Regularity for general functionals with double phase

Abstract

We prove sharp regularity results for a general class of functionals of the typefeaturing non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integralThis changes its ellipticity rate according to the geometry of the level set {a(x) = 0} of the modulating coefficient a(·). We also present new methods and proofs, that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon.

Lipschitz Bounds and Nonuniform Ellipticity

“…For instance, in [45] the bound (1.14) q p < 1 C 2 n is shown to guarantee the local Lipschitz continuity of minimizers of F when p 2, D 1, and f 2 L 1 . In the uniformly elliptic case p D q the assumptions in (1.12) coincide with the classical ones considered by Ladyzhenskaya and Ural 0 tseva [37], otherwise they are known as .p; q/-growth conditions and have been the object of intensive investigation; see, for instance, [2,9,10,18,19,26,38,44,45,48,58]. As it is natural, conditions as (1.14) also play a role in our setting, as shown in the following: THEOREM 1.2 (Scalar .p; q/-estimates).…”

Nonlinear obstacle problems with double phase in the borderline case

“…Indeed, the required regularity of the modulating function in directly links to the gap between p and q , as proved in . We refer to for the recent improvements on the regularity theory for double phase problems. The energy functional of the model equation of is The correct condition on is the log‐Hölder continuity which is exactly dual to the size of the phase transition.…”

“…Indeed, the required regularity of the modulating function (⋅) in (1.2) directly links to the gap between and , as proved in [3,27,30]. We refer to [4,5,12,[17][18][19] for the recent improvements on the regularity theory for double phase problems. The energy functional of the model equation of (1.3) is…”

Double phase image restoration

“…This condition has proved to be of central importance when considering bounded solutions, cf. [6,16,32]. In this sense, the assumption in Lemma 3.5 is probably essentially sharp.…”