A class of Orlicz-Sobolev spaces with applications to variational problems involving nonlinear Hill's equations

Abstract

Necessary and sufficient conditions are proved for a b (2)-Young function G (with independent variable t) to be convex (resp. concave) in t 2 in terms of inequalities between the second derivative of G and the first derivative of its Legendre transform G̃ (with independent variable s). It is then proven that a Young function G is convex (resp. concave) in t 2 if and only if G̃ is concave (resp. convex) in s 2. These results, along with another set of inequalities for functions G convex (resp. concave) in t 2, allow the proof of the uniform convexity and thereby of the reflexivity with respect to Luxemburg's norm ∥f∥ G = inf{k > 0: ∝ Ω dξ G(f (ξ) k ) ⩽ 1} of the Orlicz space L G(Ω) over an open domain Ω ∋ R N with Lebesgue measure dξ. When applied to G(t) = ¦t¦ p p and G ̃ (s) = ¦s¦ p′ p′ with p −1 + ( p′) −1 = 1, the preceding results lead to the shortest proof to date of two Clarkson's inequalities and of the reflexivity of L p -spaces for 1 < p < +∞. Finally, some of these results are used to solve by direct methods variational problems associated with the existence question of periodic orbits for a class of nonlinear Hill's equations; these variational problems are formulated on suitable Orlicz-Sobolev spaces W mL G(Ω) and thereby allow for nonlinear terms which may grow faster than any power of the variable.