Rectifiability of the jump set of locally integrable functions

Sharp interface limit of a multi-phase transitions model under nonisothermal conditions

“…In particular, it coincides with F 0 (see Definition 1.3). Notice that F 0 is well defined since J u is countably H N −1 -rectifiable for all u ∈ L 1 loc ( ; R M ) (see [26]). On the other hand, if (H.2) does not hold, i.e., if…”

“…Moreover, we use the phrase sequence of almost minimizing geodesics to denote a sequence in A( p, q) for which the infimum is achieved in the limit. Let us remark that the existence of minimizing geodesics for (26) has been previously investigated by many authors. We mention here the work of Zuniga and Sternberg [62], where existence of solutions to the minimization problem is shown under very general assumptions on the conformal factor F. Of particular interest for our analysis is the special case where the conformal factor is given by F(z):=2 W (x, z).…”

“…As the proofs of our main results rely on a precise understanding of minimizing geodesics for (26), and in particular on their dependence on the variable x when F is chosen as in ( 27), compared to [62] we require more stringent assumptions on the behavior of the potential W near the wells (see (H.3)). In turn, our approach is in spirit closer to that of Sternberg [59], where the author considered a singular perturbation of the conformal factor which renders the associated Riemannian metric conformal to the Euclidean metric and proceeded to prove a uniform bound with respect to the perturbation parameter.…”

“…We begin by presenting a compactness criterion for almost minimizing geodesics. The result states that the existence of a sequence of almost minimizing geodesic for (26) with a uniform bound on the Euclidean length of each element in the sequence implies the existence of a minimizing geodesic which enjoys the same bound. The proof is adapted from the classical result on the existence of shortest paths, i.e., minimizers of the length functional (28) (see, for example, [44,Theorem 5.38]).…”

“…and let d F : R M × R M → [0, ∞) be given as in (26). Then for every p, q ∈ R M there exists a minimizing geodesic γ ∈ A( p, q) for d F ( p, q) such that…”