Electron. J. Diff. Equ., Vol. 2016 (2016), No. 132, pp. 1-14.

Perturbation of the free boundary in elliptic problem with discontinuities

Sabri Bensid

Abstract:
We study the discontinuous elliptic problem
$$\displaylines{ -\Delta u = \lambda H(u-\mu ) \quad \text{in } \Omega,\cr u =h \quad \text{on }\partial \Omega, }$$
where $\Omega$ is a regular bounded domain of $\mathbb{R}^n$, $H$ is the Heaviside function, $\lambda, \mu$ are a positive real parameters and h is a given function. We prove the existence of solutions, and characterize the free boundaries $\{x\in \Omega: u(x)=\mu\}$ using the perturbation of the boundary condition and smooth boundary of the domain.

Submitted April 14, 2016. Published June 7, 2016.
Math Subject Classifications: 34R35, 35J25.
Key Words: Perturbation; implicit function theorem; free boundary problem.

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Sabri Bensid
Department of Mathematics
Faculty of Sciences
University of Tlemcen, B.P. 119
Tlemcen 13000, Algeria
email: edp_sabri@yahoo.fr

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