Rearrangements and Convexity of Level Sets in PDE

Asymptotics for the best Sobolev constants and their extremal functions

“…in Ω and show that each maximum point of the distance function ρ gives rise to a minimizer of (13). We then use this latter fact to conclude that if Ω is an annulus, then there exist infinitely many positive and nonradial minimizers of (13).…”

“…Let v * ∈ W 1,p 0 (B R ) denote the Schwarz symmetrization of v (see [13]). It follows that v * is radial and radially nonincreasing and, moreover, it satisfies v * ∞ = v ∞ and ∇v * p p ≤ ∇v p p .…”

An isoperimetric inequality related to a Bernoulli problem

“…By Theorem 1.1 J λ,p has a non-trivial minimiser u ∈ W 1,p 0 (Ω). Consider its Schwarz symmetrisation u * (see [10] for a definition and properties). By well known properties of Schwarz symmetrisation u * ∈ W 1,p 0 (Ω * ), ∇u * p ≤ ∇u p and |{u * < 1}| = |{u < 1}|.…”

Note on Affine Gagliardo–Nirenberg Inequalities

“…See, for example, Kawohl [25]- [24], Sperner [41], Talenti [42], Brothers and Ziemer [5], Hilden [22]. Inequality (2.6) is a powerful tool to many problems in physics and mathematics.…”

“…Inequality (2.6) is a powerful tool to many problems in physics and mathematics. On the other hand, several variants of inequality (2.6) have been established and applied intensively, see, for example, Kawohl [24]. Especially, Lutwak, Yang and Zhang in [32], Cianchi, Lutwak, Yang and Zhang in [10] proved the following strengthened affine version of inequality (2.6).…”