A generalization of the Pólya-Szegő inequality for one-dimensional functionals
Let \(F:\mathbb{R}\times \mathbb{R}\to \overline{\mathbb{R}}_+\) be a continuous function which is even and convex with respect to the second argument. Also let \(a: A\times\mathbb{R}\to\overline{\mathbb{R}}_+\) be a continuous function, where \(A= [-1,1]\). Let the functional \(I\) be given by \[ I(u)= \int^1_{-1} F(u(x),\;|a(x,u(x))\,u'(x)|)\,dx\tag{1} \] and defined on the set \(M\) of functions \(u\in W^1_1(A)\), for which the integral converges. Monotonicity properties of functionals under rearrangements were studied by a number of authors. Motivated by a paper by \textit{F. Brock} [Calc. Var. Partial Differ. Equ. 8, No. 1, 15--25 (1999; Zbl 0947.35056)], where the behaviour of a functional \(I\) and its multidimensional analogue under the Steiner symmetrization was studied, the authors of the present paper establish the absence of the Lavrentiev phenomenon for this functional (1). The multidimensional case however remains open. For Lipschitz functions \(u\) a stronger inequality is proved under only necessary conditions on the weight function. For an arbitrary function \(u\in M\) they succeed in doing it under a certain additional restriction.
scientific article; zbMATH DE number 5998391