Electronic Journal of Differential Equations, Vol. 2019 (2019), No. 114, pp. 1-13. Title: Free boundary problems with Neuman boundary condition Authors: Abdeslem Lyaghfouri (American Univ., Ras Al Khaimah, UAE) Abderachid Saadi (Univ. of Msila, Msila, Algeria) Abstract: In this work, we study the continuity of a free boundary in a class of elliptic problems, with a Neuman boundary condition. The main idea is to use a change of variable that reduces the problem to the one studied in [16]. Submitted January 7, 2019. Published October 10, 2019. Math Subject Classifications: 35J15, 35R35. Key Words: Free boundary; Neuman boundary condition; change of variable.