Boundary asymptotics for a class of degenerate linearized Monge-Ampère equations

In this paper, we study a class of linearized Monge–Ampère equations whose coefficient matrix is generated by a degenerate Monge–Ampère Dirichlet problem. The right-hand side is allowed to have a boundary singularity of the form ℎ ⁡ ( 𝑥 ) ⁢ 𝑑 ⁢ ( 𝑥 ) − 𝑞 , where 𝑑 ⁡ ( 𝑥 ) denotes the distance to the boundary. Assuming that a classical solution exists, we derive boundary asymptotic estimates and show that different values of q lead to different boundary behaviors. In particular, the boundary asymptotic behavior is governed by the distance function 𝑑 ⁡ ( 𝑥 ) and the singularity exponent q , rather than by the degeneracy exponent of the coefficient matrix. We further establish weighted energy estimates and a corresponding 𝑊 1 , 2 -regularity result for a suitable power of the solution in the singular regime 1 < 𝑞 < 2 . Finally, we prove the non-existence of classical solutions in the strongly singular case.