本文選題：非線性橢圓方程　切入點(diǎn)：梯度項(xiàng)　出處：《蘭州大學(xué)》2016年博士論文

【摘要】：這篇博士學(xué)位論文主要研究一類(lèi)帶有梯度項(xiàng)的擬線性橢圓方程的邊值問(wèn)題.由于有非線性梯度項(xiàng),此類(lèi)方程本質(zhì)上不具有變分結(jié)構(gòu),因此經(jīng)典的變分法對(duì)這類(lèi)方程不再適用.如何得到此類(lèi)方程解的存在性、正則性等結(jié)果已成為眾多學(xué)者關(guān)注的問(wèn)題.本文對(duì)此類(lèi)方程作了嘗試性的探索.全文共有五章.第一章是緒論.簡(jiǎn)單介紹了橢圓方程,以及對(duì)帶梯度項(xiàng)的橢圓方程的國(guó)內(nèi)外研究現(xiàn)狀,介紹本文的創(chuàng)新點(diǎn)和主要的研究方法.第二章是準(zhǔn)備知識(shí).羅列了本文所需要的基本的預(yù)備知識(shí).在本文的第三章討論下面半線性橢圓方程Dirichlet邊值問(wèn)題我們假設(shè)區(qū)域Ω關(guān)于x1,x2,…,xN-1是徑向?qū)ΨQ(chēng)的,且f與區(qū)域Ω有相同的對(duì)稱(chēng)性質(zhì),則方程的解是(N-1)徑向?qū)ΨQ(chēng)的,即,u(x)=u(r,xN),其中這樣就可以把方程轉(zhuǎn)化為具有兩變?cè)臋E圓方程,然后受處理兩變?cè)獧E圓方程方法的啟發(fā),我們找到一個(gè)與已有文獻(xiàn)中均不同的方法,利用此方法,可以得到方程的(N一1)徑向?qū)ΨQ(chēng)的正的古典C2,β解.然而,文獻(xiàn)[70]中得到的解的正則性?xún)H僅是C1,α的.第四章研究如下散度型非線性橢圓Dirichlet邊值問(wèn)題我們對(duì)A和f加合適的條件,然后借助于強(qiáng)制單調(diào)算子的性質(zhì)得到方程解的存在性.在第五章研究如下具有非線性Neumann邊界條件的擬線性橢圓問(wèn)題我們嘗試把de Figueiredo和Girardi發(fā)明的新的迭代方法運(yùn)用到上述問(wèn)題中.我們先給定一個(gè)υ∈W1,p(Ω),然后證明下面方程具有變分結(jié)構(gòu)的問(wèn)題山路解的存在性,最后證明這列山路解收斂到某個(gè)函數(shù)u,即原來(lái)方程的解.第六章給出本文的總結(jié)以及研究展望.
[Abstract]:In this dissertation, the boundary value problems of a class of quasilinear elliptic equations with gradient terms are studied. So the classical variational method is no longer applicable to this kind of equation. The results of regularity have become a concern of many scholars. This paper makes a tentative exploration of this kind of equation. There are five chapters in this paper. The first chapter is the introduction. And the research status of elliptic equation with gradient term at home and abroad, This paper introduces the innovation of this paper and the main research methods. The second chapter is the preparatory knowledge. It lists the basic preparatory knowledge needed in this paper. In the third chapter, we discuss the following Dirichlet boundary value problems of semilinear elliptic equations. The region 惟 is radial symmetric with respect to x 1 X 2, 鈥，

本文編號(hào)：1684959