Paola Cavaliere

Existence results for elliptic equations

Abstract This paper is concerned with the Dirichlet problem for second-order elliptic equations in nondivergence form with discontinuous coefficients in an unbounded open subsetxa0 Ω ofxa0 R n , n⩾3 . An our recent a priori estimate combined with the Fredholm index theory and a generalized method of continuity allows us to establish some unique solvability results for the given problem.

The Dirichlet Problem for elliptic equations in unbounded domains of the plane

In this paper we prove a uniqueness and existence theorem for the Dirichlet problem in W2,p for second order linear elliptic equations in unbounded domains of the plane. Here the leading coefficients are locally of class VMO and satisfy a suitable condition at infinity.

NORMS SUPPORTING THE LEBESGUE DIFFERENTIATION THEOREM

A version of the Lebesgue differentiation theorem is offered, where the Lp norm is replaced with any rearrangement-invariant norm. Necessary and sufficient conditions for a norm of this kind to support the Lebesgue differentiation theorem are established. In particular, Lorentz, Orlicz and other customary norms for which Lebesgue’s theorem holds are characterized.

Boundedness of Nonadditive Quantum Measures

We deal with not necessarily additive functions acting on complete orthomodular posets and taking values in Hausdorff uniform spaces, where no algebraic structure is required. As a consequence, neither pseudo-additivity, nor monotonicity are meaningful notions in this setting. Conditions ensuring their boundedness are exhibited in terms of some mild continuity properties. Such conditions are satisfied, in particular, by completely additive measures on projection lattices of von Neumann algebras. Hence, among other things, our main result provides a version in the generalized nonadditive quantum setting of the so-called boundedness principle in classical and quantum measure theory.

Approximation of Finitely Additive Functions with Values in Topological Groups

We discuss the problem of the pointwise approximation of finitely additive functions, which are defined on a Boolean algebra and take values in a Hausdorff topological commutative group, via strongly continuous and exhaustive finitely additive functions. Related properties of topological groups are also investigated.