[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.

[Comments]
 Russian, 8 pp.

[Contact e-mail]
 al.il.nazarov@gmail.com