A Green's formula for quasilinear elliptic operators
The authors consider the Neumann problem (with conormal derivative given) for a quasilinear second order elliptic equation \(Au=f\) in \(\Omega\) in divergence form. Here \(\Omega\) is a bounded domain with Lipschitz boundary. If A is linear it is well known that once \(Au\in L^ 2(\Omega)\) and \(u\in W^{1,2}(\Omega)\), then \(\partial u/\partial n_ A\) is well defined in \(W^{-1/2,2}\). For quasilinear A, for \(u\in W^{1,p}(\Omega)\) and \(Au\in L^ 1(\Omega)\) (resp. \(Au\in L^ q(\Omega))\) the authors derive the natural spaces in which \(\partial u/\partial n_ A\) lies and prove a Green's formula.
scientific article; zbMATH DE number 4154997
scientific article