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

On the quotient of $L_2$-norms of partial derivatives of convex functions

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)

estimates of $L_2$-norms of partial derivatives, convex functions and surfaces, calculus of variations, Legendre equation

    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.

Russian, 8 pp.

al.il.nazarov@gmail.com