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

Abstract

We prove an estimate on the L2(Ω) -norm of the Hessian of a function u ∈ W2,q(Ω) , satisfying an oblique derivative type condition on the boundary, allowing the oblique axis to be tangential at a finite number of points of ∂Ω . Using this inequality, the solvability in Sobolev spaces W2,q(Ω) , with q close to 2 , follows for a class of nonlinear differential equations in the plane with quadratic growth. Mathematics subject classification (2000): 35J65, 35R05.