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This paper considers the Alt-Caffarelli free boundary problem in a periodicmedium. This is a convenient model for several interesting phenomena appearingin the study of contact lines on rough surfaces, pinning, hysteresis and theformation of facets. We show the existence of an interval of effective pinnedslopes at each direction $e \in S^{d-1}$. In $d=2$ we characterize theregularity properties of the pinning interval in terms of the normal direction,including possible discontinuities at rational directions. These resultsrequire a careful study of the families of plane-like solutions available witha given slope. Using the same techniques we also obtain strong, in some casesoptimal, bounds on the class of limit shapes of local minimizers in $d=2$, andpreliminary results in $d \geq 3$.

Limit shapes of local minimizers for the Alt-Caffarelli energy functional in inhomogeneous media