Symmetric Decreasing Rearrangement Can Be Discontinuous

Frederick J. Almgren and Elliott H. Lieb
Additional contact information
Frederick J. Almgren: Princeton University, Departments of Mathematics and Physics
Elliott H. Lieb: Princeton University, Departments of Mathematics and Physics

A chapter in Inequalities, 2002, pp 479-482 from Springer

Abstract: 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

Keywords: Unit Disk; Differentiable Function; Symmetrize Function; Invariant Function; Gradient Norm (search for similar items in EconPapers)
Date: 2002
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-55925-9_38

Ordering information: This item can be ordered from
http://www.springer.com/9783642559259

DOI: 10.1007/978-3-642-55925-9_38

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().