On homogenization of the Lavrent'ev–Bitsadze equation in a partially perforated domain with the third boundary condition on the boundary of the cavities. Subcritical, critical and supercritical cases

For the Lavrent'ev—Bitsadze equation in a partially perforated model domain with a characteristic size of microinhomogeneities $\varepsilon,$ we consider the problem with the third-kind boundary condition on the boundary of the cavities (the Fourier condition), which has a small parameter $\varepsilon^\alpha$ as a multiplier in the coefficients, and the Dirichlet condition on the outer part of the boundary. For this problem, we construct a homogenized problem and prove the convergence of the solutions of the original problem to the solution of the homogenized problem in three cases. The subcritical case with $\alpha>1$ is characterized by the fact that dissipation at the boundary of the cavities is negligibly small, in the critical case with $\alpha=1$ a potential appears in the equation due to dissipation, and in the supercritical case with $\alpha<1$ the dissipation plays the major role, it leads to degeneracy of the solution of the entire problem.