Regularity of ω-minimizers of quasi-convex variational integrals with polynomial growth

Boundary regularity for almost minimizers of quasiconvex variational problems

“…Evans interior regularity result was generalized by Duzaar, Gastel and Grotowski [4] (the case p = 2) and by Duzaar and the current author [5] (the case p ≥ 2) to almost minimizers of the functional (1) imposing some additional (mild) assumptions on ω, first of all that the function ω satisfies Dini's condition (for √ ω) ω(ρ) := ρ 0 ω(r) r dr < ∞ for some ρ > 0.…”

“…In particular, it was shown in [4] and [5] that u is of class C 1 outside a relatively closed singular subset of vanishing measure and the derivative Du of u has the modulus of continuity ρ → ρ α + ω(ρ).…”

Partial regularity of minimizers of higher order integrals with (p,q)-growth

“…This result was extended to the subquadratic case (1 < p < 2) in [11]. Further papers concerning this topic are for example [2,5,13,15,16,23,26] and also [31] investigating the Hausdorff-dimension of the singular set. For higher order functionals partial regularity results have been shown in [27,32] (see also [45]).…”

A Simple Partial Regularity Proof for Minimizers of Variational Integrals

“…where c depends only on p. In the remainder of this section we adapt essentially the arguments of [5,6,8,9]. We will use Lemma 4.3 and Lemma 5.1 to derive decay estimates for the excess of the minimizer.…”

“…Typically, these regularity results yield partial C 1,α -regularity, where α is the Hölder exponent of A in (x, y). In contrast, working directly with a minimizer (or almost minimizer) of (1.1) instead of a solution of a system, one typically gets only half the Hölder exponent of f in (x, y) as in Corollary 1.2 and [5,8,25,28]. This clarifies the meaning of the condition (1.9): It interpolates -in some sense -between these two situations.…”