Abstract
In this paper, we apply De Giorgi-Moser iteration to establish the Hölder regularity of quasiminimizers to generalized Orlicz functional on the Heisenberg group by using the Riesz potential, maximal function, Calderón-Zygmund decomposition, and covering Lemma on the context of the Heisenberg Group. The functional includes the p -Laplace functional on the Heisenberg group which has been studied and the variable exponential functional and the double phase growth functional on the Heisenberg group that have not been studied.
Highlights
In this paper, we concern the generalized Orlicz functional ðΦðx, j∇HujÞdx: ð1ÞΩ on the Heisenberg group, whereΦðx, j∇HujÞ ~ φðx, j∇HujÞ ∈ ΦwðΩÞ ð2Þ, ∇Hu = ðX1u, X2u,⋯,X2nuÞ, ΦwðΩÞ, is the generalized Orlicz space
While 0 ≤ Λ < 1, for p not being far from 2, Domokos and Manfredi in [5] used the Calderón-Zygmund theory on the Heisenberg group to study regularity of weak solutions; for 2 ≤ p < 4, Mingione, Zatorska-Goldstein, and Zhong in [6] concluded the C1,α regularity of weak solutions by using a double-bootstrap method, energy estimates, and interpolation inequalities; for 1 < p < ∞, Zhong in [7] got the C1,α regularity of weak solutions by using the energy estimate, the Moser iteration, and the oscillation estimate; Zhang and Niu in [8] proved the Γα regularity of the gradient of weak solutions as Φðx, τÞ ~ φðτÞ, where φðτÞ belongs to the Orlicz space including the function φðτÞ = ðΛ + τ2Þp−2/2τ
We assume that Ω ⊂ Hn is a bounded domain, Q is a cube whose side length is R in the x′ direction, R2 in the t direction, and its edge is parallel to the coordinate axis and denote diamQ ≔ ðð2nÞ2 + 1Þ1/4R > ð2nÞ1/2R
Summary
While 0 ≤ Λ < 1, for p not being far from 2, Domokos and Manfredi in [5] used the Calderón-Zygmund theory on the Heisenberg group to study regularity of weak solutions; for 2 ≤ p < 4, Mingione, Zatorska-Goldstein, and Zhong in [6] concluded the C1,α regularity of weak solutions by using a double-bootstrap method, energy estimates, and interpolation inequalities; for 1 < p < ∞, Zhong in [7] got the C1,α regularity of weak solutions by using the energy estimate, the Moser iteration, and the oscillation estimate; Zhang and Niu in [8] proved the Γα regularity of the gradient of weak solutions as Φðx, τÞ ~ φðτÞ, where φðτÞ belongs to the Orlicz space including the function φðτÞ = ðΛ + τ2Þp−2/2τ.
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