Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface

  • Yuan GaoEmail author
  • Jian-Guo Liu
  • Xin Yang Lu
  • Xiangsheng Xu


In this work we consider
$$\begin{aligned} w_t=\left[ \left( w_{hh}+c_0\right) ^{-3}\right] _{hh},\qquad w(0)=w^0, \end{aligned}$$
which is derived from a thin film equation for epitaxial growth on vicinal surface. We formulate the problem as the gradient flow of a suitably-defined convex functional in a non-reflexive space. Then by restricting it to a Hilbert space and proving the uniqueness of its sub-differential, we can apply the classical maximal monotone operator theory. The mathematical difficulty is due to the fact that \(w_{hh}\) can appear as a positive Radon measure. We prove the existence of a global strong solution with hidden singularity. In particular, (1) holds almost everywhere when \(w_{hh}\) is replaced by its absolutely continuous part.

Mathematics Subject Classification

35D99 35K65 35R06 49J40 47J05 47H05 



We would like to thank the support by the National Science Foundation under Grant No. DMS-1514826 and KI-Net RNMS11-07444. We thank Jianfeng Lu for helpful discussions. Part of this work was carried out when Xin Yang Lu was affiliated with McGill University. Xin Yang Lu acknowledges the support of Lakehead Funding 1466319.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Yuan Gao
    • 1
    Email author
  • Jian-Guo Liu
    • 2
  • Xin Yang Lu
    • 3
  • Xiangsheng Xu
    • 4
  1. 1.Department of MathematicsHong Kong University of Science and TechnologyKowloonHong Kong
  2. 2.Department of Mathematics and Department of PhysicsDuke UniversityDurhamUSA
  3. 3.Department of Mathematical SciencesLakehead UniversityThunder BayCanada
  4. 4.Department of Mathematics and StatisticsMississippi State UniversityMississippi StateUSA

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