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

The simple product formulae for derivatives of scalar functions raised todifferent powers are generalized for functions which take values in the set ofsymmetric positive definite matrices. These formulae are fundamental inderivation of various non-linear estimates, especially in the PDE theory. Toget around the non-commutativity of the matrix and its derivative, we applysome well-known integral representation formulas and then we make anobservation that the derivative of a matrix power is a logarithmically convexfunction with respect to the exponent. This is directly related to the validityof a seemingly simple inequality combining the integral averages and the innerproduct on matrices. The optimality of our results is illustrated on numerousexamples.

Sharp nonlinear estimates for multiplying derivatives of positive definite tensor fields