Abstract
In this work we consider
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.
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Acknowledgements
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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Gao, Y., Liu, JG., Lu, X.Y. et al. Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface. Calc. Var. 57, 55 (2018). https://doi.org/10.1007/s00526-018-1326-x
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DOI: https://doi.org/10.1007/s00526-018-1326-x