一類具抽象邊界條件的遷移算子的譜分布

吳紅星,王勝華,袁鄧彬

(上饒師范學(xué)院數(shù)學(xué)與計(jì)算機(jī)科學(xué)學(xué)院,江西上饒 334001)

一類具抽象邊界條件的遷移算子的譜分布

吳紅星,王勝華,袁鄧彬

(上饒師范學(xué)院數(shù)學(xué)與計(jì)算機(jī)科學(xué)學(xué)院,江西上饒 334001)

利用線性算子半群理論,研究了板幾何中具抽象邊界條件的各向異性、連續(xù)能量、非均勻介質(zhì)的遷移方程.在假設(shè)邊界算子H部分光滑和擾動(dòng)算子K正則的條件下,采用豫解方法,得到了該遷移算子A的譜在區(qū)域Γ中由至多可數(shù)個(gè)具有限代數(shù)重?cái)?shù)的離散本征值組成等結(jié)果.

遷移算子;抽象邊界條件;非均勻;部分光滑算子;離散本征值

DO I：10.3969/j.issn.1008-5513.2013.05.008

1 引言

自文獻(xiàn)[1]對(duì)無(wú)限平行板幾何中的遷移方程研究工作以來,遷移方程解的漸近性態(tài)和該遷移算子的譜分析研究已成為數(shù)學(xué)、物理、生物和社會(huì)科學(xué)等領(lǐng)域都非常感興趣的課題[2-8].文獻(xiàn)[3]對(duì)Lp空間板幾何中具反射邊界條件非均勻介質(zhì)的遷移方程進(jìn)行研究,得到了其相應(yīng)遷移算子產(chǎn)生C0群V(t)(t≥0)和其Dyson-Phillips二階余項(xiàng)的緊性.文獻(xiàn)[4]對(duì)Lp(1≤p＜+∞)空間板幾何中具抽象邊界條件各向異性、粒子單量、均勻介質(zhì)的遷移方程進(jìn)行研究,討論了譜的存在性和抽象的Cauchy問題.那么,對(duì)抽象條件中具非均勻介質(zhì)的結(jié)果又將如何?引起了筆者的關(guān)注.本文在Lp(1≤p＜+∞)空間中研究了板幾何中具抽象邊界條件的各向異性、連續(xù)能量、非均勻介質(zhì)的遷移方程,得到了該遷移算子A的譜在區(qū)域Γ中由至多可數(shù)個(gè)具有限代數(shù)重?cái)?shù)的離散本征值組成等結(jié)果.下面研究板幾何中一類具抽象邊界條件下各向異性、連續(xù)能量、非均勻介質(zhì)的遷移方程的初邊值問題:

2 算子與空間

3 主要的定理

在Γ上同時(shí)成立.由輔理3.1和輔理3.2知(20)式與(22)式成立,又由于(21)式與(20)式有相同的結(jié)構(gòu),(23)式與(22)式有相同的結(jié)構(gòu),故同理可證(21)式與(23)式也成立.故定理3.1獲證.

4 主要的結(jié)果

致謝作者衷心感謝審稿專家和編輯部對(duì)本文提出有益的修改意見.

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The spectrum d istribu tion of a transport operator w ith
abstract boundary cond ition

Wu Hongxing,Wang Shenghua,Yuan Dengbin

(College of M athem atics and Com puter Science,Shangrao Norm al University,Shangrao 334001,China)

In this paper,transport operator equation of anisotrop ic con tinuous energy nonhom ogeneous w ith abstract boundary condition in slab geometry is studied.Supposing that the H is partly smooth and K is regular,the spectrum of the transport operator A consists of,atmost,isolated eigenvaluesw ith fnite algebraic multip licities in the tripΓis obtained.Themain method relies on theory of sem igroups of linear operators and solution m ethod.

transport operator,abstract boundary cond ition,nonhom ogeneous,partly sm ooth operator, discrete eigenvalues

O177.2

A

1008-5513(2013)05-0489-09

2012-03-15.

江西省自然科學(xué)基金(20132BAB 201002);江西省教育廳科技項(xiàng)目(G JJ13706,G JJ13703).

吳紅星(1980-),碩士,講師,研究方向：遷移方程.

王勝華(1956-),博士,教授,研究方向：遷移方程.

2010 MSC:47D 06