Besov regularity for a class of elliptic obstacle problems with double-phase Orlicz growth

Abstract

In this paper, we consider local Besov regularity of to the elliptic variational inequality with double-phase Orlicz growth:∫Ω〈A(x,Du),D(u−φ)〉dx≤∫Ω〈B(x,F),D(u−φ)〉dx∀φ∈Kψ(Ω), where Kψ(Ω) is an admissible set constrained by an obstacle function ψ(x). Here, each of nonlinearities A and B is supposed to be (Φ1,Φ2) double-phase with Orlicz type growth, respectively, under the assumption supt∈R+⁡Φ2(t)Φ1(t)+Φ11+α/n(t)<+∞ for a Hölder exponent α∈(0,1] of the coefficient a(x). We prove that a fractional differentiability of Du is reflected by extra differentiability assumption on the obstacle function and the external force, under a suitable regularity on the coefficient 0≤a(x)∈C0,α(Ω). This is an extension of recent work [44] from the elliptic problems of (p,q) growth to the setting of double phase Orlicz growth.