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1 Correction to: The Journal of Geometric Analysis (2019) 29:3055–3097 https://doi.org/10.1007/s12220-018-00104-z
In the sketch of the proof of Theorem 4 we claimed that a weak solution \((\xi , \kappa , \sigma )\) can be obtained by following the same lines as in the proof of [1, Proposition 3.4 and Theorem 3]. However, an appropriate argument should be slightly different due to different boundary conditions. The aim of this addendum is to fill that gap.
More precisely, in [1] at the free end one has \(\sigma ^\epsilon (t,0)=\kappa ^\epsilon (t,0)=0\) for all \(t\in [0,T]\). Therefore, the uniform (w.r.t. to \(\epsilon \)) \(L^2\) bound for \((\sigma ^\epsilon , \kappa ^\epsilon )\) follows directly from the Poincaré inequality and the uniform \(L^2\) bound for \((\partial _s\sigma ^\epsilon ,\partial _s\kappa ^\epsilon )\). In our case, the vanishing boundary conditions for \(\sigma ^\epsilon \) and \(\kappa ^\epsilon \) are no longer satisfied. We thus need some additional estimates to obtain the uniform \(L^2\) bound.
We first estimate the spatial average \(\overline{\sigma ^\epsilon }(t):=\int _{\mathbb {S}^1} \sigma ^{\epsilon }(s,t)\ \mathrm{{d}}s\). Indeed, an integration by parts and Cauchy–Schwarz yield
Thus
where the right-hand side is uniformly bounded (cf. [1, Proposition 3.1]). Thus from Poincaré inequality \(\Vert \sigma ^\epsilon - \overline{\sigma ^\epsilon }\Vert _{L^2(Q_T)}\le C(T) \Vert \partial _s\sigma ^\epsilon \Vert _{L^2(Q_T)}\) we obtain \(\Vert \sigma ^\epsilon \Vert _{L^2(Q_T)}\le C\) with C independent of \(\epsilon \).
It remains to show that \(\kappa ^\epsilon \) is uniformly bounded in \(L^2(Q_T)\). To this end, we note that from the definition of \(\kappa ^\epsilon \) one has \(\partial _s\xi ^\epsilon =\epsilon \kappa ^\epsilon + \frac{\kappa ^\epsilon }{\sqrt{\epsilon + |\kappa ^\epsilon |^2}}\), whence
Observing that
provided \(|\kappa ^\epsilon |\ge 1\), we infer that \(|\kappa ^\epsilon |\le \sigma ^\epsilon \) when \(|\kappa ^\epsilon |\ge 1\). Thus,
where C does not depend on \(\epsilon \).
Reference
Shi, W., Vorotnikov, D.: The gradient flow of the potential energy on the space of arcs. Calc. Var. 58, 59 (2019)
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Shi, W., Vorotnikov, D. Correction to: Uniformly Compressing Mean Curvature Flow. J Geom Anal 32, 116 (2022). https://doi.org/10.1007/s12220-021-00862-3
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DOI: https://doi.org/10.1007/s12220-021-00862-3