Concentrated terms and varying domains in elliptic equations: Lipschitz case

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In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior which is uniformly Lipschitz and nonlinear terms are concentrated in a region which neighbors the boundary domain. We prove that this family of solutions converges to the solutions of a limit problem in H¹, an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.