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Symmetric Decreasing Rearrangement Can Be Discontinuous

In: Inequalities

Author

Listed:
  • Frederick J. Almgren Jr.

    (Princeton University, Departments of Mathematics and Physics)

  • Elliott H. Lieb

    (Princeton University, Departments of Mathematics and Physics)

Abstract

Suppose f(xl,x2) ≥ 0 is a continuously differentiable function supported in the unit disk in the plane. Its symmetric decreasing rearrangement is the rotationally invariant function f*(xl,x2) whose level sets are circles enclosing the same area as the level sets of f. Such rearrangement preserves Lp norms but decreases convex gradient integrals, e.g. ||∇||*||p ≤ ||∇/||p (1 ≤ p 0 (j = 1,2,3,…) is a sequence of infinitely differentiable functions also supported in the unit disk which converge uniformly together with first derivatives to f. The symmetzed functions also converge uniformly. The real question is about convergence of the derivatives of the symmetrized functions. We announce that the derivatives of the symmetrized functions need not converge strongly, e.g. it can happen that ||∇fj*—∇f*||p →* 0 for every p. We further characterize exactly those f’s for which convergence is assured and for which it can fail

Suggested Citation

  • Frederick J. Almgren Jr. & Elliott H. Lieb, 2002. "Symmetric Decreasing Rearrangement Can Be Discontinuous," Springer Books, in: Michael Loss & Mary Beth Ruskai (ed.), Inequalities, pages 479-482, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55925-9_38
    DOI: 10.1007/978-3-642-55925-9_38
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