Partial Regularity Under Anisotropic (p, q) Growth Conditions

Partial Regularity under Anisotropic (p,q) Growth Conditions

Regularity under Sharp Anisotropic General Growth Conditions

We prove boundedness of minimizers of energy-functionals, for instance of the anisotropic type (1.1) below, under sharp assumptions on the exponents pi in terms of p ∗: the Sobolev conjugate exponent of p; i.e., p∗ = np n−p , 1 p = 1 n Pn i=1 1 pi . As a consequence, by mean of regularity results due to Lieberman [21], we obtain the local Lipschitz-continuity of minimizers under sharp assumptio...

Partial Regularity For Higher Order Variational Problems Under Anisotropic Growth Conditions

We prove a partial regularity result for local minimizers u : Rn ⊃ Ω → RM of the variational integral J(u,Ω) = ∫ Ω f(∇ku) dx, where k is any integer and f is a strictly convex integrand of anisotropic (p, q)–growth with exponents satisfying the condition q < p(1 + 2 n). This is some extension of the regularity theorem obtained in [BF2] for the case n = 2.

Partial Regularity For Anisotropic Functionals of Higher Order

Higher order variational functionals, emerging in the study of problems from materials science and engineering, have attracted a great deal of attention in last few years (e.g. [4], see [5]). In particular, the regularity of minimizers of such functionals has been studied very recently. In [15] and [16] the partial Ck,α regularity has been established for quasiconvex integrals with a p-power gr...

Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p,q)-Growth

We consider autonomous integrals F [u] := ∫ Ω f (Du)dx for u : R ⊃ Ω →R in the multidimensional calculus of variations, where the integrand f is a strictly quasiconvex C2-function satisfying the (p,q)-growth conditions γ|A| ≤ f (A)≤ Γ (1+ |A|) for every A ∈R with exponents 1 < p ≤ q < ∞. We examine the Lebesgue-Serrin extension Floc[u] := inf { liminf k→∞ F[uk] : W 1,q loc ∋ uk −−−⇀ k→∞ u weakl...