Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Perturbation formula for the two-phase membrane problem | Bozorgnia Advances in Difference Equations 2011 2011 19 http content 2011 1 19 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Perturbation formula for the two-phase membrane problem Farid Bozorgnia Correspondence bozorg@. Faculty of Sciences Persian Gulf University Bushehr 75168 Iran SpringerOpen0 Abstract A perturbation formula for the two-phase membrane problem is considered. We perturb the data in the right-hand side of the two-phase equation. The stability of the solution and the free boundary with respect to perturbation in the coefficients and boundary value is shown. Furthermore continuity and differentiability of the solution with respect to the coefficients are proved. Keywords Free boundary problems Two-phase membrane Perturbation Introduction Let l O R be non-negative Lipschitz continuous functions where o is a bounded open subset of Rn with smooth boundary. Assume further that g e W1 2 O n L O and g changes sign on do. Let K v e w 1 2 O v g e W 2 u . Consider the functional I v i Vv 2 Ấ max v 0 Ấ min v 0 dx Q 2 which is convex weakly lower semi-continuous and hence attains its infimum at some point u e K. The Euler-Lagrange equation corresponding to the minimizer u is given by Weiss 1 and is called the two-phase membrane problem i Au Ấ x ỉl 0 Ấ X U 0 in Q 2 Ịu g on d . where CA denotes the characteristic function of the set A and r u d x e o u x 0 u d x e o u x 0 n o is called the free boundary. The free boundary consists of two parts r u r u n x e o Vu x 0 and r u r u n v u x 0 . By o u and o- u we denote the sets x e O u x 0 and x e O u x 0 respectively. Also A u denotes the set x e O u x 0 . 2011 Bozorgnia licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http licenses by which permits unrestricted use distribution and reproduction in any medium provided the original .
