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Strutture subriemanniane in alcuni problemi di Analisi

Ermanno Lanconelli (2005)

Bollettino dell'Unione Matematica Italiana

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.

Su alcune questioni connesse con il problema di derivata obliqua regolare per le funzioni armoniche

Enrico Magenes (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

— Vengono riconsiderati il problema di derivata obliqua regolare e quello misto di Dirichlet-derivata obliqua regolare per le funzioni armoniche in un dominio di R 3 e le questioni di completezza hilbertiana connesse già studiate in un precedente lavoro e viene data una nuova dimostrazione di un teorema di unicità.

Subduals and tensor products of spaces of harmonic functions

Ian Reay (1974)

Annales de l'institut Fourier

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, Ω , is given. Under certain hypothesis this is seen to be a function space on the Martin boundary of Ω . 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...

Subharmonic functions in sub-Riemannian settings

Andrea Bonfiglioli, Ermanno Lanconelli (2013)

Journal of the European Mathematical Society

In this paper we furnish mean value characterizations for subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution Γ . These characterizations are based on suitable average operators on the level sets of Γ . Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach, Reade and Saks. We analyze as well the notion of subharmonic function...

Subharmonicity in von Neumann algebras

Thomas Ransford, Michel Valley (2005)

Studia Mathematica

Let ℳ be a von Neumann algebra with unit 1 ℳ . Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by μ t ( x ) t ≥ 0 the generalized s-numbers of x, defined by μ t ( x ) = inf||xe||: e is a projection in ℳ i with τ ( 1 ℳ - e ) ≤ t (t ≥ 0). We prove that, if D is a complex domain and f:D → ℳ is a holomorphic function, then, for each t ≥ 0, λ ↦ ∫ 0 t l o g μ s ( f ( λ ) ) d s is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.

Sul problema pluriarmonico in un campo sferico di 𝐂 n per n ≥ 3

Maria Adelaide Sneider (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let Σ be the boundary of the unit ball Ω of 𝐂 n . A set of second order linear partial differential operators, tangential to Σ , is explicitly given in such a way that, for n ≥ 3 , the corresponding PDE caractherize the trace of the solution of the pluriharmonic problem (either “in the large” or “local”), relative to Ω .

Superharmonic extension and harmonic approximation

Stephen J. Gardiner (1994)

Annales de l'institut Fourier

Let Ω be an open set in ℝ n and E be a subset of Ω . We characterize those pairs ( Ω , E ) which permit the extension of superharmonic functions from E to Ω , or the approximation of functions on E by harmonic functions on Ω .

Superharmonicity of nonlinear ground states.

Peter Lindqvist, Juan Manfredi, Eero Saksman (2000)

Revista Matemática Iberoamericana

The objective of our note is to prove that, at least for a convex domain, the ground state of the p-Laplacian operatorΔpu = div (|∇u|p-2 ∇u)is a superharmonic function, provided that 2 ≤ p ≤ ∞. The ground state of Δp is the positive solution with boundary values zero of the equationdiv(|∇u|p-2 ∇u) + λ |u|p-2 u = 0in the bounded domain Ω in the n-dimensional Euclidean space.

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