R, R + 1), S R := {x : |x| = R}, ∇u := grad u, u(R) := (2πR) ⁻¹

W ₂ ¹ (Q(R 0 , R))

∂x _i

∂x _j dx + Z

v ∈ W ₂ ¹ (Q(R ₀ , R))

S _R

 h ⁻¹ Z

a _ij ∂u

∂x _i x _j

a _ij ∂u

∂x _i x _j

R > R ₀

∂ν = P ₂

i,j=1 a _ij _∂x ^∂u

S _R

R ₀ 6 r < R

h ₁ > 0

h ₂ > 0

r + h ₁ < |x| < R

Φ(r) = Φ(R + h ₂ ) = 0

r < x ₁ < r + h ₁

R < x ₁ < R + h ₂

h ⁻¹ ₁ Z

a _ij ∂u

∂x _i x _j

|x| dx − h ⁻¹ ₂ Z

a _ij ∂u

∂x _i x _j

h ₁

h ₂

R > R ₀

v ² ds 6 cR ² Z

|∇v| ² ds

v ∈ W ₂ ¹ (S _R )

v ² dx 6 c(a, b)R ²

|∇v| ² dx + v ² (R)

v ∈ W ₂ ¹ (Q(aR, bR))

|v(R) − v(R ₁ )| 6 (2π) ^−1/2 ln ^1/2 R