Asymptotic characterization of generalized problems in Sobolev spaces with variable exponents

Abstract

This thesis addresses a stationary nonlinear elasticity contact problem in a thin three-dimensional domain, characterized by Tresca friction and variable exponent nonlinearities. The model in-corporates a generalized Lamé-type reaction term within the framework of variable exponent Sobolev spaces W 1,p(x). We first establish the existence and uniqueness of the three-dimensional weak solution using the Minty-Browder theorem and monotonicity arguments. Subsequently, we perform a rigorous asymptotic analysis as the thickness parameter ζ tends to zero. By employing a rescaling technique and deriving uniform a priori estimates, we justify the convergence toward an effective two-dimensional limit model. The results show that both the material’s heterogeneous structure and the non-smooth contact laws are preserved in the limit, providing a solid mathematical foundation for reduced-order modeling in nonlinear media with non-standard growth.