Closed curves of prescribed curvature and a pinning effect

Matteo Novaga; Enrico Valdinoci; Matteo Novaga; Enrico Valdinoci

doi:10.3934/nhm.2011.6.77

Networks and Heterogeneous Media

2011, Volume 6, Issue 1: 77-88. doi: 10.3934/nhm.2011.6.77

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Closed curves of prescribed curvature and a pinning effect

Matteo Novaga ^{1
,},
Enrico Valdinoci ^{2
,}

1.
Università di Padova, Via Trieste 63, 35121 Padova
2.
Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1, I-00133 Roma

Received: 01 February 2010 Revised: 01 May 2010
Primary: 53A10, 34B15; Secondary: 58E99.

We prove that for any $H: R^2 \to R$ which is $Z^2$-periodic, there exists $H_\varepsilon$, which is smooth, $\varepsilon$-close to $H$ in $L^1$, with $L^\infty$-norm controlled by the one of $H$, and with the same average of $H$, for which there exists a smooth closed curve $\gamma_\varepsilon$ whose curvature is $H_\varepsilon$. A pinning phenomenon for curvature driven flow with a periodic forcing term then follows. Namely, curves in fine periodic media may be moved only by small amounts, of the order of the period.
- Prescribed curvature,
- pinning phenomena,
- heterogeneous media
Citation: Matteo Novaga, Enrico Valdinoci. Closed curves of prescribed curvature and a pinning effect[J]. Networks and Heterogeneous Media, 2011, 6(1): 77-88. doi: 10.3934/nhm.2011.6.77

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Abstract

We prove that for any $H: R^2 \to R$ which is $Z^2$-periodic, there exists $H_\varepsilon$, which is smooth, $\varepsilon$-close to $H$ in $L^1$, with $L^\infty$-norm controlled by the one of $H$, and with the same average of $H$, for which there exists a smooth closed curve $\gamma_\varepsilon$ whose curvature is $H_\varepsilon$. A pinning phenomenon for curvature driven flow with a periodic forcing term then follows. Namely, curves in fine periodic media may be moved only by small amounts, of the order of the period.

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