Bump solutions for the mesoscopic Allen–Cahn equation in periodic media

AbstractGiven 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 Hε which is ε-close to H for which the following two problems have solutions: 1.Find a set Eε,R whose boundary is uniformly close to ∂ BR and has mean curvature equal to −Hε at any point,2.Find u = uε,R,δ solving $$ -\delta\,\Delta u + \frac{F'(u)}{\delta} +\frac{c_0}{2} H_\varepsilon = 0, $$such that uε,R,δ goes from a δ-neighborhood of + 1 in BR to a δ-neighborhood of −1 outside BR.

[1]  E. Giusti.  Minimal surfaces and functions of bounded variation , 1977 .

[2]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[3]  P. Rabinowitz,et al.  On a class of infinite transition solutions for an Allen-Cahn model equation , 2008 .

[4]  P. Rabinowitz,et al.  Mixed states for an Allen-Cahn type equation, II , 2004 .

[5]  L. Jeanjean,et al.  EXISTENCE OF INFINITELY MANY STATIONARY LAYERED SOLUTIONS IN R 2 FOR A CLASS OF PERIODIC ALLEN–CAHN EQUATIONS , 2002 .

[6]  N. Dirr,et al.  Pinning and de-pinning phenomena in front propagation in heterogeneous media , 2006 .

[7]  P. Rabinowitz,et al.  Mixed states for an Allen‐Cahn type equation , 2003 .

[8]  Rafael de la Llave,et al.  Planelike minimizers in periodic media , 2001 .

[9]  M. Novaga,et al.  MULTIBUMP SOLUTIONS AND ASYMPTOTIC EXPANSIONS FOR MESOSCOPIC ALLEN-CAHN TYPE EQUATIONS ∗ , 2009 .

[10]  Maurizio Paolini,et al.  A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem , 1997, Math. Comput..

[11]  R. Llave,et al.  Multiplicity results for interfaces of Ginzburg–Landau–Allen–Cahn equations in periodic media , 2007 .

[12]  W. Ziemer,et al.  Area minimizing sets subject to a volume constraint in a convex set , 1997, math/9811046.

[13]  M. Novaga,et al.  Gamma--convergence of the Allen-Cahn energy with an oscillating forcing term , 2006 .

[14]  Nicolas Dirr,et al.  Pulsating wave for mean curvature flow in inhomogeneous medium , 2008, European Journal of Applied Mathematics.

[15]  Enrico Valdinoci,et al.  Closed curves of prescribed curvature and a pinning effect , 2011, Networks Heterog. Media.

[16]  M. Gage,et al.  The heat equation shrinking convex plane curves , 1986 .

[17]  Ugo Bessi.  Many Solutions of Elliptic Problems on R n of Irrational Slope , 2005 .