The heat equation with the dynamic boundary condition as a singular limit of problems degenerating at the boundary

We derive the dynamic boundary condition for the heat equation as a limit of boundary layer problems. We study convergence of their weak and strong solutions as the width of the layer tends to zero. We also discuss $\mathrm{\Gamma }$-convergence of the functionals generating these flows. Our analysis of strong solutions depends on a new version of the Reilly identity.