The Dirichlet problem for singular elliptic equations

Özet

Bu makalede $L=-sum_{i=1}^n frac{partial^2}{partial x_i^2} - sum_{i=k+1}^n frac{2mu_i}{x_i} frac{partial}{partial x_i}$; $2mu_i>1$, i=k+1,...,n formunda singüler katsayılı eliptik operatör, $L^2$ metodu ile çalışıldı ve bu operatör için Dirichlet formu elde edildi. Dirichlet formu ile Dirichlet problemi kuruldu. Bu problemin çözümünün tekliği Lax-Milgram lemmasindan yararlanılarak incelendi. Ayrıca singüler katsayılı eliptik denklemin daha genel denklemi olan $L_2u=sum_{i,j=1}^n frac{partial^2 u}{partial x_i partial x_j} + sum_{i=1}^k b_i frac{partial u}{partial x_i} + sum_{i=k+1}^n frac{h_i}{x_i} frac{partial u}{partial x_i} + cu = f$ denklemi ele alındı. Ve bu denklemin çözümünün tekliği için barrier fonksiyonlar yardımıyla eliptik denklemlerin maximum özelliğinden yararlanıldı.

Abstract

In this article, the linear elliptic operator with singular coefficients $L=-sum_{i=1}^n frac{partial^2}{partial x_i^2} - sum_{i=k+1}^n frac{2mu_i}{x_i} frac{partial}{partial x_i}$; $2mu_i>1$, i=k+1,...,n is studied by the $L^2$ method, and the Dirichlet form is obtained for this operator. The Dirichlet problem is constructed with the Dirichlet form.. Using the Lax-Milgram lemma, the uniqueness of the solution for this problem is investigated. Furthermore, the more general equation $L_2u=sum_{i,j=1}^n frac{partial^2 u}{partial x_i partial x_j} + sum_{i=1}^k b_i frac{partial u}{partial x_i} + sum_{i=k+1}^n frac{h_i}{x_i} frac{partial u}{partial x_i} + cu = f$ is considered. To find the uniqueness of the solution of this equation, the maximum properties of elliptic equations, with the help of barrier functions, are used.