On optimal \(C^{1 , \alpha}\) estimates for \(p ( x )\)-Laplace type equations (Q2201752)

Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain, and let \(p,q\in C(\overline{\Omega})\) with \(\inf_{\overline{\Omega}}p>1\) and \(\inf_{\overline{\Omega}}q>n\). The authors consider the following \(p(x)\)-Laplace elliptic equation \[ \text{div}(a(x)|\nabla u|^{p(x)-2}\nabla u)=\text{div}\mathbf{h}(x)+f(x), \quad\text{in } \Omega,\tag{1}\] where \(f\in L^{q(\cdot)}(\Omega)\) and \(a:\overline{\Omega}\rightarrow \mathbb{R}\), \(\mathbf{h}:\overline{\Omega}\rightarrow \mathbb{R}^n\) satisfy the following conditions \vspace{2mm} 1) there exist \(\lambda, \Lambda>0\) such that \(\lambda \leq a(x)\leq \Lambda\) for all \(x\in \overline{\Omega}\), \vspace{2mm} 2) there exist \(C_1,C_2>0\) and \(\sigma_1,\sigma_2\in (0,1)\) such that \(|a(x)-a(y)|+|p(x)-p(y)|\leq C_1|x-y|^{\sigma_1}\) and \(|\mathbf{h}(x)-\mathbf{h}(y)|\leq |x-y|^{\sigma_2}\) for all \(x,y\in \overline{\Omega}\), \vspace{2mm} and establish an optimal local \(C^{1,\alpha}\)-estimate for the local weak solutions of equation \((1)\).