Limit Behaviour of Solutions to a Class of Boundary Value Problems with Equivalued Surface in a Domain with Thin Layer

This paper mainly study a class of boundary value problems with equivalued surface of nonlinear elliptic equations on a domain with thin layer arising in resistivity well-logging in petroleum exploitation. Under the assumptions that the coefficient of the zero order term and source term both belong to L1, the existence and uniqueness of bounded solutions to such problems are proved. Furthermore if the coefficients of the equation satisfy certain convergence conditions, the limit behaviour of solutions is studied as the thin layer converges to a surface. This result shows that in practical calculation, the boundary value problem with equivalued surface on the thin layer can be approximately replaced by the boundary value problem with equivalued interface.