An existence result for a nonconvex variational problem via regularity

Abstract

Abstract. Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations isobtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The x-dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.Mathematics Subject Classification.

A regularity result for a convex functional and bounds for the singular set

“…In the first step we assume that f (·, ξ) ∈ C 2 but we are able to establish the estimates (1.6) and (1.9) independently of the C 2 norm of the integrand f , by adopting an argument first used in [14]. In the second step we remove the assumption f (·, ξ) ∈ C 2 using an approximation procedure introduced in [14] and developed in [7,10,15]. More precisely we approximate f by a sequence {f h } of C 2 functions which are strictly elliptic (and the ellipticity constant is precisely the ν appearing in (H2)).…”

“…The following statement has been proved in [15]. It states that the condition of uniform convexity of the functional F is equivalent to the ellipticity condition for the Euler system of F .…”

Existence and regularity of minimizers of nonconvex integrals withp-qgrowth

“…A solution to this problem is to require stronger assumptions on L ensuring local Lipschitz continuity of minimizers of the relaxed problem, a strategy which has been successfully exploited in [2] and more recently in [15] for nonconvex, nonautonomous integrands L(x , ∇u). Moreover, once a minimizer u of the nonconvex problem (P) has been obtained from a local Lipschitz continuous minimizer v of the relaxed problem, then u itself has to be locally Lipschitz continuous as well as it is a minimizer of (1.2).…”

“…holds for every ξ and ζ (see [15]). Later on, this condition has been weakened by assuming the so-called puniform convexity condition at infinity which means that the previous condition holds only when the segment joining ξ and ζ lies entirely outside some fixed ball B R (0).…”

“…By approximating f with smooth functions, it is then possible to prove local Lipschitz regularity of local minimizers if L = f (ξ) or local α-Hölder continuity for all α < 1 in the general case, see e.g. [8,12,15]. As regards the regularity of minimizers of integrals with p − q growth, this was first studied by Marcellini in [24] and many contributions have been given since then, see [1,13] for a broad list of references.…”