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

Abstract

We determine a class of Carathéodory functions G for which the minimum formulated in the problem (1.1) below is achieved at a Schwarz symmetric function satisfying the constraint. Our hypotheses about G seem natural and, as our examples show, they are optimal from some points of view.

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

“…Our result concerning (P c ) in [18] gives natural conditions ensuring that (E c ) has a Schwarz symmetric ground state. Now, a natural question is, Under which conditions are all ground states of (E c ) Schwarz symmetric?…”

“…Theorem 6.3 answers us. Namely, suppose for example that G satisfies hypotheses of theorem 3.1 of [18] and theorem 6.3, then, we can easily prove that all ground states of (E c ) are Schwarz symmetric.…”

Extended Hardy-Littlewood inequalities and some applications

“…Closely following the proof of Example 5 of [3] whose principal ingredient is (1.1) with m = 1 , we obtain: Assume that H : (0, ∞) × R 2 −→ R is a 2-Carathéodory function such that: …”

“…For dealing with (1.1) (and cases of equality in (1.1)) in the calculus of variations and in some other domains, it is fundamental to establish it for integrands H which are not necessarily continuous with respect to the distance |x| (see the introduction of [3] for more details). In Theorem 4.2 of [2], we proved (1.1) under minimal regularity assumptions when m = 2; our approach set out in [2] still applies to m > 2, thus we can easily extend Proposition 4.1 of [2] (and consequently Theorem 4.2).…”

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].…”