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ÐеждÑнаÑÐ¾Ð´Ð½Ð°Ñ ÐºÐ¾Ð½ÑеÑенÑÐ¸Ñ ÑÑÑденÑов, аÑпиÑанÑов и молодÑÑ ... ÐеждÑнаÑÐ¾Ð´Ð½Ð°Ñ ÐºÐ¾Ð½ÑеÑенÑÐ¸Ñ ÑÑÑденÑов, аÑпиÑанÑов и молодÑÑ ...
98 EHFHGHKHD H;H>GHCKFR:GGHCA:>:Q >EYMJ: −1\u ε 1 1 ihegy_lky g_jZ\_gkl\h | ∇u | ≤ c | x − x( d) | ]^_ x → x(d) b d = a n beb d = b n , n = 1, , N . 1 1 Γ : ∗ *>hdeZ^hlf_q_g`xjbdZdh^bgbaemqrbogZih^k_dpbb
Ih^k_dpbyfZl_fZlbdb 99 < hij_^_e_gbb deZkkZ K nmgdpbb u(x) b ∇u(x) g_ij_ju\gh ijh^he`bfu gZ 1 1 jZaj_au Γ \ X ke_\ZbkijZ\Zghfh]mlbf_lvkdZqhdijbi_j_oh^_q_j_a Γ \ X . AZ^ZqZU. GZclbnmgdpbx u(x) badeZkkZK, dhlhjZym^h\e_l\hjy_lmjZ\g_gbx =_evf]hevpZ b]jZgbqgufmkeh\byf 2 1 ∆u ( x) + k u( x) = 0, x ∈ D \ Γ , k = const, Im k > 0 ∂u( x) + & = F ( s) , ∂n + x ( 1) x( s) ∈Γ ∂u( x) − & = F ( s) , u( x) 2 = F( s) . (1) x ( s) ∈ Γ ∂n − x ( 1) x( s) ∈Γ keb D —\g_rgyyh[eZklv^h[Z\bfmkeh\bygZ[_kdhg_qghklb u(x) ]^_ − 2 ( | | 1/ − 2 u = o x ), | ( ) | ( | | 1/ 2 ∇ x = o x ) | = x1 + − 0, λ 1 1, 2 L_hj_fZkeb F ( s), F ( s) ∈ C ( Γ ), F( s) ∈C λ ( Γ ) 2 u , | x + x → ∞ . lhdeZkkbq_kdh_j_r_gb_ aZ^ZqbU kms_kl\m_lb_^bgkl\_gghHgh\ujZ`Z_lkynhjfmehc i u [ µ ]( x) = w [ ]( x) w [ ]( ) d , (2) + − (1) ∫[ F ( s) −F ( s) ] H ( k | x − y( σ ) | ) 0 1 µ + 2 µ σ + σ 4 1 Γ Γ ( k | x − y( σ ) | ) (1) i ∂H 0 i w1[ µ ]( x) = ∫µ ( σ ) & dσ , w 2[ µ ]( x) = 4 2 ∂n ∫ µ ( σ ) V ( x, σ ) dσ , 4 1 ( k | x− y( ξ) | ) σ (1) ∂H0 V ( x, σ) = ∫ & dξ , ∈ 1 [ a ] 1 n , b n y 1 ∂n σ , a y n (1) p 1 0 2 ]^_ H 0 ( z) — nmgdpby OZgd_ey i_j\h]h jh^Z [1], µ ( s) ∈C1/ 2( Γ ) C ( Γ ) ( p = min{ 1/ 2, λ} ) — j_r_gb_ kbkl_fu bgl_]jZevguo mjZ\g_gbc ihemqZxsboky ijb ih^klZgh\d_\bm^h\e_l\hjyxs__mkeh\byf LZdh_j_r_gb_kms_kl\m_lb_^bgkl\_ggh 1 n b ∫ µ ( σ ) dσ = 0 , n = 1, N1 . 1 n a 1. Gbdbnhjh\ :N M\Zjh\ D BAMQGB>JCN:DHGLJ:KLGUOKLJMDLMJ
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Ih^k_dpbyfZl_fZlbdb 99<br />
< hij_^_e_gbb deZkkZ K nmgdpbb u(x)<br />
b ∇u(x)<br />
g_ij_ju\gh ijh^he`bfu gZ<br />
1<br />
1<br />
jZaj_au Γ \ X ke_\ZbkijZ\Zghfh]mlbf_lvkdZqhdijbi_j_oh^_q_j_a Γ \ X .<br />
AZ^ZqZU. GZclbnmgdpbx u(x)<br />
badeZkkZK, dhlhjZym^h\e_l\hjy_lmjZ\g_gbx<br />
=_evf]hevpZ<br />
b]jZgbqgufmkeh\byf<br />
2<br />
1<br />
∆u ( x)<br />
+ k u(<br />
x)<br />
= 0, x ∈ D \ Γ , k = const, Im k > 0<br />
∂u(<br />
x)<br />
+<br />
& = F ( s)<br />
,<br />
∂n<br />
+<br />
x<br />
(<br />
1)<br />
x(<br />
s)<br />
∈Γ<br />
∂u(<br />
x)<br />
−<br />
& = F ( s)<br />
, u( x)<br />
2 = F(<br />
s)<br />
. (1)<br />
x ( s)<br />
∈ Γ<br />
∂n<br />
−<br />
x<br />
(<br />
1)<br />
x(<br />
s)<br />
∈Γ<br />
keb D —\g_rgyyh[eZklv^h[Z\bfmkeh\bygZ[_kdhg_qghklb<br />
u(x)<br />
]^_<br />
− 2<br />
( | |<br />
1/<br />
− 2<br />
u = o x ), | ( ) | ( | |<br />
1/<br />
2<br />
∇ x = o x ) | = x1<br />
+ −<br />
0, λ 1<br />
1, 2<br />
L_hj_fZkeb F ( s),<br />
F ( s)<br />
∈ C ( Γ ), F( s)<br />
∈C<br />
λ ( Γ )<br />
2<br />
u , | x + x → ∞ .<br />
lhdeZkkbq_kdh_j_r_gb_<br />
aZ^ZqbU kms_kl\m_lb_^bgkl\_gghHgh\ujZ`Z_lkynhjfmehc<br />
i<br />
u [ µ ]( x)<br />
= w [ ]( x)<br />
w [ ]( )<br />
d , (2)<br />
+<br />
− (1)<br />
∫[ F ( s)<br />
−F<br />
( s)<br />
] H ( k | x − y(<br />
σ ) | )<br />
0<br />
1<br />
µ +<br />
2<br />
µ σ +<br />
σ<br />
4 1<br />
Γ<br />
Γ<br />
( k | x − y(<br />
σ ) | )<br />
(1)<br />
i ∂H<br />
0<br />
i<br />
w1[<br />
µ ]( x)<br />
= ∫µ<br />
( σ ) & dσ<br />
, w<br />
2[<br />
µ ]( x)<br />
=<br />
4 2<br />
∂n<br />
∫ µ ( σ ) V ( x,<br />
σ ) dσ<br />
,<br />
4 1<br />
( k | x−<br />
y(<br />
ξ)<br />
| )<br />
σ (1)<br />
∂H0<br />
V ( x,<br />
σ)<br />
= ∫ & dξ<br />
, ∈ 1<br />
[ a ]<br />
1 n<br />
, b n<br />
y<br />
1 ∂n<br />
σ ,<br />
a<br />
y<br />
n<br />
(1)<br />
p 1 0 2<br />
]^_ H<br />
0<br />
( z)<br />
— nmgdpby OZgd_ey i_j\h]h jh^Z [1], µ ( s)<br />
∈C1/<br />
2( Γ ) C<br />
( Γ )<br />
( p = min{ 1/ 2, λ}<br />
) — j_r_gb_ kbkl_fu bgl_]jZevguo mjZ\g_gbc ihemqZxsboky ijb<br />
ih^klZgh\d_\bm^h\e_l\hjyxs__mkeh\byf<br />
LZdh_j_r_gb_kms_kl\m_lb_^bgkl\_ggh<br />
1<br />
n<br />
b<br />
∫ µ ( σ ) dσ = 0 , n = 1,<br />
N1<br />
.<br />
1<br />
n<br />
a<br />
1. Gbdbnhjh\ :N M\Zjh\ D<br />
BAMQGB>JCN:DHGLJ:KLGUOKLJMDLMJ<br />
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