  The Helfrich-Hurault (HH) elastic instability is a well-known mechanism
behind patterns that form as a result of strain upon liquid crystal systems
with periodic ground states. In the HH model, layered structures undulate and
buckle in response to local, geometric incompatibilities, in order to maintain
the preferred layer spacing. Classic HH systems include cholesteric liquid
crystals under electromagnetic field distortions and smectic liquid crystals
under mechanical strains, where both materials are confined between rigid
substrates. However, richer phenomena are observed when undulation
instabilities occur in the presence of deformable interfaces and variable
boundary conditions. Understanding how the HH instability is affected by
deformable surfaces is imperative for applying the instability to a broader
range of materials. In this review, we re-examine the HH instability and give
special focus to how the boundary conditions influence the mechanical response
of lamellar systems to geometrical frustration. We use lamellar liquid crystals
confined within a spherical shell geometry as our model system. Made possible
by the relatively recent advances in microfluidics within the past 15 years,
liquid crystal shells are composed entirely of fluid interfaces and have
boundary conditions that can be dynamically controlled at will. We examine past
and recent work that exemplifies how topological constraints, molecular
anchoring conditions, and boundary curvature can trigger the HH instability in
liquid crystals with periodic ground states. We then end by identifying similar
phenomena across a wide variety of materials, both biological and synthetic.
With this review, we aim to highlight that the HH instability is a generic and
often overlooked response of periodic materials to geometrical frustration.
