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p- hp hp 1.1 (a1(x, u)u) = f(x, u), x ;

(a2(x, u)u) = F (x, u), x ; (1) (x)n(a1u n) + (x)n (a2u n) = g1, x , (2) (A1(x, u) u) = f(x, u), x ;

(A2(x, u) u) + (A3(x, u) u) = F (x, u), x ; (3) (x)n((A1 u) n) + (A2 u)) + (x)n (A3 u) n) = g2, x . (4) (a1(x, u)u) + (A1(x, u) u) = f(x, u), x ;

(a2(x, u)u) + (A2(x, u) u) + (A3(x, u) u) = F (x, u), x ; (5) (x)n((a1u n) + (A1 u) n) + (A2 u)) + (x)n ((a2u + A3 u) n) = g3, x . (6) |aiu| lim = 0, i = 1, 2, (7) |x| |x| |Ai u| |A2 u| lim = 0, i = 1, 3, lim = 0, (8) |x| |x| |x| |x| (a1(x, u)u) = f(x), (a2(x, u)u) = F (x), x ;

(1(x)u) = f1(x), (1(x)u) = F1(x), x 1 (R3 \ ); (9) n (a1u - 1u) = 0, n (a2u - 1u) = 0, x ;

lim 1u = 0.

|x| 1 f L2() f1 L2(1) F (L2())3 F1 (L2(1))3 1 5,6 a1u, x , a2u, x , V = Z = U = V - Z.

1u, x 1, 1u, x 1, x Z = (x, V )V V = k(x, Z)Z f = 0 A V = A 1 1 RA A - U(A) dy = Af (x), x . (10) 4 r U(A) = (1 - (x, A)) A r = |x - y| x y Af(x) Af (L2())3 0 < < 1 = const H(rot, ) = {u : u (L2())3, u (L2())3} (u, v) = (u, v) + ( u, v), = const > 0 u v H(rot, ) (, ) (L2())3 (L2())3 H(rot, ) |U1 - U2| (1 - )|V1 - V2|, x . (11) R : H(rot, ) H(rot, ) A1, A2 H(rot, ) 2 (R(A1) - R(A2), A1 - A2) 1 A1 - A2, , 1 A1, A2 H(rot, ) R(A1) - R(A2) l1 A1 - A2, , l1 R (A)A+ A+ - QU, U (E - Z (V ))V+ Z (V ) = {Zi/Vj}i,j=1,2,3 V+ = A+ 1 J1() = {u : u (W2 ())3, u = 0, u n| = 0}, 1 u n| W2 () (u, v)J1 = ( u, v) u = u u, v J1() J1 J1() PJ1(R(A) - Af ) = 0, 1 PJ R : J1() J1() A1, A2 J1() 2 (PJ1(R(A1) - R(A2)), A1 - A2)J1 A1 - A2, J1 PJ1(R(A1) - R(A2)) (2 - ) A1 - A2.

J1 J1 PJ1R (A) : J1() J1() A A, A+ J1() 1 PJ R (A)A+ A+, J1 J1 1 PJ (R (A1) - R (A2)) A1 - A2, J1 J1 = const 1 PJ (R(Am) - Af ) = 0, m 1 Jm m J1() J1() u1,..., um m Am = iui.

i=1 J1() m 1 Am uidx + n ui (UJ n)dSydSx = (n ui) AfdS, m 4r i = 1, 2,..., m;

UJ (1 - PJ ) Am PJ Am m m m 1 Jm J() = {u : u (W2 ())3 u = 0}.

F = 0 Z = 1 1 T () (x) + U() dy = f(x), x . (12) 4 r U() = (k(x, ) - 1) f f L2() 1 < k k k = const 1 G1() G1() = {u : u = , W2 ()} A0 : G1() G1() 1 (A0u)(x) u dy.

4r A0 1 A0 2 1, k = const, k > 0, k = 1, 2.

A0 G1() (u, v) = (A-1u, v).

0 u u -1 u.

1 1 W2 () (, ) = (, ) + (, ), 1 2 = const > 0 , W2 () = (, ) U1 - U2 (k - 1) Z1 - Z2, (13) (U1 - U2, Z1 - Z2) 0, (14) Ui, Zi (L2())3, i = 1, 2 1 1, 2 W2 () 2 (T (1) - T (2), 1 - 2) 2 1 - 2, , 2 1 1, 2 W2 () T (1) - T (2) l2 1 - 2, , l2 1 1 T ()+ + + (V (Z) - 1)Z+ dy, 4 r Z+ = + V (Z) = {Vi/Zj}i,j=1,2,3 1 H1() W2 () dS = 0, dl = 0, 0 L0 0 L0 (, )H = (, ).

e 1 1 Hm H1 1,..., m H1 m m = bii i=1 PH (T (m) - f) = 0, (15) 1 em 1 PH H1 Hm 1 em 1 kmidx + ni (U(G )mn)dSydSx = infdS, 1 4r i = 1, 2,..., m, U(G )m (P(G )mk - 1)m P(G )mkm 1 1 1 1 PH T : H1 H1 PH T : Hm 1 e 1 em 1 Hm PH T : H1 H1 PH T :

1 e 1 em 1 1 Hm Hm V Z Z = F, V = f, V = kZ, x , (16) k k = k(x, V ) k = k(x, Z) 1 V Z V n Z n g(x) = - dSy + dSy + V, x 1, (17) 4r 4r n F1 f1 g(x) = dy - dy.

4r 4r 1 1 x V Z V n Z n 1 g1(x) = - dSy + dSy + (n(V n) + n (Z n)), x . (18) 4r 4r 2 1 f n (F n) g1(x) = g(x) - (nf + F n) + dSy + dSy, 2 4r 4r f = f - f1 F = F - F1 P P = 0, P = 0, x . (19) P n P n (x)P (x) = dSy + dSy, (20) r r (x) = 4 x (x) = 2 x P P n = Z n P n = V n (V - P ) n P n = Z n, (P + dS) n = g1(x) n, (21) 4r B + P (V - P ) n P n = Z n, ( + dS) n = g1(x) n, (22) 2 4r Z + P (Z - P ) n P n = V n, ( + dS) n = g1(x) n, (23) 2 4r (Z - P ) n P n = V n, (P + dS) n = g1(x) n, (24) 4r h-, p- hp- (p 2) h-, p- hp 1 Z() = {u (W2 ())3 : u = 0, u = 0};

U() = {u (L2())3 : u = 0, u (L2())3};

V () = {u (L2())3 : u L2(), u = 0}.

Z() uh m uh(x) = uhNi(x), (25) i i=1 uh i = 1,..., m {xi}i=1,...,m i Ni(x) i = 1,..., m Ni(x) Ni = 0 m Ni(x) = a(i)fg(j)(x) a(i) g(j) j j j=1 j = 1,..., m fg(j) k k pn+1,k+1(x) = dnk(r/r0)n cos(k)Pn (cos), qn+1,k+1(x) = dnk(r/r0)n sin(k)Pn (cos) (26) x = (r, , ) dnk = (2n + 1)(n - k)!/(n + k)! r0 m a(i)fg(j)(xl) = (xi, xl), l = 1,..., m.

j j=1 g(j) Ni(x) fg(j) j = 1,..., m m Ni(x) = b(i)fj(x) fj j j=1 m b(i) fjfldS = LifldS, l = 1, 2,..., m, j j=1 Li Ni(x) Li i = 1,..., m Z() V () U() u Wj(k) 3 m 3 m u(x) = ik( uk,jNj(x)) = uk,jWj(k)(x), (27) k=1 j=1 k=1 j=1 uk,j {xj}j=1,...,m 3m (k,i) Wi(k)(x)= a(k,i)fg(j)(x) aj fg(j) j j=1 g(j) 3m 3m 3m fg(j) fg(j) fg(j) a(k,i) (xl) = il, a(k,i) (xl) = 0, a(k,i) (xl) = 0, (28) j j j xk xk1 xk2 j=1 j=1 j=1 xl , k = k1, k = k2, k1 = k2, 1 k, k1, k2 3, l = 1, 2,..., m.

g a(k,i) j 3m Wi(k)(x) = b(k,i)fg(j)(x) j j=1 b(k,i) j 3m fg(l) (k,i) bj fg(j) fg(l)dS = Li dS, l = 1, 2,..., 3m. (29) xk j=1 g(j) = j + 1 j = 1, 2,..., 3m f1 = 1 Li (1 i m) U() V () V () u uk,j {xj}j=1,...,m Wi(k)(x) = 3m a(k,i)hg(j)(x), a(k,i) g(j) hg(j) j j j=1 j = 1,..., m {1; x1; x2; x3; x1x2; x1x3; x2x3; x1x2x3; x2, x2, x2,...}. (30) 1 2 3 3m Wi(k)(x) = b(k,i)hg(j)(x), j j=1 b(k,i) j Wi(k)(x) a(k,i) b(k,i) j j U() Wi(k) V () i = 1,..., m k = 1, 2, 3 m d S = {xi 0, i = 1, 2, 3, x1 + x2 + x3 1} T = {x1, x2 0, x1 + x2 1; -1 x3 1} [-1, 1]3 S T m m d d (1) N 0.1 10-4 (1v) = f, v = F, x ;

(31) v = 0, x .

w = f, (2w) = F, x ;

(32) w = 0, x .

1, 2 1 = 1(x, v) f L2() F (L2()3 F = 0 1 W2,0()3 1 W2,0()3 = {u (L2())3 : u/xk (L2())3, k = 1, 2, 3; u| = 0}.

3 u 2 = (u/xk, u/xk)(L ())3; u 2 = u 2 + u 2.

1 1 W2,0()3 2 W2,0()3 (L2())3 (L2())3 k=1 F1(v) (1/2) [( (1v) - f)2 + ( v - F )2]d.

c1 v (1v) c2 v, (33) L2() L2() L2() 1 (1(x, v1)v1) - (1(x, v2)v2) c3 v1 - v2 W2,0()3, (34) L2() 1 ci > 0 ci = const i = 1, 2, 3 v, v1, v2 W2,0()3 1 v+ = tv2 + (1 - t)v1 t (0, 1) v1, v2 W2,0()3 1 1(x, v+)v+ = t1(x, v2)v2 + (1 - t)1(x, v1)v1, 1 f L2() F (L2())3 F1 W2,0()3 (u (1, v, u)u) c4 u 1, L2() W2,0()3 1 u (1, v, u) {(1(x, v)vi)/uj}i,j=1,2,3 v, u W2,0()3 c4 = const F1 (A1v, u)(L ())3 = [( (1(x, v)v) - f) (u (1, v, u)u) + ( v - F ) u]d, 2 1 1 W2,0()3 (W2,0()3) 1 (A1v, u) = 0, u W2,0()3, 1 v W2,0()3 n 1 1 un (W2,0)n()3 un (W2,0)n()3 F = 0 V () 2 = 2(x, w) f L2() F 1 (L2())3 F = 0 W2,0()3 F2(v) (1/2) [( (w) - f)2 + ( (2w) - F )2]d.

c5 w (L2())3 (2w) (L2())3 c6 w (L2())3, (35) (2(x, w1)w1) - (2(x, w2)w2) (L2())3 c7 w1 - w2 1, (36) W2,0()3 1 ci > 0 ci = const i = 5, 6, 7 w, w1, w2 W2,0()3 1 w+ = tw2 + (1 - t)w1 t (0, 1) w1, w2 W2,0()3 2 2(x, w+)w+ = t2(x, w2)w2 + (1 - t)2(x, w1)w1, 1 f L2() F (L2())3 F2 W2,0()3 F2 (v (2, w, u)u) c8 u 1, L2() W2,0()3 1 v (2, w, u) {(2(x, w)wi)/uj}i,j=1,2,3 w, u W2,0()3 c8 = const F2 (A2w, u)(L ())3 = [( w - f) u + ( (2(x, w)w) - F ) (v (2, w, u)u)]d, 2 1 1 W2,0()3 (W2,0()3) 1 (A2w, u) = 0, u W2,0()3, 1 w W2,0()3 n 1 1 un (W2,0)n()3 wn (W2,0)n()3 f = 0 U() V () V = 0, Z = 0, V = pZ, x ;

V = 0, Z = F (x), V = Z, x ; (37) [n V ] = 0, [n Z] = 0, x ; V = Z = 0, x 0.

0 p p = p(x, V ) p = p(x, Z) [n V ] [n Z] x V () R3 1 1 = u, u1 3 (ai(x, u, u)) + a(x, u, u) = 0, x ;

xi i=1 3 u1 3 u1 (ij(x) ) + bi = f, x 1; (38) xi xj i=1 xi i,j=1 3 3 u1 u - u1 = 0, (ai(x, u, u) - ij(x) - biu1) i = 0, x ;

xj i=1 j=1 3 u1 3 u = u(x), x \ ; ( ij(x) + biu1) i = u1(x), x 1 \ , xj i=1 i,j=1 u u i, i f, u, u1 1 G Gi (i = 1, 2, 3) 3 3 (ij(x) Gj) + bi Gi = f, x 1, xi i,j=1 i=1 = G 1 1 u 1 = u1 - 1 1 1 G f 1 u, 1 1 W2 () = {u : u L2(), u (L2())2}, s t n u = su/s + tu/t.

1 1 W2 () W2,0() udS = 0 udl = 0, 0 l0 0 l0 1 u, v W2,0() (u, v)W 1 = ( u, v)(L ())2, u 1 = ((u, u)W 1 )1/2.

() W2,0() () 2 2,0 2,0 1 W2 () 1 vdS + (G, ||)( - G ) vdS = G vdS, v W2,0(), 1 2 1 1 2 G K = { : L2(), (L2())3)} |Gn| (G, ||) = 2 -.

|G - | B1, B2 1 1 (B1u1, v1)W (1) = u1 v1dS, u1, v1 W2,0(1);

2,0 1 1 (B2u2, v2)W 1 = (G, | u2|)(u2 - G) v2dS, u2, v2 W2,0(2).

(2) 2,0 2 1 1 1 1 B1 : W2,0(1) W2,0(1) B2 : W2,0(2) W2,0(2) B2 1 1 B2 : W2,0(2) W2,0(2) 1 1 (B2u - B2v, u - v)W (2) u - v 2, u, v W2,0(2);

1 W2,0(2) 2,0 1 B2u - B2v 1 7 u - v 1, u, v W2,0(2), W2,0(2) W2,0(2) 7 = 2 + max(1/|Gn|) G K x2 B : H H H = 1 1 1 W2,0() {W2 (1) W2 (2)} (Bu, v)H = (B1u1, v1)W 1 + (B2u2, v2)W 1 (1) (2) 2,0 2,0 u = (u1, u2) v = (v1, v2) H S k, k = 1, 2,..., m S = HS(x), x k, k = 1, 2,..., m, (39) S(yk) = S yk k S(y0) y0 k k, k = 1, 2,..., m yk, k = 1, 2,..., m y0 Lk, k = 1, 2,..., m S S v k v d = ( HS + ) d, k = 1, 2,..., m, Lk Lk v k y0 (x) Lk S vdS = (HS + k) vdS, k = 1, 2,..., m, (40) k k v k v k hp R3 wi i = 1, 2,..., N i Ni,k(x) Ni,k(x(j)) = (x(k), x(j)) x(k), x(j) wi k k k = suppNi,k (C0 (k))3 wi:x(k)wi k v (C0 (k))3 | v(x) - v(y)| c1d0, | v(x) - v(y)| c2d0, x, y k d0 = max |x - z| c1, c2 xk,zk x y x k v(x) = ( v(x))I0(x) + ( v(x)) I0(x) + R0v, -1 r I0(x) = dy, |I0(x)| d0, |R0v| c3d2, 0 4 r3 k 3 c3 = max{1, c2} r = ip(xp - yp) r = |x - y| c p=1 v (C(wk))3 wk v (C(k))3 | v(x) - v(y)| c4d, | v(x) - v(y)| c5d, x, y k d = max |x - z| c4, c5 xk,zk x y x k v(x) = ( v(x))I(x) + ( v(x)) I(x) + P v + Rv, 3 -1 r I(x) = dy, |I(x)| d, |P v| max ( |vp|)(y), |Rv| c6d2, yk 4 r3 p=1 k c6 = max{4, c5} c wi 1 i N (I(l)B, w) = (| I(l)Bdw| + | I(l)Bdw|)/(|w|B0), w w w |w| w B0 I(l)B ml I(l) I(l)B = BiNi(l)(x) l e e i=1 ml Bi B i Ni(l)(x) i (l) (Ie B, w) C hl+1 |Dl+1B(x)|, x w, 1 2 3 C = const, C > 0 h w Dm = |m|/m x1m x2m x3 |m| = m1 + m2 + m3 J B (I(k)B, w0) 1 k 4 p K() 0 c

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