The $ {L^\infty } $ estimate of the spatial gradient of the solution to a variational inequality problem originates from the financial contract problem with advanced implementation clauses.

The present study investigates a class of variational inequality problems under the framework of the parabolic Kirchhoff operator from the financial contract problem. This particular issue stems from the financial contract problem. By utilizing the energy inequality of the obtained solutions, the energy inequality of the solution gradients, and the Caffarelli–Kohn–Nirenberge inequality, an estimation of the infinite norm of the solution gradients is obtained. [ABSTRACT FROM AUTHOR]