Journal of Mathematical Study (JMS), formerly known as “数学研究”, is a comprehensive mathematical research journal published by Global Science Press and Xiamen University. It publishes original research and survey papers, in English, of high scientific value in all major fields of mathematics, particularly the research areas represented by its editorial board members. JMS will be published quarterly which is a refereed international journal edited by a group of leading scientists in diverse fields and aims to become a premier mathematical research journal in China. All papers in “Journal of Mathematical Study" should be of high quality and contains original research which has not been published or submitted elsewhere (except for the survey papers which should be by invitation only).

We consider the Cahn-Hilliard equation for solving the binary image inpainting problem with emphasis on the recovery of low-order sets (edges, corners) and enhanced edges. The model consists in solving a modified Cahn-Hilliard equation by weighting the diffusion operator with a function which will be selected locally and adaptively. The diffusivity selection is dynamically adopted at the discrete level using the residual error indicator. We combine the adaptive approach with a standard mesh adaptation technique in order to well approximate and recover the singular set of the solution. We give some numerical examples and comparisons with the classical Cahn-Hillard equation for different scenarios. The numerical results illustrate the effectiveness of the proposed model.