An intrinsic approach to manifold constrained variational problems

Manifold Constrained Variational Problems

The integral representation for the relaxation of a class of energy functionals where the admissible fields are constrained to remain on a C m-dimensional manifold M ⊂ R is obtained. If f : Rd×N → [0,∞) is a continuous function satisfying 0 ≤ f(ξ) ≤ C(1 + |ξ|), for C > 0, p ≥ 1, and for all ξ ∈ Rd×N , then F(u,Ω) : = inf {un}  lim inf n→∞ Z Ω f(∇un) dx : un ⇀ u in W , un(x) ∈M a.e. x ∈ Ω, n ∈ ...

Manifold Constrained Variational Mixtures

In many data mining applications, the data manifold is of lower dimension than the dimension of the input space. In this paper, it is proposed to take advantage of this additional information in the frame of variational mixtures. The responsibilities computed in the VBE step are constrained according to a discrepancy measure between the Euclidean and the geodesic distance. The methodology is ap...

Homogenization of Variational Problems under Manifold Constraints

Abstract. Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna, Fonseca, Malý & Trivisa [18]. For energies with superlinear or linear growth, a Γ-convergence result is established i...

A Variational Approach to Recovering a Manifold from Sample Points

A variational approach to double-porosity problems

In this paper we outline an approach by Γ-convergence to some problems related to ‘double-porosity’ homogenization. Various such models have been discussed in the mathematical literature, the first rigorous result for a linear double-porosity model having been obtained by Arbogast, Douglas and Hornung in [7]. The two-scale convergence approach to double-porosity problems was developed by Allair...