Abstract
In this paper, we study the evolution equation derived by Xu and Xiang (SIAM J Appl Math 69(5):1393–1414, 2009) to describe heteroepitaxial growth in \(2+1\) dimensions with elastic forces on vicinal surfaces is in the radial case and uniform mobility. This equation is strongly nonlinear and contains two elliptic integrals and defined via Cauchy principal value. We will first derive a formally equivalent parabolic evolution equation (i.e., full equivalence when sufficient regularity is assumed), and the main aim is to prove existence, uniqueness and regularity of strong solutions. We will extensively use techniques from the theory of evolution equations governed by maximal monotone operators in Banach spaces.
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Auchmuty, G.: Divergence \(L^2\)-coercivity inequalities. Numer. Funct. Anal. Opt. 27(5–6), 499–515 (2006)
Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoof, Leydon (1976)
Browder, F.: Problèmes Nonlinéaires, Les Presses de l’Université de Montréal. (1966)
Dal Maso, G., Fonseca, I., Leoni, G.: Analytical validation of a continuum model for epitaxial growth with elasticity on vicinal surfaces. Arch. Ration. Mech. Anal. 212, 1037–1064 (2014)
Duport, C., Politi, P., Villain, J.: Growth instabilities induced by elasticity in a vicinal surface. Journal de Physique I 1(5), 1317–1350 (1995)
Fonseca, I., Leoni, G., Lu, X.Y.: Regularity in time for weak solutions of a continuum model for epitaxial growth with elasticity on vicinal surfaces. Commun. Partial Differ. Equ. 40(10), 1942–1957 (2015)
Gao, Y., Liu, J.-G., Lu, J.: Continuum limit of a mesoscopic model of step motion on vicinal surfaces. Preprint arXiv:1606.08060 (2016)
Kačur, J.: Method of Rothe in Evolution Equations. Teubner Verlaggesellschaft, Leipzig (1985)
Minty, G.: Monotone (nonlinear) operators in Hilbert spaces. Duke Math. J. 29, 341–346 (1962)
Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5(1), 286–292 (1960)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
Suryanarayana, M.B.: Monotonicity and upper semicontinuity. Bull. Am. Math. Soc. 82(6), 936–938 (1976)
Tersoff, J., Phang, Y.H., Zhang, Z., Lagally, M.G.: Step-bunching instability of vicinal surfaces under stress. Phys. Rev. Lett. 75, 2730–2733 (1995)
Xiang, Y.: Derivation of a continuum model for epitaxial growth with elasticity on vicinal surface. SIAM J. Appl. Math. 63, 241–258 (2002)
Xiang, Y., Weinan, E.: Misfit elastic energy and a continuum model for epitaxial growth with elasticity on vicinal surfaces. Phys. Rev. B 69, 035409-1–035409-16 (2004)
Xu, H., Xiang, Y.: Derivation of a continuum model for the long-range elastic interaction on stepped epitaxial surfaces in 2+1 dimensions. SIAM J. Appl. Math. 69(5), 1393–1414 (2009)
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The author warmly thanks Xu Xiang for useful comments and suggestions.
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Communicated by Irene Fonseca.
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Lu, X.Y. On the Solutions of a \(2+1\)-Dimensional Model for Epitaxial Growth with Axial Symmetry. J Nonlinear Sci 28, 807–831 (2018). https://doi.org/10.1007/s00332-017-9428-8
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DOI: https://doi.org/10.1007/s00332-017-9428-8