Local $L^{\mathrm{\infty }}$ norm estimates for the gradient solutions of variational inequalities arising from the mortgage problems

This paper investigates local estimates for the spatial gradient of solutions to variational inequalities within the framework of a parabolic Kirchhoff operator, which arises from mortgage problems. By utilizing the integral inequality for the gradient of the solutions derived in this study, together with the Caffarelli–Kohn–Nirenberg inequality, we establish an $L^{\mathrm{\infty }}$ norm estimate for the gradient of the solution in a local cylindrical region. This $L^{\mathrm{\infty }}$ estimate is formulated in terms of the $L^p$ norm of the solution.