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 investigate the problem of finding minimal graphs in$M^n\times\mathbb R$ with general boundary conditions using a variationalapproach. We look at so called generalized solutions of the Dirichlet Problemthat minimize a functional adapted from the area functional. We constructbarriers to show that for certain conditions on our boundary data, $\phi(x)$,the solutions obtain the boundary data $\phi(x)$. Following Oliker-Ural'tsevawe also consider solutions $u^{\epsilon}$ of a perturbed mean curvature flowfor $\epsilon > 0$. We show that there are subsequences $\epsilon_i$ where$u^{\epsilon_i}$ converges to a function $u$ satisfying the mean curvatureflow, and subsequences $u(\cdot, t_i)$ converge to a generalized solution $\baru$ of the Dirichlet problem. Furthermore, $\bar u$ depends only on the choiceof sequence $\epsilon_i$.

Minimal Graphs and Graphical Mean Curvature Flow in $M \times \mathbb R$