\(\Gamma \) -Convergence for the Bi-Laplace-Beltrami Equation on Hypersurfaces

A mixed boundary value problem for the bi-Laplacian equation in a thin layer around a surface \(\mathcal {C}\) with the boundary is investigated. We track what happens in \(\Gamma \) -limit when the thickness of the layer converges to zero. It is shown how the mixed type boundary value problem (BVP) for the bi-Laplace equation in the initial thin layer transforms in the \(\Gamma \) -limit into an appropriate Dirichlet BVP for the bi-Laplace-Beltrami equation on the surface. For this we apply the variational formulation and the calculus of Günter’s tangential differential operators on a hypersurface and layers. This approach allow global representation of basic differential operators and of corresponding BVPs in terms of the standard cartesian coordinates of the ambient Euclidean space \(\mathbb {R}^n\) .