This paper gives some very elementary proofs of results of Aupetit, Ransford and others on the variation of the spectral radius of a holomorphic family of elements in a Banach algebra. There is also some brief discussion of a notorious unsolved problem in automatic continuity theory.
In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem.
We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain non-uniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete non-compact manifolds having non-negative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically...
In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder () where denotes the cylindrical co-ordinates in is considered. The motion is with swirl (i.e. the -component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. that for the problem without swirl ( in (f)) in the whole space, as the flux constant tends to , 1) ; ; 2) converges to a vortex cylinder (see...
Vengono presentati alcuni problemi, idee e tecniche sorte nell'ambito della teoria delle equazioni alle derivate parziali del secondo ordine, con forma caratteristica semidefinita positiva e con soggiacenti strutture sub-riemanniane. Se ne traccia lo sviluppo a partire dalla classica teoria delle funzioni armoniche e caloriche, attraverso la teoria del potenziale negli spazi armonici astratti e la teoria della regolarità locale delle soluzioni.
— Vengono riconsiderati il problema di derivata obliqua regolare e quello misto di Dirichlet-derivata obliqua regolare per le funzioni armoniche in un dominio di e le questioni di completezza hilbertiana connesse già studiate in un precedente lavoro e viene data una nuova dimostrazione di un teorema di unicità.
Working in the axiomatic potential theory of M. Brelot, a description of the subdual of the vector space generated by the cone of positive harmonic functions on a harmonic space, , is given. Under certain hypothesis this is seen to be a function space on the Martin boundary of . Some ancillary results are proved. Next, it is shown, using this result and the theory of tensor products of simplexes, that the cone of positive separately harmonic functions is the tensor product of the cones of positive...