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I will discuss aÌýdegenerateÌýform of theÌýspecialÌýLagrangianÌýequationÌýthat arises as the geodesicÌýequationÌýfor the space of positiveÌýLagrangians. Considering graphÌýLagrangiansÌýin Euclidean space, the equation reduces to a second order fully non-linear PDE for a single real function. I will explain how to prove existence and uniqueness for Lipschitz solutions to the Dirichlet problem on a convex domain times the unit interval. The proof uses the subequation theory of Harvey-Lawson. Existence of solutions on general manifolds with sufficient regularity would imply a version of the strong Arnold conjecture in Hamiltonian dynamics as well as uniqueness forÌýspecialÌýLagrangians. This talk is based on joint work with Yanir Rubinstein.