Regularity results for mixed local and nonlocal double phase functionals

Abstract

We investigate the De Giorgi-Nash-Moser theory for minimizers of mixed local and nonlocal functionals modeled afterwhere 0 < s < 1 < p ≤ q and a(•) ≥ 0. In particular, we prove Hölder regularity and Harnack's inequality under possibly sharp assumptions on s, p, q and a(•).

Local behavior of the mixed local and nonlocal problems with nonstandard growth

“…To some extent, our results in this work offset the scenario 𝑝 ⩽ 𝑠𝑞 that repels the center assumptions on growth exponents 𝑝, 𝑞 in [10,21]. Note that the hypothesis required in (1.5) and the center assumption on growth exponents 𝑝, 𝑞 in [11,22] are mutually exclusive. Finally, let us mention that given the condition 𝑝 ≠ 𝑞, the functional (1.1) is closely related to a large number of anisotropic local or nonlocal problems with nonstandard growth, see, for instance, [2,3,10,18,23,26,35].…”

“…Lately, Byun-Lee-Song [11] further investigated the case that local term provides no regularizing effects, where the Hölder regularity and Harnack inequality were discussed for the minimizers to the functionals modeled by…”

Wolff potentials and measure data vectorial problems with Orlicz growth

“…Moreover, given any boundary data g ∈ W 1,p (R n ), we say that u is a weak solution to the problem (4), if u is a weak solution to (14) and, in addition, u − g ∈ X 1,p 0 (Ω).…”

“…Studies on regularity results for mixed local-nonlocal p-Laplacian type operators were initiated in the paper [36], where the De Giorgi-Nash-Moser theory was investigated by combining the techniques for the fractional p-Laplacian [28,29] with those for the classical p-Laplacian. We refer to [2,11,14,37,38,51] and references therein for further results concerning mixed local-nonlocal nonlinear problems. We would like to single out the paper [27] where maximal regularity, including (local) gradient Hölder regularity, was proved for a larger class of mixed problems related to −∆ p + (−∆ γ ) s with p, γ > 1 and s ∈ (0, 1) satisfying p > sγ.…”

Lavrentiev gap for some classes of generalized Orlicz functions

“…Mixed problems of p, s-Laplacian type were studied by De Filippis and Mingione [DM22]. Mixed nonlocal-local problems of the type II with the were considered by Byun, Lee, and Song [BLS23], where they obtained Hölder regularity results for such problems under the assumption s ∈ (0, 1) and 1 < p ≤ q.…”