橢圓方程解的正則性研究


目 錄

第一章引言………………………………………………………………………………………1
1.1 橢圓方程介紹及解的正則性的研究進(jìn)展……………………………………………1
1.2 預(yù)備知識(shí)………………………………………………………………………………1
1.3 主要結(jié)論………………………………………………………………………………2
第二章定理證明過(guò)程…………………………………………………………………………4
2.1 定理1.1的證明………………………………………………………………………4
2.2 定理1.2的證明………………………………………………………………………11
2.3 定理1.3的證明………………………………………………………………………18
結(jié)論……………………………………………………………………………………………23
致謝………………………………………………………………………………………………24
參考文獻(xiàn)…………………………………………………………………………………………25



摘要 橢圓型方程是數(shù)學(xué)物理中一類(lèi)非常重要的方程，在彈性力學(xué)、流體力學(xué)、幾何學(xué)、電磁學(xué)和變分法中都有應(yīng)用。本論文關(guān)心橢圓方程的角點(diǎn)正則性，由橢圓方程的凝固系數(shù)法知，本質(zhì)上我們只需要研究橢圓型方程最典型的代表——Laplace方程。就橢圓方程在非光滑區(qū)域中的解的正則性問(wèn)題方向，已有許多研究結(jié)果，但是其理論涉及到橢圓方程中許多復(fù)雜和深入的理論和技巧。本論文則采用完全初等的技巧，如分離變量法、Sturm-Liuville定理及一些簡(jiǎn)單的分部積分的技巧，得到角點(diǎn)正則性的相關(guān)結(jié)論。本文討論了二維Laplace方程在相同的角狀區(qū)域中，由于Dirichlet條件，混合邊界條件，Neumann邊界條件的不同，分別形成問(wèn)題（D）（M）（N）,其角點(diǎn)正則性也是不同的。問(wèn)題（D）中，為得到解的最佳正則性，我們首先考慮原問(wèn)題的一個(gè)簡(jiǎn)化問(wèn)題，運(yùn)用極坐標(biāo)變換和分離變量法得到相應(yīng)的特征值和特征向量，從而得到解的最佳正則性。然后同樣運(yùn)用極坐標(biāo)變換和分離變量法，得到解。接下來(lái)論證級(jí)數(shù)一致收斂，級(jí)數(shù)允許關(guān)于兩個(gè)變量逐項(xiàng)微分兩次，證明了級(jí)數(shù)為定解問(wèn)題在角形區(qū)域上的古典解。最后利用標(biāo)準(zhǔn)的Scaling技巧，得到解的估計(jì)，這里。問(wèn)題（M）類(lèi)似求解得到估計(jì)，這里。問(wèn)題（N）類(lèi)似得，這里。
關(guān)鍵詞：橢圓方程 正則性 分離變量法 加權(quán)空間

The regularity of the solutions of elliptic equations

Abstract Elliptic equation is a very important equation in mathematical physics, which has many applications in elasticity, fluid mechanics , geometry, electromagnetism and calculus of variations. This paper is about the most typical exception elliptic equations - Laplace equation expanded the study of the regularity of its solution. There are many studies on issues about regularity of elliptic equations in non-smooth regions of solutions, but the theory of elliptic equations involving many complex and in-depth theory and techniques, this paper is a fully elementary skills, such as separation variable method,the Sturm-Liuville theorem and some simple tips segment integral to obtain the relevant conclusions about the regularity on the corner. This article discusses the two-dimensional equation in the same angular region, due to the different conditions, mixed boundary conditions, boundary conditions, namely the formation of the problem (D)(M)(N), whose corner regularity is different, but the solution of the problem of the three regular discuss ways are similar. In the problem (D), in order to get the best solution regularity, firstly,to consider the deformation of the original problem, use the polar coordinate transformation and separation of variables corresponding eigenvalues ​​and eigenvectors, to get the best the regularity of solution. Then use the same polar coordinate transformation and separation of variables, get the solution. The next argument is consistent series converges, the series allows itemized differential twice on two variables, the series proved to be the classical solution of the problem solution on the angular region. Finally using Scaling standard techniques to obtain estimates of the solution , here. Problem (M) obtained similar estimates here. Problem (N) obtained similar estimates, here.

Key words：Elliptic equations; Regularity; Separation variable method; The weighted space