On the approximation of the elastica functional in radial symmetry

Abstract

Abstract. We prove a result concerning the approximation of the elastica functional with a sequence of second order functionals, under radial symmetry assumptions. This theorem is strictly related to a conjecture of De Giorgi [8].

Sharp-interface limit of the Allen-Cahn action functional in one space dimension

“…On the analytical side, [1] proves the -convergence for ⊂ R 2 with an assumption of radial symmetry; [19] proves -convergence for ⊂ R 3 (with a technical assumption that u x 3 ≥ 0). These results may be useful in proving the -convergence of the action functional in d ≥ 2.…”

“…In Sect. 1.3, we comment on the higher dimensional problem and links to recent work by Bellettini and Mugnai [1] and Moser [19].…”

Phase-field models for the approximation of the willmore functional and flow

“…The contributions on this point [3,20,22,24,26] culminated with the proof by Röger and Schätzle [24] in space dimensions N = 2, 3 and, independently, by Nagase and Tonegawa [22] in dimension N = 2, that the result holds true for smooth sets. More precisely, given u = 1 E the characteristic function of a set E ∈ C 2 (Ω), and u ε converging to u in L 1 (Ω) with a uniform control of the approximating perimeter…”

“…See [7] Claim 3.2. In a suitable regime provided by the method of matched asymptotic expansions, the normal velocity of the 1 2 -front Γ(t) = ∂E(t) associated with a solution u ε (x, t) to Bellettini's phase field model (2) is the Willmore velocity V = ∆ Γ H + A 2 H − H 3 2 , and one has u ε (x, t) = q( d(x,E(t))…”

“…In a suitable regime provided by the method of matched asymptotic expansions, the normal velocity of the 1 2 -front Γ(t) = ∂E(t) associated with a solution (u α ε , µ α ε , ξ α ε ) to Esedoḡlu-Rätz-Röger's phase field model (4) in both cases α = 0 and α = 1 is the Willmore velocity V = ∆ Γ H + A 2 H − H 3 2 . In addition, for α = 0:…”