Continuity estimates for $p$-Laplace type operators in Orlicz–Zygmund spaces

Abstract

We study the Dirichlet problem

in a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^N$, with $Ngeq2$. The vector field $\mathcal{A}:\Omega \times {\mathbb{R}}^N\to {\mathbb{R}}^N$ satisfies the typical growth and coercivity conditions of the $p$-Laplacian type operator with $p>1$. We prove existence and uniqueness results in the case the vector field $f$ belongs to the Orlicz–Zygmund space ${\mathcal{L}}^q{\log }^{–\alpha }\mathcal{L}(\log \log \mathcal{L}{)}^{–\beta }(\Omega ,{\mathbb{R}}^N),q=\frac{p}{p-1},\alpha >0$ and $\beta \in \mathbb{R}$ or $\alpha =0$ and $\beta >0$. In particular, the gradient of the solution belongs to ${\mathcal{L}}^q{\log }^{–\alpha }\mathcal{L}(\log \log \mathcal{L}{)}^{–\beta }(\Omega ,{\mathbb{R}}^N)$. Further, we provide estimates implying the continuity of the operator which carries any given $f$ into the gradient field $\nabla u$ of the solution.