Irregular Double Obstacle Problems with Orlicz Growth
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Irregular Double Obstacle Problems with Orlicz Growth Sun-Sig Byun1 · Shuang Liang2 · Jihoon Ok3
© Mathematica Josephina, Inc. 2020
Abstract We study an irregular double obstacle problem with Orlicz growth over a nonsmooth bounded domain. We establish a global Calderón–Zygmund estimate by proving that the gradient of the solution to such a nonlinear elliptic problem is as integrable as both the nonhomogeneous term in divergence form and the gradient of the associated double obstacles. We also investigate minimal regularity requirements on the relevant nonlinear operator for the desired regularity estimate. Keywords Nonlinear elliptic equation · Orlicz growth · Double obstacles · Calderón–Zygmund estimate Mathematics Subject Classification 35J60 · 35B65
1 Introduction In this paper, we study a nonlinear elliptic double obstacle problem with Orlicz growth. Let ⊂ Rn , n ≥ 2, be a bounded open domain. Given two functions ψ1 , ψ2 ∈ W 1,G () belonging to the Orlicz–Sobolev space W 1,G () that will be specified later in Sect. 2 with an N-function G defined in (1.5), let us suppose that these double obstacles ψ1 and ψ2 also satisfy that ψ1 ≤ ψ2 a.e. in and ψ1 ≤ 0 ≤ ψ2 on ∂
B B
Sun-Sig Byun [email protected] Jihoon Ok [email protected] Shuang Liang [email protected]
1
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
2
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China
3
Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, Yongin 17104, Republic of Korea
123
S.-S. Byun et al.
in the sense that max{ψ1 , 0}, min{ψ2 , 0} ∈ W01,G (). We then consider a convex admissible set A0 () = ϕ ∈ W01,G () : ψ1 ≤ ϕ ≤ ψ2 a.e. in . We deal with a weak solution u ∈ A0 () to the double obstacle problem which means that it satisfies the variational inequality ˆ
ˆ A(x, Du) · D(u − ϕ) dx ≤
H (x, F) · D(u − ϕ) dx
(1.1)
for all ϕ ∈ A0 (). Here the nonlinearity A = A(x, ξ ) : Rn ×Rn → Rn is measurable for each ξ ∈ Rn and differentiable for almost every x ∈ Rn , and there exist constants 0 < ν ≤ 1 ≤ L < ∞ such that for all x, ξ, η ∈ Rn ,
Dξ A(x, ξ )η · η ≥ νg (|ξ |)|η|2 , |A(x, ξ )| + |ξ ||Dξ A(x, ξ )| ≤ Lg(|ξ |),
(1.2)
where Dξ denotes the differentiation in ξ . In addition, the term H (x, ξ ) : Rn × Rn → Rn satisfies |H (x, ξ )| ≤ Lg(|ξ |)
x, ξ ∈ Rn ,
(1.3)
where g(t) : [0, +∞) → [0, +∞) has the following properties: ⎧ ⎪ ⎨ g(0) = 0 if and only if t = 0, g ∈ C 1 (R+ ), ⎪ ⎩ i g ≤ tg (t) ≤ sg for every t > 0, where 0 < i g ≤ 1 ≤ sg < ∞. g(t)
(1.4)
We also define ˆ G = G(t) =
t
g(s) ds for t ≥ 0.
(1.5)
0
The existence and uniqueness of weak solution to the variational inequality (1.1) has been proved in [32]. In fact, if A(x, ξ ) = a(x)g(|ξ |) |ξξ | with 0 < ν ≤ a(·) ≤ L and F ≡ 0, then the unique weak solution to the corresponding variational problem (1.1) is a unique minimizer of the following functional ˆ A0 () v
→
a(x)G(|Dv|
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