Elliptic estimates for the Dirichlet–Neumann operator on a corner domain

Article type: Research Article

Authors: Ming, Mei^{a; *} | Wang, Chao^b

Affiliations: [a] School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China. E-mail: mingm@mail.sysu.edu.cn | [b] School of Mathematical Sciences, Peking University, Beijing 100871, China. E-mail: wangchao@math.pku.edu.cn

Correspondence: [*] Corresponding author. E-mail: mingm@mail.sysu.edu.cn.

Abstract: We consider the elliptic estimates for the Dirichlet–Neumann operator related to the water waves problem on a two-dimensional corner domain in this paper. Due to the singularity of the boundary, there will be singular parts in the solution of the elliptic problem for the D–N operator. To begin with, we study elliptic problems with mixed boundary conditions to derive singularity decompositions and estimates. Based on this analysis, we present the estimates for both the D–N operator and its shape derivative with the existence of singular parts.

Keywords: Elliptic estimate, Dirichlet–Neumann operator, corner domains, water waves problem

DOI: 10.3233/ASY-171427

Journal: Asymptotic Analysis, vol. 104, no. 3-4, pp. 103-166, 2017