Continuity estimates for \(p\)-Laplace type operators in Orlicz-Zygmund spaces (Q530639)

In this paper the author considers the PDE \[ \nabla\cdot\mathcal{A}(x,Du)=\nabla\cdot f, \] for \(x\in\Omega\subseteq\mathbb{R}^{N}\), where \(N\geq2\) and \(\Omega\) is assumed to be a bounded Lipschitz domain. The map \(\mathcal{A}\;: \;\Omega\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}\) is a Carathéodory vector field satisfying some standard conditions, including a continuity-type condition of the form \[ \big|\mathcal{A}(x,\xi)-\mathcal{A}(x,\eta)\big|\leq b|\xi-\eta|\big(|\xi|+|\eta|\big)^{p-2}. \] In addition, the function \(u\) is required to satisfy the boundary data \(u=0\) on \(\partial\Omega\). The author considers the existence of a unique solution to this problem as well as a quantitative estimate regarding the norm of the derivative of \(u\). More precisely, given the Young function \(\Phi\), let \(L^{\Phi}(\Omega)\) denote the Orlicz space of all \(L^1(\Omega)\) functions satisfying \[ \int_{\Omega}\Phi\left(\frac{\big|f(x)\big|}{\lambda}\right)\;dx<+\infty, \] for some \(\lambda>0\). Now consider the Orlicz space generated by the function \(\Phi(t):=t^q\log^{-\alpha}(a+t)\) for \(t\geq0\) and some sufficiently large \(a\geq e\). Then this generates the Zygmund space \(\mathcal{L}^q\log^{-\alpha}\mathcal{L}(\Omega)\), for \(1<q<+\infty\) and \(\alpha>0\). Similarly, if one puts \(\Phi(t):=t^q\log^{-\alpha}(a+t)\big(\log\log(a+t)\big)^{-\beta}\), for \(\beta\in\mathbb{R}\) and the same restrictions on \(t\) and \(\alpha\) as earlier, then one generates the Zygmund space \(\mathcal{L}^q\log^{-\alpha}\mathcal{L}\big(\log\log\mathcal{L}\big)^{-\beta}\big(\Omega;\mathbb{R}^{N}\big)\). With this notation in hand, then, the author's main result is that given \(1<p<+\infty\), with \(p\neq2\), for each \(f\in\mathcal{L}^q\log^{-\alpha}\mathcal{L}\big(\log\log\mathcal{L}\big)^{-\beta}\big(\Omega;\mathbb{R}^{N}\big)\), with \(pq=p+q\) and \(0<\alpha<\frac{p}{|p-2|}\), it follows that the PDE above admits a unique solution such that the derivative, \(Du\), satisfies \(Du\in\mathcal{L}^{p}\log^{-\alpha}\mathcal{L}\big(\log\log\mathcal{L}\big)^{-\beta}\big(\Omega;\mathbb{R}^{N}\big)\). Finally, the author provides a similar result in case the forcing term, \(f\), belongs instead to the space \(\mathcal{L}^{q}\big(\log\log\mathcal{L}\big)^{-\beta}\big(\Omega;\mathbb{R}^{N}\big)\).