A RELAXATION RESULT FOR AN INHOMOGENEOUS FUNCTIONAL PRESERVING POINT-LIKE AND CURVE-LIKE SINGULARITIES IN IMAGE PROCESSING

In this paper we address a relaxation theorem for a new integral functional of the calculus of variations defined on the space of functions in [Formula: see text] whose gradient is an Lp-vector field with distributional divergence given by a Radon measure. The result holds for integrand of type f(x, Δu) without any coerciveness condition, with respect to the second variable, and C1-continuity assumptions with respect to the spatial variable x.

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

SynopsisUsing 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)) + β0(x, u) where β (resp β0) is a maximal monotone subdifferential on ℝN(resp ℝ) depending smoothly on x in a bounded domain Ω of ℝNThese operators are shown to be the subdifierentials over Lp(Ω) of convex functional of the following type:where j is a normal convex integrand over Ω×ℝN+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+.