English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

This paper continues the study of the mixed problem for the Laplacian. Weconsider a bounded Lipschitz domain $\Omega\subset \reals^n$, $n\geq2$, withboundary that is decomposed as $\partial\Omega=D\cup N$, $D$ and $N$ disjoint.We let $\Lambda$ denote the boundary of $D$ (relative to $\partial\Omega$) andimpose conditions on the dimension and shape of $\Lambda$ and the sets $N$ and$D$. Under these geometric criteria, we show that there exists $p_0>1$depending on the domain $\Omega$ such that for $p$ in the interval $(1,p_0)$,the mixed problem with Neumann data in the space $L^p(N)$ and Dirichlet data inthe Sobolev space $W^ {1,p}(D) $ has a unique solution with the non-tangentialmaximal function of the gradient of the solution in $L^p(\partial\Omega)$. Wealso obtain results for $p=1$ when the Dirichlet and Neumann data comes fromHardy spaces, and a result when the boundary data comes from weighted Sobolevspaces.

The mixed problem in Lipschitz domains with general decompositions of the boundary