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 show local and global scale invariant regularity estimates forsubsolutions and supersolutions to the equation $-{\rm div}(A\nablau+bu)+c\nabla u+du=-{\rm div}f+g$, assuming that $A$ is elliptic and bounded.In the setting of Lorentz spaces, under the assumptions $b,f\in L^{n,1}$,$d,g\in L^{\frac{n}{2},1}$ and $c\in L^{n,q}$ for $q\leq\infty$, we show that,with the surprising exception of the reverse Moser estimate, scale invariantestimates with "good" constants (that is, depending only on the norms of thecoefficients) do not hold in general. On the other hand, assuming a necessarysmallness condition on $b,d$ or $c,d$, we show a maximum principle and Moser'sestimate for subsolutions with "good" constants. We also show the reverse Moserestimate for nonnegative supersolutions with "good" constants, under nosmallness assumptions when $q<\infty$, leading to the Harnack inequality fornonnegative solutions and local continuity of solutions. Finally, we show that,in the setting of Lorentz spaces, our assumptions are the sharp ones toguarantee these estimates.

Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces