Nash-Moser techniques for nonlinear boundary-value problems

Abstract

A new linearization method is introduced for smooth short-time solvability of initial boundary value problems for nonlinear evolution equations. The technique based on an inverse function theorem of Nash-Moser type is illustrated by an application in the parabolic case. The equation and the boundary conditions may depend fully nonlinearly on time and space variables. The necessary compatibility conditions are transformed using a Borel's theorem. A general trace theorem for normal boundary conditions is proved in spaces of smooth functions by applying tame splitting theory in Frechet spaces. The linearized parabolic problem is treated using maximal regularity in analytic semigroup theory, higher order elliptic a priori estimates and simultaneous continuity in trace theorems in Sobolev spaces.