On an inequality of Opial type in two variables (Q1200042)

The following two-dimensional analogue of the Opial inequality is proved: Let \(u=u(x,y)\) be a function on \(\Delta=[a,b]\times[c,d]\) such that \(u\), \(u_ x\), \(u_ y\) and \(u_{xy}\) are continuous on \(\Delta\), \(u(a,y)=u(b,y)=u_ x(x,c)=u_ x(x,d)=0\) for \((x,y)\in\Delta\), and let \(1\leq p_ i<\infty\), \(i=1,\dots,4\). Then the inequality \[ \int^ b_ a\int^ d_ c| u|^{p_ 1}| u_ x|^{p_ 2}| u_ y|^{p_ 3}| u_{xy}|^{p_ 4}dx dy\leq M\prod^ 4_{i=1}\left(\int^ b_ a\int^ d_ c| u_{xy}|^{2p_ i}dx dy\right)^{1/2} \] holds with \(M=2^{-(2p_ 1+p_ 2+p_ 3)}(b- a)^{p_ 1+p_ 3-1}(d-c)^{p_ 1+p_ 2-1}\).