C1,α-solutions to non-autonomous anisotropic variational problems

Abstract

We establish several smoothness results for local minimizers of nonautonomous variational integrals with anisotropic growth conditions.

Variational integrals with a wide range of anisotropy

“…From [LU] (see the discussion in Remark 2.3 of [BF4]) we deduce that u δ is in the space W 2 t,loc (B R (x 0 )) for any t < ∞, therefore we may test the differentiated Euler equation valid for u δ with the function η 2 ∂ γ u δ Γ β/2 δ , where β ≥ 0, η ∈ C ∞ 0 (B R (x 0 )) and γ runs from 1 to n − 1. Since we consider the scalar case, it is easy to check that (from now on summation w.r.t.…”

“…Having established (4.8), the proof of C 1,α -regularity can be obtained following for example [Bi], proof of Theorem 5.22, or [BF4] Lemma 2.9, where it is shown that from (4.8) we can deduce ∇u δ L ∞ (Bρ(x 0 )) ≤ c(ρ) < ∞. Uniform Hölder continuity of ∇u δ then follows as outlined in [BF4], end of Section 2.1.…”

“…Uniform Hölder continuity of ∇u δ then follows as outlined in [BF4], end of Section 2.1. Alternatively we may quote [LU], Chap.4, Sec.6, or [Ma1], Theorem D, as references for the step from Lipschitz regularity to Hölder continuity of the gradient.…”

“…ii) We may also consider functionals as discussed above with an additional x-dependence (compare [BF4]). Note that in the situation at hand we do not have to expect a Lavrentiev phenomenon.…”

Local boundedness of quasi-minimizers of integral functionals with variable exponent anisotropic growth and applications

“…The regularity for variational problems and differential equations with nonstandard growth has been studied extensively and many interesting results have been obtained, for example, we refer to [8,12,34] for general (p, q) growth case, refer to [1,[4][5][6][19][20][21][30][31][32][33] for − → p growth case and refer to [2,9,10,[13][14][15][16][17]25,41,44,45] for p(x) growth case. To our knowledge, the regularity for the − → p (x) growth case has not yet been studied specially and systematically.…”