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

We investigate the Monge-Amp\`ere equation subject to zero boundary value andwith a positive right-hand side unction assumed to be continuous or essentiallybounded. Interior estimates of the solution's first and second derivatives areobtained in terms of moduli of continuity. We explicate how the estimatesdepend on various quantities but have them independent of the solution'smodulus of convexity. Our main theorem has many useful consequences. One ofthem is the nonlinear dependence between the H\"older seminorms of the solutionand of the right-side function, which confirms the results of Figalli, Jhaveriand Mooney (J. Func. Anal. 2016). Our technique is in part inspired by Jian andWang (SIAM J. Math. Anal. 2007) which includes using a sequence of so-calledsections.

Interior Estimates for Monge-Amp\`ere Equation in Terms of Modulus of Continuity