Semigeostrophic equations in physical space with free upper boundary

We define Lagrangian solutions in physical space for 3-d incompressible semigeostrophic system with free upper boundary under various conditions for initial data, then prove their existence via the minimization with respect to a geostrophic functional, generalizing the results of Cullen and Feldman (J Math Anal 37(5): 1371–1395, 2006) and Feldman and Tudorascu (Arch Ration Mech Anal 218(1): 527–551 2015) to the situation of free upper boundary. As a byproduct of our proof, we obtain the existence of measure-valued dual space solutions when the initial measure ν₀∈ P₂(R³) and is supported on {-1δ≤y3≤-δ}.