Overdetermined Hardy inequalities
The aim of the paper is to find necessary and sufficient conditions on the weights \(w\) and \(w_0\) for the validity of the higher-order Hardy inequality \[ \Biggl(\int^1_0| u|^qw_0\Biggr)^{1/q}\leq C\Biggl(\int^1_0| u^{(k+ 1)}|^p w\Biggr)^{1/p} \] on the class of all solutions of certain overdetermined boundary value problems. In Section 2 the case when \(k=0\) and \(u\) satisfies \(u'= f\) in \((0,1)\), \(u(0)= u(1)= 0\), is treated. The corresponding result is formulated as Theorem 2.1 and extends a result of P. Gurka (on the other hand, Theorem 2.1 is a particular case of Theorems 8.8, 8.17 and Remark 9.10 (ii) of \textit{B. Opic} and \textit{A. Kufner} [``Hardy-type inequalities'' (1990; Zbl 0698.26007)]). The core of the paper represents Section 3, where higher-order Hardy's inequality \((k\in\{1,2,\dots\})\) is investigated.
scientific article; zbMATH DE number 1115559