Abstract
We consider the inhomogeneous Allen–Cahn equation
where \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with smooth boundary \(\partial \Omega \) and V(x) is a positive smooth function, \(\epsilon >0\) is a small parameter, \(\nu \) denotes the unit outward normal of \(\partial \Omega \). For any fixed integer \(N\ge 2\), we will show the existence of a clustered solution \(u_{\epsilon }\) with N-transition layers near \(\partial \Omega \) with mutual distance \(O(\epsilon |\ln \epsilon |)\), provided that the generalized mean curvature \(\mathcal {H} \) of \(\partial \Omega \) is positive and \(\epsilon \) stays away from a discrete set of values at which resonance occurs. Our result is an extension of those (with dimension two) by Malchiodi et al. (Pac. J. Math. 229(2):447–468, 2007) and Malchiodi et al. (J. Fixed Point Theory Appl. 1(2):305–336, 2007).
Similar content being viewed by others
References
Alikakos, N.D., Bates, P.W.: On the singular limit in a phase field model of phase transitions. Ann. Inst. H. Poincaré Anal. Non Linéaire 5(2), 141–178 (1988)
Alikakos, N.D., Bates, P.W., Chen, X.: Periodic traveling waves and locating oscillating patterns in multidimensional domains. Trans. Am. Math. Soc. 351(7), 2777–2805 (1999)
Alikakos, N.D., Bates, P.W., Fusco, G.: Solutions to the nonautonomous bistable equation with specified Morse index, I. Existence. Trans. Am. Math. Soc. 340(2), 641–654 (1993)
Alikakos, N.D., Chen, X., Fusco, G.: Motion of a droplet by surface tension along the boundray. Calc. Var. Partial Differ. Equ. 11(3), 233–305 (2000)
Alikakos, N.D., Simpson, H.C.: A variational approach for a class of singular perturbation problems and applications. Proc. R. Soc. Edinb. Sect. A 107(1–2), 27–42 (1987)
Allen, S., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metall. 27, 1085–1095 (1979)
Angenent, S., Mallet-Paret, J., Peletier, L.A.: Stable transition layers in a semilinear boundary value problem. J. Differ. Equ. 67(2), 212–242 (1987)
Dancer, E.N., Yan, S.: Multi-layer solutions for an elliptic problem. J. Differ. Equ. 194(2), 382–405 (2003)
Dancer, E.N., Yan, S.: Construction of various types of solutions for an elliptic problem. Calc. Var. Partial Differ. Equ. 20(1), 93–118 (2004)
del Pino, M.: Layers with nonsmooth interface in a semilinear elliptic problem. Commun. Partial Differ. Equ. 17(9–10), 1695–1708 (1992)
del Pino, M.: Radially symmetric internal layers in a semilinear elliptic system. Trans. Am. Math. Soc. 347(12), 4807–4837 (1995)
del Pino, M., Kowalczyk, M., Wei, J.: Concentration on curves for nonlinear Schrödinger equations. Commun. Pure Appl. Math. 60(1), 113–146 (2007)
del Pino, M., Kowalczyk, M., Wei, J.: Resonance and interior layers in an inhomogeneous phase transition model. SIAM J. Math. Anal., 38(5), 1542–1564 (2006/07)
del Pino, M., Kowalczyk, M., Wei, J.: The Toda system and clustering interface in the Allen–Cahn equation. Arch. Ration. Mech. Anal. 190(1), 141–187 (2008)
del Pino, M., Kowalczyk, M., Wei, J., Yang, J.: Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature. Geom. Funct. Anal. 20(4), 918–957 (2010)
do Nascimento, A.S.: Stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in \(N\)-dimensional domains. J. Differ. Equ. 190(1), 16–38 (2003)
Du, Y., Nakashima, K.: Morse index of layered solutions to the heterogeneous Allen–Cahn equation. J. Differ. Equ. 238(1), 87–117 (2007)
Du, Y.: The heterogeneous Allen–Cahn equation in a ball: solutions with layers and spikes. J. Differ. Equ. 244(1), 117–169 (2008)
Du, Z., Gui, C.: Interior layers for an inhomogeneous Allen–Cahn equation. J. Differ. Equ. 249(2), 215–239 (2010)
Du, Z., Wang, L.: Interface foliation for an inhomogeneous Allen–Cahn equation in Riemannian manifolds. Calc. Var. Partial Differ. Equ. 47, 343–381 (2013)
Du, Z., Wei, J.: Clustering layers for the Fife–Greenlee problem in \({\mathbb{R}}^{n}\). Proc. R. Soc. Edinb. Sect. A 146(1), 107–139 (2016)
Fan, X., Xu, B., Yang, J.: Phase transition layers with boundary intersection for an inhomogeneous Allen–Cahn equation. J. Differ. Equ. 266(9), 5821–5866 (2019)
Fife, P.: Boundary and interior transition layer phenomena for pairs of second-order differential equations. J. Math. Anal. Appl. 54(2), 497–521 (1976)
Fife, P., Greenlee, M.W.: Interior transition Layers of elliptic boundary value problem with a small parameter. Russ. Math. Surv. 29(4), 103–131 (1974)
Flores, G., Padilla, P.: Higher energy solutions in the theory of phase transitions: a variational approach. J. Differ. Equ. 169(1), 190–207 (2001)
Hale, J., Sakamoto, K.: Existence and stability of transition layers. Jpn. J. Appl. Math. 5(3), 367–405 (1988)
Kohn, R.V., Sternberg, P.: Local minimizers and singular perturbations. Proc. R. Soc. Edinb. Sect. A 111(1–2), 69–84 (1989)
Kowalczyk, M.: On the existence and Morse index of solutions to the Allen–Cahn equation in two dimensions. Ann. Mat. Pura Appl. 184(1), 17–52 (2005)
Li, F., Nakashima, K.: Transition layers for a spatially inhomogeneous Allen–Cahn equation in multi-dimensional domains. Discrete Contin. Dyn. Syst.-A 32, 1391–1420 (2012)
Mahmoudi, F., Malchiodi, A., Wei, J.: Transition layer for the heterogeneous Allen–Cahn equation. Ann. Inst. H. Poincar\(\acute{e}\) Anal. Non Lin\(\acute{e}\)aire 25(3), 609–631 (2008)
Malchiodi, A., Ni, W.-M., Wei, J.: Boundary clustered interfaces for the Allen–Cahn equation. Pac. J. Math. 229(2), 447–468 (2007)
Malchiodi, A., Wei, J.: Boundary interface for the Allen–Cahn equation. J. Fixed Point Theory Appl. 1(2), 305–336 (2007)
Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98(2), 123–142 (1987)
Morgan, F.: Manifolds with density. Notices Am. Math. Soc. 52, 853–858 (2005)
Nakashima, K.: Multi-layered stationary solutions for a spatially inhomogeneous Allen–Cahn equation. J. Differ. Equ. 191(1), 234–276 (2003)
Nakashima, K., Tanaka, K.: Clustering layers and boundary layers in spatially inhomogeneous phase transition problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(1), 107–143 (2003)
Pacard, F., Ritoré, M.: From constant mean curvature hypersurfaces to the gradient theory of phase transitions. J. Differ. Geom. 64(3), 359–423 (2003)
Padilla, P., Tonegawa, Y.: On the convergence of stable phase transitions. Commun. Pure Appl. Math. 51(6), 551–579 (1998)
Rabinowitz, P.H., Stredulinsky, E.: Mixed states for an Allen–Cahn type equation, I. 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. Partial Differ. Equ. 21(2), 157–207 (2004)
Sakamoto, K.: Existence and stability of three-dimensional boundary-interior layers for the Allen–Cahn equation. Taiwan. J. Math. 9(3), 331–358 (2005)
Sternberg, P., Zumbrun, K.: Connectivity of phase boundaries in strictly convex domains. Arch. Ration. Mech. Anal. 141(4), 375–400 (1998)
Tang, F., Wei, S., Yang, J.: Phase transition layers for Fife–Greenlee problem on smooth bounded domain. Discrete Contin. Dyn. Syst.-A 38(3), 1527–1552 (2018)
Wei, J., Yang, J.: Toda system and cluster phase transition layers in an inhomogeneous phase transition model. Asymptot. Anal. 69(3–4), 175–218 (2010)
Wei, S., Yang, J.: Connectivity of boundaries by clustering phase transition layers of Fife–Greenlee problem on smooth bounded domain. J. Differ. Equ. 269(3), 1745–1795 (2020)
Wei, S., Yang, J.: Clustering phase transition layers with boundary intersection for an inhomogeneous Allen–Cahn equation. Commun. Pure Appl. Anal. 19(5), 2575–2616 (2020)
Yang, J., Yang, X.: Clustered interior phase transition layers for an inhomogeneous Allen–Cahn equation in higher dimensional domains. Commun. Pure Appl. Anal. 12(1), 303–340 (2013)
Acknowledgements
J. Yang and S. Wei are supported by National Natural Science Foundation of China (Nos. 11771167, 11831009 & 12001203).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Manuel del Pino.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendices
The computations of (3.30)
The main objective in this section is to compute the quantities in (3.30). For the cases \(n=3,\ldots ,N \), from the expressions of \({\mathbf {b}}_{1n}, {\mathbf {b}}_{2n} \) as in (3.20)–(3.21), we can obtain that
Specially, when \( n=1, 2\), we obtain
and
Similarly, for the cases \( n =1, \ldots , N-2\), we can also obtain
And specially, when \( n=N-1, N\), recalling the notation \(f_{N+1} = + \infty \), we have
and
By combining the above formulas, we obtain the following:
Case 1: When \(n=3,\ldots ,N-2\), (3.30) holds.
Case 2: When \(n=1, 2\), we obtain that
and
Case 3: When \(n=N-1, N\), we get that
and
Linear problem
We first set
and provide the following lemma in [13].
Lemma B.1
(Lemma 4.2 in [13]) For a given function \(\Phi _*(x, z)\in L^2({\mathcal {S}})\) with
let us consider the following problem
with the conditions
The problem (B.1)–(B.3) has a unique solution \(\phi _{*} \in H^2({\mathcal {S}})\). \(\square \)
Then, by using the above lemma, we can obtain following result:
Lemma B.2
For a given function \(\Phi ^*(x, z)\in L^2({\mathcal {S}})\) with
consider the following problem
with the conditions
There exists a unique solution \(\phi ^{*} \in H^2({\mathcal {S}})\) to problem (B.4)–(B.6), which satisfies
Proof
Let
Here, the map
is a diffeomorphism, where \( \hat{\ell } = \int _0^\ell \beta (r) {\mathrm {d}}r. \)
It is easy to derive that
while differentiation in x does not change. Therefore, problem (B.4)–(B.6) can be rewritten as
with the conditions
From Lemma B.1, we can know that problem (B.8)–(B.11) has a unique solution \(\tilde{\phi }^{*}(x, {\tilde{z}}) \). The result follows by transforming \(\tilde{\phi }^{*}(x, \iota (z) )\) into \( \phi ^{*}(x, z)\) via change of variables. By using the method of sub-supersolutions, we can get the estimate (B.7). This concludes the proof of the lemma. \(\square \)
Rights and permissions
About this article
Cite this article
Duan, L., Wei, S. & Yang, J. Clustering of boundary interfaces for an inhomogeneous Allen–Cahn equation on a smooth bounded domain. Calc. Var. 60, 70 (2021). https://doi.org/10.1007/s00526-020-01913-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-020-01913-3