Suppose \(f(x_ 1,x_ 2)\geq 0\) is a continuously differentiable function supported in the unit disk in the plane. Its symmetric decreasing rearrangement is the rotationally invariant function \(f^*(x_ 1,x_ 2)\) whose level sets are circles enclosing the same area as the level sets of f. Such rearrangement \(f\to f^*\) preserves \(L^ p\) norms and generally decreases convex gradient integrals, e.g., \(\| \nabla f^*\|_ p\leq \| \nabla f\|_ p\quad (1\leq p<\infty).\) This definition of the rearrangement operator \({\mathcal R}: f\to f^*\) can be naturally extended to the Sobolev space \(W^{1,p}({\mathbb{R}}^ n)\) and the author shows that in general \({\mathcal R}\) is not continuous for \(n\geq 2\). He characterizes the functions f at which \({\mathcal R}\) is continuous introducing thus a new property called co-area regularity and shows that every sufficiently differentiable function is co-area regular and the set of co-area regular functions is dense in the space \(W^{1,p}({\mathbb{R}}^ n)\).

Reviewer: L.Janos

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