On the Ratio of the L₂-Norms of Partial Derivatives for Concave Functions

For a convex domain Ω=\ (x,y) a ≤ x ≤ b,\,φ₁(x) ≤ y ≤ φ₂(x)\ in the plane with Cartesian coordinates $$(x,y)$$ , we estimate the quantity M_Ω :=ᵤ (\|uₓ\|₂/\|uy\|₂) in the class of concave functions u∈ W¹₂(Ω) vanishing on the boundary of Ω . In [1], it was proved that M_Ω < ∞ if and only if φ₁(a)=φ₂(a) , φ₁(b)=φ₂(b) , and the one-sided derivatives φᵢ' , $$i=1,2$$ , at a and b are finite. In the present paper, it is proved that if M_Ω is finite, then M_Ω ≤ a₀⁻¹m+|k| , where k=φ₁(b)-φ₁(a)/b-a, m=max \ |φᵢ'(x)-k|,\,i=1,2,\,x=a,b \, and a₀ ≈ 0.83 is the zero of the Legendre function of the second kind Q₁(y)= y2log (1+y/1-y)-1 on the interval $$(0,1)$$ . In the special case of $$k=0$$ , this estimate is sharp; the equality is attained on the rhombus Ω=\ (x,y) m|x|+|y| ≤ 1 \ .