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

GIUFFRE', Sofia
2005

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.
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: http://hdl.handle.net/20.500.12318/3720
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