A symmetry problem related to Wirtinger's and Poincaré's inequality
The authors consider the minimizers of \[ J(u;-1, 1)= \|u'\|_p/\|u\|_q,\quad p,q\in(1, \infty), \] among periodic functions in \(W^{1,p}(-1, 1)\) with mean value zero. It was known that the minimizers of \(J(u;-1,1)\) are scaled in such a way that their maximum is \(+1\) and their minimum is \(-m\in (0,1]\). A minimizer is called symmetric if for it holds \(m=1\). Let \(A\) be the subset of \(W^{1,p}(- 1,1)\) consisting of all mean value zero functions \(v\) such that there exists \(b\in (-1,1)\) such that \(\text{supp }v_+= (-1,b)\) and \(v_+\), \(v_-\) are minimizers of \(J(u;-1, b)\) and \(J(u; b,1)\), respectively. The main results of the paper are: Theorem 1. The minimizer of \(J(u;- 1,1)\) is symmetric for \(q\leq 2p+1\). Theorem 2. If the class of admissible functions is further restricted to a set \(A\), then the minimizer of \(J(u;-1,1)\) is symmetric for \(q\leq 4p-1\). The authors indicate that, in some cases, the above Theorem 1 is optimal, and these results obtained yield the best constant in Poincaré's inequality in a certain situation. The present paper is closely related to some known results given by \textit{B. Dacorogna}, \textit{W. Gangbo} and \textit{N. Subía} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9, No. 1, 29-50 (1992; Zbl 0764.49009)], \textit{, Yu. V. Egorov} [C. R. Acad. Sci., Paris, Sér. I 324, No. 5, 503-507 (1997; Zbl 0891.47046)], and \textit{B. Kawohl} [Math. Intell. 20, No. 2, 16-22 (1998)].
scientific article; zbMATH DE number 1339364
scientific article