On the Ratio of the $$L_2$$-Norms of Partial Derivatives for Concave Functions

For a convex domain $$\Omega=\{ (x,y)\colon a \le x \le b,\,\varphi_1(x) \le y \le \varphi_2(x)\} $$ in the plane with Cartesian coordinates $$(x,y)$$ , we estimate the quantity $$M_\Omega :=\sup_u ({\|u_x\|_2}/{\|u_y\|_2})$$ in the class of concave functions $$u\in W^1_2(\Omega)$$ vanishing on the boundary of $$\Omega$$ . In [1], it was proved that $$M_\Omega < \infty$$ if and only if $$\varphi_1(a)=\varphi_2(a)$$ , $$\varphi_1(b)=\varphi_2(b)$$ , and the one-sided derivatives $$\varphi_i'$$ , $$i=1,2$$ , at $$a$$ and $$b$$ are finite. In the present paper, it is proved that if $$M_\Omega$$ is finite, then $$M_\Omega \le a_0^{-1}m+|k|$$ , where $$k=\frac{\varphi_1(b)-\varphi_1(a)}{b-a}, \qquad m=\max \{ |\varphi_i'(x)-k|,\,i=1,2,\,x=a,b \},$$ and $$a_0 \approx 0.83$$ is the zero of the Legendre function of the second kind $$Q_1(y)=\frac y2\log \biggl(\frac {1+y}{1-y}\biggr)-1$$ on the interval $$(0,1)$$ . In the special case of $$k=0$$ , this estimate is sharp; the equality is attained on the rhombus $$\Omega=\{ (x,y)\colon m|x|+|y| \le 1 \}$$ .