林娟,謝碧華

(1.福建商業(yè)高等專科學(xué)?；A(chǔ)部,福建福州 350012; 2.福建師范大學(xué)數(shù)學(xué)與計(jì)算機(jī)科學(xué)學(xué)院,福建福州 350007)

雙解析函數(shù)的一般復(fù)合邊值問題關(guān)于邊界曲線的穩(wěn)定性

林娟1,謝碧華2

(1.福建商業(yè)高等?？茖W(xué)?；A(chǔ)部,福建福州 350012; 2.福建師范大學(xué)數(shù)學(xué)與計(jì)算機(jī)科學(xué)學(xué)院,福建福州 350007)

開口弧段Γ上的雙解析函數(shù)的Riemann邊值問題與單位圓周L上雙解析函數(shù)的Hilbert邊值問題復(fù)合而成的一般復(fù)合邊值問題,當(dāng)L與Γ發(fā)生微小的光滑攝動(dòng)后,借助于推廣的拉甫倫捷夫近似于圓的共形映射,將星形域映為單位圓域,從而得出攝動(dòng)后的問題的解的表達(dá)式,同時(shí)討論了解的穩(wěn)定性情況,并給出誤差估計(jì).

雙解析函數(shù);復(fù)合邊值問題;光滑攝動(dòng);共形映射;穩(wěn)定性

1 引言

解析函數(shù)邊值問題不僅在理論上有著重要意義,而且在實(shí)際應(yīng)用中也有著重要意義,自以Muskhelishvili為首的前蘇聯(lián)學(xué)派在這一領(lǐng)域做出大量杰出的開創(chuàng)性工作以來,得到了許多重要結(jié)果,國(guó)內(nèi)尤為突出的是文[1],不少研究者還把經(jīng)典的解析函數(shù)邊值問題理論向各種函數(shù)類上推廣[24].當(dāng)邊界曲線發(fā)生微小的光滑攝動(dòng),解析函數(shù)的邊值問題的解的穩(wěn)定性問題,近年來得到許多學(xué)者關(guān)注.文[5-8]也研究了相關(guān)攝動(dòng)穩(wěn)定性的問題,文[9]研究了雙解析函數(shù)一般復(fù)合邊值問題(簡(jiǎn)稱為B-RH問題)的解的情況,本文將討論當(dāng)G1(z)=G2(z)時(shí),它的解關(guān)于邊界攝動(dòng)的穩(wěn)定性.

設(shè)L是復(fù)平面C中的單位圓周,D為L(zhǎng)所界定的單位圓域,Γ=?ab是一條開口的光滑弧段,且Γ?D,記C2(L+Γ)為L(zhǎng)+Γ上具有二階連續(xù)導(dǎo)數(shù)的函數(shù)類,定義

其中ω∈C2(L+Γ).對(duì)于充分小的ρ＞0,記B(ρ)={ω∈C2(L+Γ):‖ω‖2＜ρ}.L經(jīng)過擾動(dòng)ω∈B(ρ)后得到曲線

設(shè)曲線Lω是Lyapunov曲線,它所界定的區(qū)域關(guān)于原點(diǎn)是星形域Dω(近似于單位圓盤).Γ經(jīng)過擾動(dòng)ω∈B(ρ)后得到曲線

記E1為從a沿Γ至b的左側(cè)與從a沿Γω至b的右側(cè)所形成的區(qū)域,E2為從a沿Γ至b的右側(cè)與從a沿Γω至b的左側(cè)所形成的區(qū)域,D(D)為L(zhǎng)ω所圍成的內(nèi)(外)部區(qū)域.

2 B-RH問題的提出

3 B-RH問題的解的穩(wěn)定性

3.1 當(dāng)K1≥0且K2≥0時(shí),B-RH問題的解的穩(wěn)定性

3.2 當(dāng)K1＜0且K2＜0時(shí),B-RH問題的解的穩(wěn)定性

3.3 當(dāng)K1≥0且K2＜0或K1＜0且K2≥0時(shí),B-RH問題的解的穩(wěn)定性

討論同3.1,3.2.

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[3]趙楨.雙解析函數(shù)與復(fù)調(diào)和函數(shù)以及它們的基本邊值問題[J].北京師范大學(xué)學(xué)報(bào):自然科學(xué)版,1997,31(2):175-179.

[4]李玉成,蘭文華.廣義解析函數(shù)的帶位移的非線性邊值問題[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2008,24(1):34-40.

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The stability of the general compound boundary value problem for bianalytic functions about boundary curve

LIN Juan1,XIE Bi-hua2

(1.Department of Foundation,Fujian Commercial College,Fuzhou350012,China;
2.College of Mathematics and Computer Science,Fujian Normal University,Fuzhou350007,China)

For the general compound boundary value problems combining Riemann boundary value problem for bianalytic functions on an open arc Γ and Hilbert boundary value problem for bianalytic functions on a unit circle L,when smooth perturbation happens for Γ and L,by extending Lavrentjev’s conformal mapping on a region approximating to a unit disc from a star-like domain onto a unit disc,the authors show the solutions of the perturbed problem.They also discuss the stability of the solutions and give error estimates.

bianalytic functions,compound boundary value problem,smooth perturbation,conformal mapping,stability

O175.8

A

1008-5513(2009)04-0816-06

2008-09-14.

福建省自然科學(xué)基金(2008J0187),福建省教育廳科技項(xiàng)目(JA08255).

林娟(1965-),碩士,研究方向：邊值問題與積分方程.

2000MSC:30E20,30E25