St. Petersburg Mathematical Society

Препринты Санкт-Петербургского математического общества
Preprints of the St. Petersburg Mathematical Society

Аннотация к препринту 2026-03

Author

Nazarov, A. I.; Plakhov, A. Yu.

Title

On the quotient of

L_2

-norms of partial derivatives of convex functions

AMS Subj-class

26D10 Inequalities involving derivatives and differential and integral operators
49K21 Optimality conditions for problems involving relations other than differential equations
52A15 Convex sets in 3 dimensions (including convex surfaces)

Keywords

estimates of

L_2

-norms of partial derivatives, convex functions and surfaces, calculus of variations, Legendre equation

Abstract

In a convex planar domain with Cartesian coordinates $x$ and $y$ , we are interested in estimating the quantity $\sup_u ({||u_x||_2}/{||u_y||_2})$ in the class of upward convex functions vanishing on the boundary. Plakhov and Protasov (2025) established the necessary and sufficient geometrical condition for the finiteness of this quantity. We establish a quantitative estimate for it and show that in a particular case this estimate is sharp.