Abstract
Given a double-well potential F, a \({\mathbb{Z}^n}\)-periodic function H, small and with zero average, and ε > 0, we find a large R, a small δ and a function H ε which is ε-close to H for which the following two problems have solutions:
-
1.
Find a set E ε ,R whose boundary is uniformly close to ∂ B R and has mean curvature equal to −H ε at any point,
-
2.
Find u = u ε ,R,δ solving
$$ -\delta\,\Delta u + \frac{F'(u)}{\delta} +\frac{c_0}{2} H_\varepsilon = 0, $$such that u ε,R,δ goes from a δ-neighborhood of + 1 in B R to a δ-neighborhood of −1 outside B R .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alessio F., Jeanjean L., Montecchiari P.: Existence of infinitely many stationary layered solutions in \({\mathbb{R}^2}\) for a class of periodic Allen–Cahn equations. Commun. Part. Differ. Eq. 27(7–8), 1537–1574 (2002)
Bessi U.: Many solutions of elliptic problems on \({\mathbb{R}^n}\) of irrational slope. Commum. Part. Differ. Eq. 30(10–12), 1773–1804 (2005)
Caffarelli L.A., de la Llave R.: Planelike minimizers in periodic media. Commun. Pure Appl. Math. 54(12), 1403–1441 (2001)
Crandall M., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)
Dirr N., Lucia M., Novaga M.: Γ-convergence of the Allen–Cahn energy with an oscillating forcing term. Interfaces Free Bound. 8(1), 47–78 (2006)
de la Llave R., Valdinoci E.: Multiplicity results for interfaces of Ginzburg–Landau–Allen–Cahn equations in periodic media. Adv. Math. 215(1), 379–426 (2007)
Dirr N., Yip N.K.: Pinning and de-pinning phenomena in front propagation in heterogeneous media. Interfaces Free Bound. 8(1), 79–109 (2006)
Dirr N., Yip N.K., Karali G.: Pulsating wave for mean curvature flow in inhomogeneous medium. Eur. J. Appl. Math. 19, 661–699 (2008)
Gage M., Hamilton R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23(1), 69–96 (1986)
Giusti, E.: Minimal surfaces and functions of bounded variation. In: Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel (1984)
Novaga M., Valdinoci E.: The geometry of mesoscopic phase transition interfaces. Discrete Contin. Dyn. Syst. 19(4), 777–798 (2007)
Novaga M., Valdinoci E.: Multibump solutions and asymptotic expansions for mesoscopic Allen–Cahn type equations. ESAIM Control Optim. Calc. Var. 15(4), 914–933 (2009)
Novaga, M., Valdinoci, E.: Closed curves of prescribed curvature and a pinning effect. Netw. Heterog. Media (to appear)
Paolini M.: A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem. Math. Comput. 66, 45–67 (1997)
Rabinowitz P.H., Stredulinsky E.: Mixed states for an Allen–Cahn type equation. Dedicated to the memory of Jürgen K. Moser. Commun. Pure Appl. Math. 56(8), 1078–1134 (2003)
Rabinowitz P.H., Stredulinsky E.: Mixed states for an Allen–Cahn type equation. II. Calc. Var. Part. Differ. Eq. 21(2), 157–207 (2004)
Rabinowitz P.H., Stredulinsky E.: On a class of infinite transition solutions for an Allen–Cahn model equation. Discrete Contin. Dyn. Syst. 21(1), 319–332 (2008)
Stredulinsky E., Ziemer W.P.: Area minimizing sets subject to a volume constraint in a convex set. J. Geom. Anal. 7(4), 653–677 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Malchiodi.
Rights and permissions
About this article
Cite this article
Novaga, M., Valdinoci, E. Bump solutions for the mesoscopic Allen–Cahn equation in periodic media. Calc. Var. 40, 37–49 (2011). https://doi.org/10.1007/s00526-010-0332-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-010-0332-4