Regularity of solutions of elliptic problems with a curved fracture

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• 作者：S. Ariche^a ; ^{Sadjia_14@hotmail.fr" class="auth_mail" title="E-mail the corresponding author} ; C. De Coster^b ; ^{Colette.DeCoster@univ-valenciennes.fr" class="auth_mail" title="E-mail the corresponding author} ; S. Nicaise^b ; ^{serge.nicaise@univ-valenciennes.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Laplace equation ; Dirac measure ; Regularity
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：908-932
• 全文大小：764 K

文摘

We consider the Laplace equation with a right-hand side concentrated on a curved fracture of class C^m+2$C^{m+2}$ for some nonnegative integer m (i.e., a sort of Dirac mass). We show that the solution belongs to a weighted Sobolev space of order m, the weight being the distance to this fracture. Our proof relies on a priori estimates in a dihedron or a cone with singularities for elliptic operators with variable coefficients. In both cases, such an estimate is obtained using a dyadic covering of the domain.