Symmetrization Inequalities for Composition Operators of Carathéodory type

Rearrangement inequalities for functionals with monotone integrands

“…The purpose of this paper is to dispense with the continuity assumptions on F in the theorems of Brock and Draghici, and to characterize the equality cases in some relevant situations. This continues prior work of the second author [19][20][21][22][23].…”

Existence and non-existence of Schwarz symmetric ground states for elliptic eigenvalue problems

“…This lemma is crucial for the use of symmetrization in connection with problem (1.1) and the essential ingredient is the inequality (1.4) which we have justified in [10] under hypotheses which we believe are natural, and, as our examples in Section 3 show, more or less optimal, in the context of (1.1) (see also [11] for some extensions). For example, the proof of (1.4) in [5] [22], [23] requires the assumption that, for some p ≥ 1, …”

“…Note that in the case N = 1 our hypotheses about G are well suited to problems concerning symmetric planar waves guides composed of layers of self-focusing media where discontinuities in x may occur at the interfaces between adjacent layers, [21]. The inequality (1.4) has also been discussed in [5] and [22] [23], but under stronger regularity assumptions than ours, [10], [11]. Our main result establishes the existence of a Schwarz symmetric minimizer for (1.1).…”

“…Then we have that (1.6) provided that G(s) ≤ G(|s|) for all s ∈ R and G(0) = 0, which justifies (1.3) in this special case. In [10] (see also [11] for some extensions), we established (1.4) for functions G of Carathéodory type having appropriate monotonicity properties and, as we show below, these results now allow us to use symmetrization to treat (1.1) under rather general and natural assumptions about the function G. Here we discuss the minimization problem (1.1) under the following basic assumptions about the function G:…”

Cases of equality and strict inequality in the extended Hardy–Littlewood inequalities

“…However, this condition can be relaxed when v is a step function. In this case, all results established in [15][16][17] remain valid for cases of equality and strict inequality, adding a strict monotonicity assumption and an integrability condition.…”