Global integrability for solutions to some anisotropic problem with nonstandard growth.

Gao, Hongya; Jia, Miaomiao

This paper deals with the problem u ∈ 𝒦 u * , ψ ⁢ (Ω) , <graphic></graphic> \displaystyle u\in{\cal K}_{u_{*},\psi}(\Omega), ∀ v ∈ 𝒦 u * , ψ ⁢ (Ω) : ∫ Ω ∑ i = 1 n [ a i ⁢ (x , D ⁢ u) - f i ] ⁢ D i ⁢ (u - v) ⁢ d ⁢ x ⩽ ∫ Ω f ⁢ (u - v) ⁢ 𝑑 x , <graphic></graphic> \displaystyle\forall v\in{\cal K}_{u_{*},\psi}(\Omega):\int_{\Omega}\sum_{i=1}% ^{n}[a_{i}(x,Du)-f^{i}]D_{i}(u-v)\,dx\leqslant\int_{\Omega}f(u-v)\,dx, where { 𝒦 u * , ψ ⁢ (Ω) = { v ∈ u * + W 0 1 , (p i) ⁢ (Ω) : ∑ i = 1 n a i ⁢ (x , D ⁢ u) ⁢ D i ⁢ v ∈ L 1 ⁢ (Ω) ⁢ and ⁢ v ⩾ ψ , a.e. ⁢ Ω } , u * ∈ W 1 , (p i) ⁢ (Ω) , θ = max ⁡ { u * , ψ } ∈ u * + W 0 1 , (p i) ⁢ (Ω) , f ∈ L (p ¯ *) ′ ⁢ (Ω) , f i ∈ L p i ′ ⁢ (Ω) , i = 1 , … , n , <graphic></graphic> \left\{\begin{aligned} &\displaystyle{\cal K}_{u_{*},\psi}(\Omega)=\biggl{\{}v% \in u_{*}+W_{0}^{1,(p_{i})}(\Omega):\sum_{i=1}^{n}a_{i}(x,Du)D_{i}v\in L^{1}(% \Omega)\text{ and }v\geqslant\psi,\text{ a.e. }\Omega\biggr{\}},\\ &\displaystyle u_{*}\in W^{1,(p_{i})}(\Omega),\quad\theta=\max\{u_{*},\psi\}% \in u_{*}+W_{0}^{1,(p_{i})}(\Omega),\\ &\displaystyle f\in L^{(\bar{p}^{*})^{\prime}}(\Omega),\quad f^{i}\in L^{p_{i}% ^{\prime}}(\Omega),\,i=1,\dots,n,\end{aligned}\right. and the Carathéodory functions a i : Ω × ℝ n → ℝ {a_{i}:\Omega\times{\mathbb{R}}^{n}\to{\mathbb{R}}} , i = 1 , … , n {i=1,\dots,n} , satisfy some coercivity condition. We assume that the function θ = max ⁡ { u * , ψ } {\theta=\max\{u_{*},\psi\}} makes a i ⁢ (x , D ⁢ θ) {a_{i}(x,D\theta)} to be more integrable than L p i ′ ⁢ (Ω) {L^{p_{i}^{\prime}}(\Omega)} , i = 1 , … , n {i=1,\dots,n} , and then we prove that the solution u enjoys higher integrability.

INTEGRABLE functions; ANISOTROPY; MATHEMATICAL functions; GRAPH theory; MATHEMATICAL analysis

Forum Mathematicum, 2018, Vol 30, Issue 5, p1237

0933-7741

Academic Journal

10.1515/forum-2017-0240