Polar factorization and pseudo-rearrangements: applications to Pólya–Szegö type inequalities

Convex rearrangement: Equality cases in the P�lya-Szeg� Inequality

“…Inequality (1.1) is strictly related to the so called Pólya-Szegö principle asserting that Dirichlet type integrals do not increase under spherical symmetrization (see [PS], [T1], [T2], [T3], [LL], [K], [BZ], [FV2]). In fact (1.1) can be regarded as a special case of this principle for the total variation of functions of bounded variation.…”

On Symmetric Functionals of the Gradient Having Symmetric Equidistributed Minimizers

“…Remark 4.4. Let us notice that σ and {σ k }, as in the statement of Lemma 5.1, certainly exist owing to [FV1].…”

“…Theorem 1.2 is essentially known in literature as a consequence of the fact that every u ∈ W 1 + (R n ) satisfies |∇u | * * (s) ≤ |∇u| * * (s) for s > 0 (1.10) (see, e.g., [CP], [F], [K]). Here, we shall present a proof of Theorem 1.2 where an intermediate step is enucleated, showing that a third quantity, involving |∇u|, always lies between the left-hand side and the right-hand side of (1.9) (see also [RT] and [FV1] for contributions in this direction). This is crucial in preparation for our main results concerning the equality case in (1.9).…”

Sobolev type inequalities for rearrangement invariant spaces

“…when u ≥ 0 is radially symmetric and radially decreasing. For such u, with c n := m n (B(0, 1)), direct calculation via (8) gives…”

“…For further results related to Pólya-Szegö type inequalities, akin to d ds u ∇ ≺ |∇u|, we refer to [8] and the references therein.…”