Minimizing \(L^ p\)-norm among the functions of bounded second derivatives (Q1191441)

The study of comparison theory in statistics leads to the problem of minimizing the integral \(\int_{{\mathbf R}} | u(x)|^ 2 dx\) among all the functions on \({\mathbf R}\) with bounded second derivatives. In this paper the above problem is solved in a more general setting. More precisely, given \(p\geq 1\), \(\Delta>0\), \(M>0\) it is proved that the integral \(\int_{{\mathbf R}} | u(x)|^ p dx\) has a unique minimizer \(u_ p\) on the set \(U=\bigl\{u\subset C^{1,1}({\mathbf R})\cap L^ p({\mathbf R}): 0<u(0)=\Delta,\;\text{Lip}(u')\leq M\bigr\}\), where \(\text{Lip}(u')\) denotes the Lipschitz constant of the derivative \(u'\). Moreover, it is proved that \(u_ p\) has the following properties: it is an even function, it consists of quadratic curves and there exists a calculable constant \(t_ p\in(1,\sqrt 2)\) such that \[ u_ p(x)= \Delta+ \Delta \int^{x\sqrt{M/\Delta}}_ 0\left\{\int^ z_ 0\sum^ \infty_{k=0} (-1)^{k+1}\chi_{[y_ k,y_{k+1}]}(y) dy\right\} dz \] for every \(x\in{\mathbf R}\), where \(y_ 0= 0\), \[ y_ k= 2t_ p \sum^{k- 1}_{m=0} \bigl(t^ 2_ p- 1\bigr)^{m/2}+\textstyle{{1\over 2}}\bigl(t^ 2_ p- 1\bigr)^{k/2} \] for every \(k\in\mathbb{N}\) and \(\chi_{[y_ k,y_{k+1}]}\) is the characteristic function of the interval \([y_ k,y_{k+1}]\). In addition, \(u_ p\in \partial U\), \[ \text{supp}(u_ p)= \left[- 2t_ p \sqrt{M/\Delta} \sum^ \infty_{m=0} \bigl(t^ 2_ p- 1\bigr)^{m/2},\;2t_ p \sqrt{M/\Delta} \sum^ \infty_{m=0} \bigl(t^ 2_ p- 1\bigr)^{m/2}\right], \] \(u_ p\) is smooth and \(u_ p^{\prime\prime\prime}= 0\) except at a set of countable points which has only two cluster points.