37 5 Ó Ä Ä Vol. 37 No. 5 014 9 ACTA MATHEMATICAE APPLICATAE SINICA Sep., 014 É Ì - Î Dirichle ÓÆ ÞÝÜ ÎÞÈÅÔÅ ÅÅ 100048 E-mail: wyin@mail.cnu.edu.cn Ñ - ƱРÑĐ» ³Æ Ð Û Ò ÌĐ Ø ÕÃ Ý Caran-Harogs ÚÆ - ƱРDirichle Ò Ì Ýß µ º Ï ÚÇĐ» Æ Ð Ò Ì ºÀ Ï Ð Ö³ Ð Dirichle Ì µå ÛÑ Û Ê Ò Ì ÀÄ - Æ±Ð Ò Ì Dirichle Caran-Harogs Ú aehler-einsein Ñ ÚÇĐ» Æ Ð MR 000 Õ È 3C17; 3F07; 35G30; 53C55 Ô ÈÁ 0174.55 1 Ð Õ ÍÞ Õ ÍÝ Ö Ø ÐÆß È Ø ÝÜ È Ù Ø ÐƳÕÆÐÆ Â Ö Õ ØÙ Ø Â ÑÖ ÅÜ À ³ Ý ¾Ð» Í Ü - ² - ² È - ²Ø 5 Í Ø Ú ¾Ó ¾ Í À Ú ¼¾ ß aehler Ç aehler-einsein Ó Ä Ð º Ö ß - ² Monge-Ampère ÒÍ È - ² ½Î Í ³ Èß Ç 80 Ê ¹ Æ ÕÕ C n ÓÎ Ü D Ä «aehler-einsein 01 Đ 10 01 5 Đ 11 Å ¹ Å Ï 11071171,1117185
5 ÅÏÝ - ƱРDirichle Ò ÌÝß 787 Ó [1,]. ÇÚ À - ² Dirichle Ø Á ³ Õ Í g Ä ³ g de = e n+1g, z D, g =, z D, Ø Á ³ ÕÍ g Í Ü D À «aehler-einsein Ó E D z := g dz i dz j. Ø Á ³ Õ g Ø Á ³ Õ g Í ½À g > 0. ÑÖ - ² «¼» Í Ù Ç Đ Î ½ Í Ä Ú Â Í Æ Ü ÖÄ Ù Â È Ð Í Æ ÉØÕ º Ü ÖÖ - ² Dirichle Í ÖÕ ÍÝ Ä Ø Ú ÍÎ ¾¼ Ó Ö - ² Õ ÍÝ Ö Åß Caran-Harogs Ü - ² Dirichle Õ Í Ý Þ º Þ Õ Ý Ë Ù Åß Caran-Harogs Ü - ² Dirichle µ» ß ¼ ÛÈ Û Ù Õ ÍÝ ÓÅ È Ö ÃÕÆ ÙÉ Æ Ù Õ ÍÝÞ Ù ØÇÎÉÉÁ Ø Ï Dirichle Í Æ ÜÒ Ë Õ ÍÝ Ì Åß Caran-Harogs Ü Y III = Y III N 3, q; := { W C N3, Z R III q : W < dei ZZ, > 0 },» R III q ¹ È Åß Ü R III q = { Z C qq 1 : I ZZ > 0 }, Ð Z q È Ö Æ Z > 0 Æ Z Æ Z, Z Æ Z ³µ de  Y III Bergman ² Õ Æ Ò ² Õ Bergman Ê Ù Y III ÓÎ Ü Ù Ñ 1 ÓÍ Þ À - ² Dirichle Õ Í g de z i z = j 1 i, j N en+1g, z Y III, g =, z Y III, 1
788 Ò Ã Ã 37» N = N 3 + qq 1 Ü Y III Õ Ù z = Z 1, Z = z 1, z,,z qq 1, z qq 1 +1, z qq 1 +,...,z N, Z 1 Z = z ij Ý z ij»çåã Ý ÁÁ Ó qq 1 Ý Z 1 = z 1, z,,z qq 1 = z 1,, z 1q, z 3,,z q,,z q 1,q, Z W Ý w 1, w,, w N3. «Z = z qq 1 +1, z qq 1 +,..., z N = w 1, w,...,w N3, Ì Y III Z, W Ò Ó N Ý z = Z 1, Z. ÇÑ Dirichle» ÛÈ Û Ì 1 Õ Íµ» ÛÈ Û Õ Í Ù ÓÅ È [3] ;»Á 1 Í Æ Ò.1 ³½ Y III N 3, q, º ³½ µ Au Y III : { W = UWdeI Z 0 Z 0 1 Z = AZ Z 0 I Z 0 Z 1 A 1, dei ZZ 0 1» A A = I Z 0 Z 0 1, Z 0 R III q, U Ò Æ ³½ Z 0, W 0, W. ß Z = AZ Z 0 I Z 0 Z 1 A 1 R III q º Ø ½ [4] I Z Z = A 1 I ZZ 0 1 I ZZ I Z 0 Z 1 A 1. dei Z Z = dei Z 0 Z 0 de I ZZ0 dei ZZ. Ù W W = WW dei Z 0 Z 0 1/ dei ZZ 0 /. Ó de I Z Z W = dei Z 0 Z 0 dei ZZ0 [ dei ZZ W ]. Çѳ½ Y III N 3, q; º.. «Le X = XZ, W = W [ dei ZZ ] 1/, X Au Y III º³ Ó XZ, W = XZ, W. ß Ò Æ ØÙ ÎÒ½ [4]. Ù X ³ Õ FX Ð Au Y III º³.3.  gz, z = g[z, W, Z, W] 1 Í ds = g dz i dz j
5 ÅÏÝ - ƱРDirichle Ò ÌÝß 789 Ü Y III aehler-einsein Ó ÑÖ Ó º³Ó º³ ³½ Z, W = FZ, W = Fz Ó g[z 0, W, Z 0, W] de = dej F de g[0, W, 0, W ] w i w j, 3» w z Fz Î N w = w 1,, w N ; J F ³½ F Jacobian Æ J F = Z Z 0 W W dej F Z 0=Z = de W / W de Z / Z Z 0=Z. Ü Æ [4], Ó de Z / Z Z 0=Z = dei ZZ q 1. Ù de W / W Z0=Z = dei ZZ N3/. Ó Ì Î Z = Z 0 Ï 3 ³ dej F Z 0=Z = dei ZZ q 1+N3/. gz, z de = dei ZZ q 1+N3/ de. g[0, W, 0, W ] w i w j. 4 Ç Í Ø g[0, W, 0, W de ] wi. 5 w j  g 1 Í 4 ÔÔ Î Ö e N+1g, dei ZZ q 1+N3/ g[0, W, 0, W de ] wi = e N+1g. 6 w j  6 Ö g = N + 1 1 log [ GXdeI ZZ q 1+N3/], 7 g[0, W, 0, W de ] wi = GX. 8 w j Ñ Ò Â 8 Ô X, GX»ÏÕ 1 - ² Ö Ç Ô ¾ É 3 ÂËÏ - Í
790 Ò Ã Ã 37 ± Ø ¹ 8 Ô W, w «W, z Ì Í Ø g[0, W, 0, W] de. 9 Ù Ñ 7 Ò g[0, W, 0, W] = g[z, W, Z, W] Z=0. g = N + 1 1 log [ GXdeI ZZ q 1+N3/]. 9 ³» Cde log[gxdei ZZ q 1+N 3 / ] log[gxdei ZZ q 1+N 3/ ] z αβ z στ z αβ w j log[gxdei ZZ q 1+N 3 / ] log[gxdei ZZ q 1+N 3/ ] w i z στ w i w j C = N + 1 N, 1 α < β q, 1 σ < τ q, 1 i, j N 3. Z=0, 10 Ê Ø log GX = M, d log GX dx = M, d log GX dx = M. 11 X z αβ Z=0 = 0, X z στ Z=0 = 0, X z αβ z στ Z=0 = 1 Xr[I αβi τσ ], X z αβ w q Z=0 = X w p z στ Z=0 = 0, X w i Z=0 = w i, X w i w j Z=0 = δ ij, log dei ZZ z αβ z στ Z=0 = ri αβ I τσ = δ ασ δ βτ, log dei ZZ z αβ w X w j Z=0 = w j, = log dei ZZ Z=0 = log dei ZZ Z=0 = 0, Z=0 w i z στ w i w j» I αβ q q Æ» α β Â Ý 1, β α Â Ý
5 ÅÏÝ - ƱРDirichle Ò ÌÝß 791 1,» Ý Ù log[gxdei ZZ q 1+N3/ ] z αβ z στ = 1 M X + q 1 + N 3 r I αβ I τσ, log[gxdei ZZ q 1+N3/ ] z αβ w j Z=0 Z=0 = log[gxdei ZZ q 1+N3/ ] w i z στ Z=0 = 0, log[gxdei ZZ q 1+N3/ ] w i w j =M w i w j + M δ ij. Ù 10 ³ [ 1 C de M X + q 1 + N ] 3 I 0 0 M I + M W. 1 W ¾ Þܳ ÑÖ M, M, M Ð º³ Õ ÙÇ ³ [ 1 C de M X + q 1 + N ] 3 I 0 0 M I + M W. 13 W Ç Ö C[ M X + ÑÖÖ α, ¼Ó Ç Ö C[ M X + Ù W W = X, Ù Ç Ö C[ M X + q 1 + N 3 ] qq 1 de[i + α α] = 1 + αα. q 1 + N 3 q 1 + N 3 Z=0 de[m I + M W W ]. ] qq 1 M N3 [ 1 + M M W W ]. ] qq 1 M N3 [ 1 + M M X ]. ² 11, Ù Ç 8 Ô 8» N + 1 N[ X dg dx + [ G dg dx + G d G dg dx dx q 1 + N 3 X ] qq 1 G ] dg dx N3 1 = G N+.
79 Ò Ã Ã 37 1 - ² ÖÇÑ 1 Ö ÑÛÈ Û [ N + 1 N qq 1 X G + G0 = q 1 + N 3 G qq 1 ; lim X 1 GX =. 1 Ø Á ³ ÕÍ» G = GX ½ Ù 14 Í ] qq 1 g = N + 1 1 log [ GXdeI ZZ q 1+N3/], GX > 0 4 ËÏ - Í Dirichle Å Ò 14 Í «[GG + GG G X] G N3 1 G N+1 = G, 14 G = GX = A1 X N+1, 15 ÜÊÁ 14 Õ A. Ø G = AN + 11 X N+, G X = AN + 11 X N+ AN + 11 X N+1, 1 G X = A N + 11 X N+ A N + 11 X N+1, «Ó = qq 1 + q 1 = q + 1 q 1, [ 1 G X + q 1 + N ] qq 1 [ qq 1 3 N + 1 G = A1 X N+], 16 G N3 1 G N+1 = N + 1 N3 1 A N3 N 1 X N+1 N+N 3 1, 17 Ø GG = N + 1A 1 X N+3, GG = N + 1N + A 1 X N+4, G = N + 1 A 1 X N+4, GG G = N + 1A 1 X N+4,
5 ÅÏÝ - ƱРDirichle Ò ÌÝß 793 Ó GG G X = N + 1A 1 X N+4 N + 1A 1 X N+3, GG + GG G X = N + 1A 1 X N+4. 18 15, 16, 17, 18 Ë ÊÁ 14 qq 1 4q 1 qq 1 1 X N+1 = q 1 X N+1 = A1 X N+1. q + A = 4q 1 qq 1 q. q + Ï GX ½ 14 Î Ù Ó [ 4q 1 qq 1 lim GX = lim X 1 X 1 q 1 X N+1] =. q + G = GX = 14 Í 19, = qq 1+ q 1 Ç Î 4q 1 qq 1 q 1 X N+1 19 q + ÊÁ 7 [ g = log 1 X 1 dei ZZ 1/ qq 1 ] N+1. 0 g = log [1 X 1 dei ZZ q 1 q q+ 4q 1 qq 1 ] N+1 q. 0 q + Ü 1 Ø Á ³ ÕÍ Ü ½ 1 - ² Ù g z i z j > 0. ÜÎ ½ Î Í 1 Ò Ê 1  GX 14 Í Ù G z i z j > 0, g = N + 1 1 log[gxdei ZZ q 1+N3/ ] 1 Ø Á ³ ÕÍ ß ÇÑ g ½ 1 - ² Ù Ñ G z i z j > 0 Ò Å g z i z j > 0. g Î ½ 1 Î Â Z, W Y III, W 0 Ï Î Z, W Y III Z, W Z, W Ï Ó X 1 1, 1 X +, Ù Ï dei ZZ W > 0. Ó g +, Z, W Y III.  Z, W Y III, Ù W= 0 Ï Î Z, W Y III Z, w Z, 0 Ï Ó 1 1 X > 1, dei ZZ 0, dei ZZ 1 +, ÎÓ g +, Z, W Y III.
794 Ò Ã Ã 37 g Î ½ 1 Î 1 Ì Ëµ 19, 0, 0 ÀÑ Ñ Đ Ê 1 Î = qq 1+ q 1 g = log [1 X 1 dei ZZ q 1 q q+ 4q 1 qq 1 ] N+1 q 0 q + ÂÇÑÛ Ò Î = qq 1+ q 1 g; Î º Ö qq 1+ q 1 Ï Ø Á ³ Õ Í Ï 1 Í Ï 1 ÍÙ 7 g ǽ» GX ËÍ Ï g 1 Í ÑÖ g GX 14 Í Ù 14 ÛÈ Û»Õ Í Ó È [3], Ì 1 Õ Í» 14 ÛÈ Û Õ Í Öà 1 14 lim GX = ÖÕ ÍÙÉ Í Ò X 1 Ñ ÝÓ Ñ 8 Ò GX Ù FX = 1 GX «Ì 14 ³ À { C [ XF + q 1 + N 3 F] qq 1 [ FF FF X + F X] F N3 1 = F N, qq 1 F0 = ; lim FX = 0. X 1» C = N + 1 N qq 1. Ì Ò Õ ÝÖ 1. Öà [5] Æ Î N 3 = 1 Ï 1 Í È N 3 Ï 1 Í Ù Ù ¹ κ Öà 3 Ð Caran-Harogs Ü ¹ Ü Ü Ó [6,7]. Ü Ó Ê¹ Ü W, Z, ¼Ä Ü º W, Z ³ W, 0. ³± Ò ¹ Ü ¹ Ü ÊÜ H C M+N Þ ºÒÐÓÎÌ Ü H W, Z,» W C M, Z C N, Ù W, Z H, W, 0 H. ÂÖ H W 0, Z 0, H º Au H Ä Ý FW, Z FW 0, Z 0 = W0, 0, Í W 0, 0 º H ¹ Ü ²» W0, 0 ¼ º³ H ¼ Ü ¼ Ü Ó± È º ¹ Æß ¹ ÜÎ µ¼ Ü H ̵¼ Ü H Ë ¼ Ü ÙÉ ¼ µõó Ø ÓÏÓ Â Ø º M = 0 Ï ¹ Ü ³ Óμ Ü Ù¹ ÜÒ ¼ Ü Å 1
5 ÅÏÝ - ƱРDirichle Ò ÌÝß 795 Öà 4 Ì Đ ² Ó Ö Monge-Ampere Õ È [8], ¹ Ö Ù [8] Õ Ý Å Ö Í º ÑÀ Ù Æ Monge-Ampere Dirichle» X ̳ ÛÈ ¼ Î Monge-Ampere Dirichle Õ Ý» ÛÈ ¼ Õ Ý Ó È [3], ٠ͽ [3] Ùº Õ Í Ü ÖÀ Å X Monge-Ampere Ò» Ù º Å ÍÎ Ò»È [8] Í ÓºÝ Galerkin Ý ¼Ëµ»Ú ³ ÖÙ Æ Monge-Ampere Õ Í Ó» Ù Æ Ö Monge-Ampere Ç Ó Ù Æ Monge-Ampere Î º Ç [1] Cheng S Y, Yau S T. On he exisence of a complee ähler meric on non-compac complex manifolds and he regulariy of Fefferman s equaion. Comm. Pure App. Mah., 1980, 33: 507?? [] Mok N, Yau S T. Compleeness of he ähler-einsein meric on bounded domain and he characerizaion of domain of holomorphy by curvaure condiions. Proc. Symposia Pure Mah., 1983, 39: 41?? [3] eller H B. Numberical Soluion of Two-Poin Boundary Value Problems. Philadelphia: Sociey for Indusrial and Applied Mahemaics, 1976 [4] ÄÞ Caran Û Bergman ± Ô ÙÚØÛ, 000, 95: 45 434 Yin Weiping. The Bergman kernel funcions on Caran-Harogs domains of he hird ype. Advances in Mahemaics, 000,95: 45-434 [5] ÄÞ Caran Û Einsein-aehler Ò ÙÚØÛ, 004, 33: 15 8 Wang An, Yin Weiping, Zhanng Wenjuan. The Einsein-aehler merics on Caran-Harogs domains of he hird ype. Advances in Mahemaics, 004, 33: 15 8 [6] ÛǻРÙÚØÛ, 007, 36: 19 15 Yin Weiping. A survey of research on Hua domains. Advances in Mahemaics, 007, 36: 19 15 [7] «² Û ÀØÇ Ü Õ ² Ç ØÜ ½, «Å É 009 4 Đ 116 149 Yin Weiping, Zhao Xiaoxia, Zhang Wenjuan. Consracions and Researches on Hua domains. Several Complex Variables in China on Is Research and Developmens Edied by Lu Qikeng and Yin Weiping, Beijing: Science Press, 009, April, 116 149 [8] Feng Xiaobing, Neilan, Michael. Analysis of Galerkin mehods for he fully nonlinear Monge-Ampère equaion. J. Sci. Compu., 011, 473: 303 37
796 Ò Ã Ã 37 Research on Numerical Soluion of Dirichle Problem of Complex Monge-ampére Equaion YIN Weiping School of Mahemaical Science, Capial Normal Universiy, Beijing 100048, China 100048 E-mail: wyin@mail.cnu.edu.cn Absrac Complex Monge-Ampère equaion is a nonlinear equaion wih high degree, herefore o ge is numerical soluion is very difficul. This paper sudies he numerical soluion of Dirichle problem of complex Monge-Ampère equaion on Caran-Harogs domain of he hird ype. Firsly, his problem is reduced o he numberical soluion of wo-poin boundary value problem of a nonlinear second-order ordinary differenial equaion. Secondly, he soluion of he above Dirichle s problem is given in explici formula under he special case, his explici formula can be used o check above numerical soluion. ey words complex Monge-Ampère equaion; numerical soluion; Dirichle s problem; Caran-Harogs domain; aehler-einsein meric; wo-poin boundary balue problems MR000 Subjec Classificaion O1.7 Chinese Library Classificaion 3C17; 3F07; 35G30; 53C55