Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications (Q1288869)

The author deals with a fully nonlinear second-order, possibly degenerate, elliptic equation (1) \(F(x,u,Du,D^2u)=0\) in \(\Omega\) in a smooth bounded domain \(\Omega\subset \mathbb{R}^N\) with fully nonlinear Neumann boundary conditions (2) \(L(x,u,Du)=0\) on \(\partial\Omega\). Here \(F\) and \(L\) are real continuous functions of all the variables \(x\in\overline\Omega\), \(u\subset\mathbb{R}\), \(Du\) and \(D^2u\) are the gradient and the Hessian matrix, respectively. The author proves comparison results between viscosity sub- and supersolutions of the elliptic problem (1)--(2) as well as of the parabolic equation \(u_t+ F(x,t,u, Du,D^2u)=0\) in \(\Omega\times (0,T)\) with the initial condition \(u(x,0)=u_0(x)\) on \(\overline\Omega\) and the boundary condition (2) or \(u_t+L (x,t,u,Du)=0\) on \(\partial\Omega \times(0,T)\). The comparison results have been obtained both for the case of a standard and a degenerate equation. Later on, they are applied by proving the existence and uniqueness of solutions for boundary value problems and extending the so-called level set approach for equations set in bounded domains with nonlinear Neumann boundary conditions.