\(H^2\) regularity for the \(p(x)\)-Laplacian in two-dimensional convex domains (Q2260378)

The paper deals with the Dirichlet problem: \(-\Delta_{p(x)}u=f\) for \(x\in \Omega\), and \(u=g \) on \(\partial\Omega\) where \(\Omega \) is a bounded domain in \(\mathbb{R}^2\) and \(\Delta_{p(x)}=\text{div}(|\nabla u(x)|^{p(x)-2}\nabla u(x))\) is the \(p(x)\)-Laplacian. The main results are the following. Assume that \(\Omega\) has \(C^2\) or convex boundary, \(p(x)\geq p_1>1\) is Lipschitzian on \(\bar{\Omega}\), \(g\in H^2(\Omega)\) and \(f\) is such that \(f\in L^{q(\cdot)}\) with \(q(x)\geq q_1>2\) on the set \(\left\{x\in\Omega:p(x)\leq 2\right\}\) and \(f\equiv 0\) on the set \(\left\{x\in\Omega:p(x)> 2\right\}\). Then the weak solution \(u\) is \(H^2(\Omega)\).