2011, 6(1): 77-88. doi: 10.3934/nhm.2011.6.77

Closed curves of prescribed curvature and a pinning effect

1. 

Università di Padova, Via Trieste 63, 35121 Padova, Italy

2. 

Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1, I-00133 Roma

Received  February 2010 Revised  May 2010 Published  March 2011

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.
Citation: Matteo Novaga, Enrico Valdinoci. Closed curves of prescribed curvature and a pinning effect. Networks & Heterogeneous Media, 2011, 6 (1) : 77-88. doi: 10.3934/nhm.2011.6.77
References:
[1]

L. A. Caffarelli and R. de la Llave, Planelike minimizers in periodic media,, Comm. Pure Appl. Math., 54 (2001), 1403. doi: 10.1002/cpa.10008.

[2]

A. Chambolle and G. Thouroude, Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem,, Netw. Heterog. Media, 4 (2009), 127. doi: 10.3934/nhm.2009.4.127.

[3]

N. Dirr, M. Lucia and M. Novaga, $\Gamma$-convergence of the Allen-Cahn energy with an oscillating forcing term,, Interfaces and Free Boundaries, 8 (2006), 47. doi: 10.4171/IFB/135.

[4]

N. Dirr and N. K. Yip, Pinning and de-pinning phenomena in front propagation in heterogeneous media,, Interfaces Free Bound., 8 (2006), 79. doi: 10.4171/IFB/136.

[5]

K. Ecker, "Regularity Theory for Mean Curvature Flow,", Progress in Nonlinear Differential Equations and their Applications, (2004).

[6]

K. Ecker and G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes,, Comm. Math. Phys., 135 (1991), 595. doi: i:10.1007/BF02104123.

[7]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves,, J. Differential Geom., 23 (1986), 69.

[8]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," volume 80 of "Monographs in Mathematics,", Birkhäuser Verlag, (1984).

[9]

M. Novaga and E. Valdinoci, The geometry of mesoscopic phase transition interfaces,, Discrete Contin. Dyn. Syst., 19 (2007), 777. doi: 10.3934/dcds.2007.19.777.

[10]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau,, Comm. Pure Appl. Math., 57 (2004), 1627. doi: 10.1002/cpa.20046.

show all references

References:
[1]

L. A. Caffarelli and R. de la Llave, Planelike minimizers in periodic media,, Comm. Pure Appl. Math., 54 (2001), 1403. doi: 10.1002/cpa.10008.

[2]

A. Chambolle and G. Thouroude, Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem,, Netw. Heterog. Media, 4 (2009), 127. doi: 10.3934/nhm.2009.4.127.

[3]

N. Dirr, M. Lucia and M. Novaga, $\Gamma$-convergence of the Allen-Cahn energy with an oscillating forcing term,, Interfaces and Free Boundaries, 8 (2006), 47. doi: 10.4171/IFB/135.

[4]

N. Dirr and N. K. Yip, Pinning and de-pinning phenomena in front propagation in heterogeneous media,, Interfaces Free Bound., 8 (2006), 79. doi: 10.4171/IFB/136.

[5]

K. Ecker, "Regularity Theory for Mean Curvature Flow,", Progress in Nonlinear Differential Equations and their Applications, (2004).

[6]

K. Ecker and G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes,, Comm. Math. Phys., 135 (1991), 595. doi: i:10.1007/BF02104123.

[7]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves,, J. Differential Geom., 23 (1986), 69.

[8]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," volume 80 of "Monographs in Mathematics,", Birkhäuser Verlag, (1984).

[9]

M. Novaga and E. Valdinoci, The geometry of mesoscopic phase transition interfaces,, Discrete Contin. Dyn. Syst., 19 (2007), 777. doi: 10.3934/dcds.2007.19.777.

[10]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau,, Comm. Pure Appl. Math., 57 (2004), 1627. doi: 10.1002/cpa.20046.

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