Minimal conditions for implications of Gronwall–Bellman type

Abstract

A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. Abstract Gronwall-Bellman type inequalities entail the following implication: if a sufficiently integrable function satisfies a certain homogeneous linear integral inequality, then it is nonpositive. We present a minimal (necessary and sufficient) condition on the Borel measure underlying the integrals for this implication to hold. The condition is also a necessary prerequisite for any nontrivial bound on solutions to inhomogeneous linear integral inequalities of GronwallBellman type.

On the Isoperimetric Inequality for the Magnetic Robin Laplacian with Negative Boundary Parameter

“…and by(3.8), joint with (3.6), we get 0 ≤ w(a + ) ≤ w(a) ≤ 0. If τ > a, then, by definition of τ , w(τ − ) ≤ 0.Using (3.8) joint with (3.5) and (3.6), we also have 0 ≤ w(τ + ) ≤ w(τ − ) ≤ 0, and the claim is proved.Therefore, since w(τ ) = 0, (3.4) reads as follows: for every s ∈ (τ, t 0 ) we havew(s) ≤ L ˆs τ w(ρ) dρ.By the integral form of the Gronwall inequality (see for instance[22, Lemma 3.2], which is a particular case of [23, Theorem 3.1]) , we get w ≤ 0 in [τ, t 0 ], which is in contradiction with (3.7).…”