Quantitative Homogenization Theory for Lam\'e-Stokes Coupled Systems

Abstract

We study a Lam\'e-Stokes coupled system arising as the incompressible-inclusion limit of periodic high-contrast elastic composites. The elastic matrix satisfies the Lam\'e equations, the fluid inclusions satisfy the Stokes equations with a local incompressibility constraint, and the two phases are coupled by continuity of displacement and traction across the interface. The main difficulty is that the incompressibility constraint is imposed only in the inclusions, while the interface structure prevents a direct application of standard elliptic homogenization theory. We develop a quantitative homogenization theory for this model. First, using the Babu\v{s}ka-Brezzi theory, we prove uniform well-posedness of the mixed variational problem, including inf-sup stability and a priori estimates independent of the microscale parameter. Second, by combining formal asymptotic expansions with two-scale convergence, we derive the homogenized effective elasticity equation and prove weak $H^1$ convergence of the microscopic displacement. The effective tensor is characterized by cell problems and is symmetric and strongly elliptic on symmetric matrices. Third, under smoothness assumptions on the interface, we establish piecewise higher-order Sobolev regularity and $L^\infty$ gradient bounds for the cell correctors. These estimates yield $O(\sqrt{\varepsilon})$ convergence rates for the displacement in $H^1$ and for the pressure in $L^2$ over the fluid region.