Solving marginals of the LDP for the directed landscape

We prove the upper-tail Large Deviation Principle (LDP) for the parabolic Airy process and characterize the limit shape of the directed landscape under the upper-tail conditioning. The LDP result answers Conjecture 10.1 in Das, Dauvergne, and Vir\'{a}g (2024). The starting point of our proof is the metric-level LDP for the directed landscape from Das, Dauvergne, and Vir\'{a}g (2024) that reduces our work to solving a variational problem. Our proof is PDE-based and uses geometric arguments, connecting the variational problem to the weak solutions of Burgers' equation. Further, our method may generalize to the setting of the upper-tail LDP for the KPZ fixed point under the multi-wedge initial data, and we prove a decomposition result in this direction.