Local Lipschitz Regularity of Minima for a Scalar Problem of the Calculus of Variations

Abstract

We consider a functional I(u) = ∫Ωf(∇ u(x)) dx on u0 + W1,1(Ω). Under the assumption that f is just convex, we prove a new Comparison Principle, we improve and give a short proof of Cellina's Comparison result for a new class of minimizers. We then extend a local Lipschitz regularity result obtained recently by Clarke for a wider class of functions f and boundary data u0 satisfying a new one-sided Bounded Slope Condition. A relaxation result follows.

The lack of strict convexity and the validity of the Comparison Principle for a simple class of minimizers

“…The arguments of the proof of Theorem 1 are similar to those used in [12], [14]; we just stress here how the less restrictive conditions (5)- (6) are used to obtain the conclusion.…”

“…The function h ± p,x 0 were first defined in [4] and then thoroughly studied in [14]. In the case where L is strictly convex the set ∂L * (p) is reduced to a point: in this case the functions h ± p,x 0 are nothing more than affine.…”

“…It is shown, under the assumption that Ω is convex and x 0 / ∈ Ω in [4], just assuming that x 0 is the limit of points out of the closure of Ω in [14], that h…”

“…It was first proven in [4], under the assumption that Ω be convex and x 0 / ∈ Ω, later in [14] by just assuming that x 0 is a limit of points that do not belong to the closure of Ω, that h …”

Hölder continuity of solutions to a basic problem in the calculus of variations

“…Si ces conditions ne sont pas satisfaites, est-ce que la seule continuité de φ implique la continuité des minimiseurs ? La réponse est donnée par le théorème suivant, valable pour des lagrangiens convexes mais pas nécessairement strictement convexes (dans ce cas, pour un lagrangien superlinéaire, on peut montrer [16] …”