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Abstract

We study a free boundary problem on the lattice whose scaling limit is a harmonic free boundary problem with a discontinuous Hamiltonian. We find an explicit formula for the Hamiltonian, prove that the solutions are unique, and prove that the limiting free boundary has a facets in every rational direction. Our choice of problem presents difficulties that require the development of a new uniqueness proof for certain free boundary problems. The problem is motivated by physical experiments involving liquid drops on patterned solid surfaces.

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Acknowledgements

Both authors benefited from conversations with Hayk Aleksanyan and Henrik Shahgholian.

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Correspondence to William M. Feldman.

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Conflict of interest

The first author was partially supported by the National Science Foundation RTG Grant DMS-1246999. The second author was partially supported by the National Science Foundation DMS-1606670 and the Alfred P Sloan foundation.

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Communicated by D. Kinderlehrer

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Feldman, W.M., Smart, C.K. A Free Boundary Problem with Facets. Arch Rational Mech Anal 232, 389–435 (2019). https://doi.org/10.1007/s00205-018-1323-4

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  • DOI: https://doi.org/10.1007/s00205-018-1323-4

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