Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 54, pp. 1-33.
Title: Nash-Moser techniques for nonlinear boundary-value problems
Author: Markus Poppenberg (Univ. Dortmund, Germany)
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.
Submitted July 14, 2002. Published May 5, 2003.
Math Subject Classifications: 35K60, 58C15, 35K30.
Key Words: Nash-Moser; inverse function theorem;
boundary-value problem; parabolic; analytic semigroup;
evolution system; maximal regularity; trace theorem.