On an estimate related to the Hessian and application to an oblique derivative problem

We prove an estimate on the $L^2(\Omega)$-norm of the Hessian of a function $u \in W^{2,q}(\Omega)$, satisfying an oblique derivative type condition on the boundary, allowing the oblique axis to be tangential at a finite number of points of $\partial \Omega$. Using this inequality, the solvability in Sobolev spaces $W^{2,q}(\Omega)$, with $q$ closed to $2$, follows for a class of nonlinear differential equations in the plane with quadratic growth.