Convergence of Sobolev spaces on varying manifolds

We deal with variational problems on varying manifolds in ℝⁿ. We represent each manifold by a positive measure μ, to which we associate a suitable notion of tangent space Tμ, of mean curvature H(μ), and of Sobolev spaces with respect to μ on an open subset Ω ⊆ ℝⁿ. We introduce the notions of weak and strong convergence for functions defined on varying manifolds, that is defined μ_h -a.e., being {μ_h} a weakly convergent sequence of measures. In this setting, we prove a strong-weak type compactness theorem for the pairs (P_{μ h} H(μ_h)), where P_{μ h} are the projectors onto the tangent spaces T_{μ h}. When μ_h belong to a suitable class of k-dimensional measures, having in particular a prescribed (k−1)-manifold as a boundary, we enforce this result to study the convergence of energy functionals, possibly with a Dirichlet condition on ∂Ω. We also address a perspective for optimization problems where the control variable is represented by a manifold with a prescribed boundary.