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Capillary drops on a rough surface. (English) Zbl 1300.76014
Summary: We study liquid drops lying on a rough planar surface. The drops are minimizers of an energy functional that includes a random adhesion energy. We prove the existence of minimizers and the regularity of the free boundary. When the length scale of the randomly varying surface is small, we show that minimizers are close to spherical caps which are minimizers of an averaged energy functional. In particular, we give an error estimate that is algebraic in the scale parameter and holds with high probability.

MSC:
76D45 Capillarity (surface tension) for incompressible viscous fluids
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35R60 PDEs with randomness, stochastic partial differential equations
35R35 Free boundary problems for PDEs
49Q10 Optimization of shapes other than minimal surfaces
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