Strong approximation inh-mass of rectifiable currents under homological constraint

Abstract

Let h : R → R+ be a lower semi-continuous subbadditive and even function such that h(0) = 0 and h(θ) ≥ α|θ| for some α > 0. The h-mass of a k-polyhedral chain PGiven such a rectifiable flat chain T with M h (T ) < ∞ and ∂T polyhedral, we prove that for every η > 0, it decomposes as T = P + ∂V with P polyhedral, V rectifiable, M h (V ) < η and M h (P ) < M h (T ) + η. In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint ∂P = ∂T . These results are motivated by the study of approximations of M h by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of T → M h (T ) + I ∂S (∂T ) with respect to the topology of the flat norm.

Space-time integral currents of bounded variation

“…The proof is essentially contained in [4, Theorem 1.2] and [5, Proposition 2.7] (for us, the easier argument of Section 1.2 in [4] suffices). An inspection of this proof yields in particular that if T is an integral current, then also P is an integral polyhedral chain (p ℓ ∈ N).…”

Approximation of rectifiable 1-currents and weak-⁎ relaxation of the h-mass

“…We concentrate on models (which we call admissible below) in which the generalized branched transport cost metrizes weak- * convergence. Similarly to [4] we now show that one may also prescribe the boundary during the relaxation.…”

“…In that case we employ our new approximation lemma for 1-rectifiable flat chains to reduce theorem 5 to the case of chains with polyhedral boundary. This case in turn has already been solved by Chambolle, Ferrari, and Merlet [4] (under the above condition on h). The proof for general transportation cost h can then be reduced to costs with h(m) ≥ αm using a representation theorem for M h (T ).…”

“…As also emphasized in [4], the latter result is particularly useful for the development of phasefield approximations of generalized branched transport or minimal h-mass problems. Indeed, when proving Γ-convergence of a phasefield functional to the minimal h-mass problem with prescribed boundary, a recovery sequence can typically only be constructed for polyhedral fluxes, in particular with polyhedral boundary.…”

“…In this note we prove the equivalence between generalized branched transport and the h-mass in full generality in theorem 5. The main tool will be a recent relaxation result by Chambolle, Ferrari, and Merlet [4] for currents with polyhedral boundary, combined with a new strong approximation result of rectifiable 1-currents by currents with polyhedral boundary and equibounded boundary mass (lemmas 7 to 9). In the remainder of the introduction we describe the above-mentioned models and corresponding notions in more detail.…”