Application des méthodes de convexité et monotonie a l'étude de certaines équations quasi linéaires

Abstract

Synopsis Using Hilbert space methods, existence and uniqueness are proved for the solution of some strongly non-linear partial differential equations of elliptic and parabolic type. They are associated with quasi-linear operators of the form: -div(β( x , grad u )) + β 0 ( x, u ) where β (resp β 0 ) is a maximal monotone subdifferential on ℝ N (resp ℝ) depending smoothly on x in a bounded domain Ω of ℝ N These operators are shown to be the subdifierentials over L p (Ω) of convex functional of the following type: where j is a normal convex integrand over Ω×ℝ N+1 satisfying a coerciveness condition. This method avoids the theory of Sobolev-Orlicz spaces. An application is given also forthe gas-diffusion equation over ℝ + .