Inequalities pp 479-482 | Cite as

Symmetric Decreasing Rearrangement Can Be Discontinuous

  • Frederick J. AlmgrenJr.
  • Elliott H. Lieb


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 < ∞). Now suppose that fj(x1,x2) > 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


Unit Disk Differentiable Function Symmetrize Function Invariant Function Gradient Norm 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Frederick J. AlmgrenJr.
    • 1
  • Elliott H. Lieb
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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