On the ``one-dimensionality'' of the extremal in the Poincaré inequality in a square (Q5947501)

The author obtains additional new results concerning the Poincaré inequality, comparing values of definite integrals of a function with that of its derivative on the interval \(I= [0,1]\) of the real line, or on the square \(I\times I\) in two dimensions. Poincaré introduced the functional \(\lambda_q=\|u'\|_{L_2}/\|u\|_{L_q}\) on the interval \(I\) which takes a minimum under the condition \(\int_I u\, dx= 0\) or, respectively, \(\int_{I\times I} u\, dx\, dy= 0\). Of course, in two dimensions the simple derivative is replaced by the gradient. The early one-dimensional version, with \(q=2\), is also known as Wirtinger's theorem. It was shown earlier that minimization of the functional \(\lambda_q\) is equivalent to finding the minimum of the functional \(\int\{\max(0,1+\mu x-|x|^q)\}^{1/2}\,dx\). In a previous work by A. P. Buslaev, V. A. Kondrat'ev, and the author it was shown that if \(q\leq 4+\varepsilon\), then the minimum of this integral is attained when \(\mu= 0\), and the graph of the function \(f_q(x)\) corresponding to the extremum of \(\lambda_q\) is symmetric about the point \(1/2\). But that is the essence of Wirtinger's theorem. However, if \(q> 6\) then the minimum is obtained when \(\mu\neq 0\) and the graph of \(f_q(x)\) lacks symmetry about the middle point. A boundary value eigenvalue problem is studied. The author proves that for \(q>2\) the smallest eigenvalue is smaller than \(-\pi^2\), and establishes bounds for \(1< q< 2\). He also studies the close relation between these eigenvalues and the eigenvalues of the operator \(-d^2/dx^2-\psi\), where \(\psi\) is a positive function.