article

Title: On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains

Authors: Tunde Jakab, Irina Mitrea and Marius Mitrea

Issue: Volume 58 (2009), Issue 5, 2043-2072

Abstract:

$Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain, whose boundary decomposes into two disjoint pieces $\Sigma_t$, $\Sigma_n \subseteq \partial\Omega$, which meet at an angle $< \pi$. Denote by $\nu$ the outward unit normal to $\Omega$. Then there exists $\varepsilon > 0$ with the property that if $|2-p| < \varepsilon$, then the following holds. Consider a vector field $u$ with components $u_1,\dots,u_n \in L^p(\Omega)$ such that \[\mathrm{div}\, u = \sum_{j=1}^n \partial_j u_j \in L^p(\Omega) \] and $\mathrm{curl} \, u = (\partial_j u_k - \partial_k u_j)_{1\leq j,k\leq n} \in L^p(\Omega)$. Set $\nu \cdot u = \sum_{j=1}^n \nu_j u_j$ and $\nu \times u = (\nu_j u_k - \nu_k u_j)_{1\leq j,k \leq n}$. Then the following are equivalent: \begin{enumerate}[label=(\roman*)] \item $(\nu \cdot u)|_{\Sigma_t} \in L^p(\Sigma_t)$ and $(\nu \times u)|_{\Sigma_n} \in L^p(\Sigma_n)$; \item $\nu \cdot u \in L^p(\partial\Omega)$; \item $\nu \times u \in L^p(\partial\Omega)$. \end{enumerate} Moreover, if either condition holds, then $u$ belongs to the Besov space $B^{p,\max(p,2)}_{1/p}(\Omega)$. In fact, similar results are valid for differential forms of arbitrary degree. This generalizes earlier work dealing with the case when $\Sigma_t = \emptyset$ or $\Sigma_n = \emptyset$.$