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In this article, we develop the formalism for singular hypersurfaces andjunction conditions in generalized coupling theories using a variationalapproach. We then employ this formalism to examine the behavior of sharp matterdensity gradients in generalized coupling theories. We find that such gradientsdo not necessarily lead to the pathologies present in other theories of gravitywith auxiliary fields. A detailed example, based on a simple instance of ageneralized coupling theory called the MEMe model, is also provided. In thestatic case, we show that sharp boundaries do not generate singularities in thedynamical frame despite the presence of an auxiliary field. Instead, in thecase of a collapsing spherical density distribution with a general profile anadditional force compresses over-densities and expands underdensities. Theseresults can also be used to deduce additional constraints on the parameter ofthis model.

Junction conditions and sharp gradients in generalized coupling theories