Regularity for multi-phase variational problems

Abstract

We prove C 1,ν regularity for local minimizers of the multi-phase energy:under sharp assumptions relating the couples (p, q) and (p, s) to the Hölder exponents of the modulating coefficients a(·) and b(·), respectively.

Regularity for minimizers for functionals of double phase with variable exponents

“…For y ∈ B T , we write Ω r := B r (y) ∩ B + T . Then, we have the following proposition on the higher integrability near the boundary, independently proved in [58,Lemma 5] , see also [59,Lemma 5] for the manifold constrained case. Let R be a positive constant satisfying R ≤ (T − S)/2.…”

Removable sets in non-uniformly elliptic problems

“…which turn out to be useful in dealing with the regularity theory for double/multi phase functionals, see [3,11,12,15,16]. In the following, we will also consider the double phase integrand…”

“…They appeared originally in the context of homogenization and the Lavrentiev phenomenon [52]. Recently, regularity theory in this setting is getting increasing attention [2,3,11,12,16,17,49], see also [13][14][15] for the manifold-constrained case and [44] for a reasonable survey on older results. The growth of the operator we investigate is trapped between two power-type functions following the ideas of [39,40].…”

Optimal Lipschitz criteria and local estimates for non-uniformly elliptic problems

“…Indeed, while considering the ratio in (2) as the coefficient a(x) is allowed to be zero. For such energies, ad hoc methods can be developed; see for instance [2,4,15,14,16,20]. Another class of relevant non-autonomous functionals is the one involving a variable growth exponent.…”