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We prove a strong law of large numbers for directed last passage times in anindependent but inhomogeneous exponential environment. Rates for theexponential random variables are obtained from a discretisation of a speedfunction that may be discontinuous on a locally finite set of discontinuitycurves. The limiting shape is cast as a variational formula that maximises acertain functional over a set of weakly increasing curves.  Using this result, we present two examples that allow for partial analyticaltractability and show that the shape function may not be strictly concave, andit may exhibit points of non-differentiability, flat segments, andnon-uniqueness of the optimisers of the variational formula. Finally, in aspecific example, we analyse further the macroscopic optimisers and uncover aphase transition for their behaviour.

Last passage percolation in an exponential environment with discontinuous rates