In a first result, we prove that every continuous local triple derivation on a JB * -triple is a triple derivation. We also give an automatic continuity result, that is, we show that local triple derivations on a JB * -triple are continuous even if not assumed a priori to be so. These results provide positive answers to the conjectures posed by Mackey (Bull. London Math. Soc. 45 (2013) 811-824). In particular, every local triple derivation on a C * -algebra is a triple derivation. We also explore the connections between (bounded local) triple derivations and generalized (Jordan) derivations on a C * -algebra.
We study the following elliptic problem −A(u) = λu q with Dirichlet boundary conditions, where A(u)(x) = ∆u(x)χ D 1 (x) + ∆pu(x)χ D 2 (x) is the Laplacian in one part of the domain, D 1 , and the p−Laplacian (with p > 2) in the rest of the domain, D 2 . We show that this problem exhibits a concaveconvex nature for 1 < q < p − 1. In fact, we prove that there exists a positive value λ * such that the problem has no positive solution for λ > λ * and a minimal positive solution for 0 < λ < λ * . If in addition we assume that p is subcritical, that is, p < 2N/(N − 2) then there are at least two positive solutions for almost every 0 < λ < λ * , the first one (that exists for all 0 < λ < λ * ) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every 0 < λ < λ * ) comes from an appropriate (and delicate) mountain pass argument.
Abstract. We study when a local triple derivation on a real JB * -triple is a triple derivation. We find an example of a (real linear) local triple derivation on a rank-one Cartan factor of type I which is not a triple derivation. On the other hand, we find sufficient conditions on a real JB * -triple E to guarantee that every local triple derivation on E is a triple derivation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.