Advanced Search
Article Contents
Article Contents

Closed curves of prescribed curvature and a pinning effect

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 53A10, 34B15; Secondary: 58E99.


    \begin{equation} \\ \end{equation}
  • [1]

    L. A. Caffarelli and R. de la Llave, Planelike minimizers in periodic media, Comm. Pure Appl. Math., 54 (2001), 1403-1441.doi: 10.1002/cpa.10008.


    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), 127-152.doi: 10.3934/nhm.2009.4.127.


    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), 47-78.doi: 10.4171/IFB/135.


    N. Dirr and N. K. Yip, Pinning and de-pinning phenomena in front propagation in heterogeneous media, Interfaces Free Bound., 8 (2006), 79-109.doi: 10.4171/IFB/136.


    K. Ecker, "Regularity Theory for Mean Curvature Flow," Progress in Nonlinear Differential Equations and their Applications, 57. Birkhäuser Boston Inc., Boston, MA, 2004.


    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), 595-613.doi: i:10.1007/BF02104123.


    M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.


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


    M. Novaga and E. Valdinoci, The geometry of mesoscopic phase transition interfaces, Discrete Contin. Dyn. Syst., 19 (2007), 777-798.doi: 10.3934/dcds.2007.19.777.


    E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.doi: 10.1002/cpa.20046.

  • 加载中

Article Metrics

HTML views() PDF downloads(90) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint