Second-order Derivatives and Rearrangements

Inequality (1.2) is classical and very well known. It has been the object of extensions and variants which can be found in a number of papers and monographs, including [AFLT], [Bae], [BBMP], [BH], [Bro], [BZ], [CP], [E], [H], [Ka2], [Kl], [Ma], [Sp1], [Sp2], [Spi], [Ta1], [Ta5], and [Ta6]. Even if not as popular as (1.2), inequality (1.1) has also been known for a long time, and versions of it appear in [Ci1], [Du], [Ga], [Ka1], [Ma], [RT], and [Ry]. Various are the applications of PólyaSzegö-type inequalities. In particular, they have proved to be crucial for such results as isoperimetric inequalities of mathematical physics (see [PS] and [Ta4]) and Sobolevtype inequalities in optimal form (see, e.g., [AFT], [A], [Ci2], [CF], [EKP], [L], [Ko], [Mo], [Ta1], and [Ta5]). They are also strictly connected to a priori sharp estimates for solutions to second-order elliptic and parabolic boundary value problems (see [ALT], [Ba], [Di], [Ka1], [Ke], [Ta2], and [Ta3]). On the other hand, the effect of rearrangements on Dirichlet-type functionals depending on higher-order derivatives seems to be still unknown. The present paper is aimed at giving a contribution on this subject. Indeed, we are concerned with inequalities in the spirit of (1.1) and (1.2) involving second-order derivatives. An evident obstacle in attacking this question is that very smooth functions may have a decreasing (and symmetric) rearrangement whose first-order derivative is not even weakly differentiable (see Section 2, Remark 3, for an example). Thus, unlike W 1,p(0, l), membership of a function in the second-order Sobolev space W 2,p(0, l) need not be preserved after rearranging it in decreasing order. This shortcoming can be overcome by enlarging the class of admissible functions. Actually, our main result—Section 2,