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

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