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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we consider the PDE system of vanishing normal projection ofthe Laplacian for $C^2$ maps $u : \mathbb{R}^n \supseteq \Omega \longrightarrow\mathbb{R}^N$: \[ [\![\mathrm{D} u]\!]^\bot \Delta u = 0 \ \, \text{ in}\Omega. \] This system has discontinuous coefficients and geometricallyexpresses the fact that the Laplacian is a vector field tangential to the imageof the mapping. It arises as a constituent component of the $p$-Laplace systemfor all $p\in [2,\infty]$. For $p=\infty$, the $\infty$-Laplace system is thearchetypal equation describing extrema of supremal functionals in vectorialCalculus of Variations in $L^\infty$. Herein we show that the image of asolution $u$ is piecewise affine if either the rank of $\mathrm{D} u$ is equalto one or $n=2$ and $u$ has the additively separated form $u(x,y)=f(x)+g(y)$.As a consequence we obtain corresponding flatness results for the images of$p$-Harmonic maps, $p\in [2,\infty]$.

Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian