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.

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

#### Abstract

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.
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Hessian estimate; nonlinear elliptic equations; tangential oblique derivative problem; strong solutions
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12318/3720
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