For an analytic variety V in ℂⁿ containing the origin which satisfies the local Phragmén-Lindelöf condition it is shown that for each real simple curve γ and each d ≥ 1 the limit variety satisfies the strong Phragmén-Lindelöf condition (SPL).
We give a simplified proof of J. P. Rosay's result on plurisubharmonicity of the envelope of the Poisson functional [10].
Let L be a symmetric second order uniformly elliptic operator in divergence form acting in a bounded Lipschitz domain Ω of RN and having Lipschitz coefficients in Ω. It is shown that the Rellich formula with respect to Ω and L extends to all functions in the domain D = {u ∈ H01(Ω); L(u) ∈ L2(Ω)} of L. This answers a question of A. Chaïra and G. Lebeau.
For complex algebraic varieties V, the strong radial Phragmén-Lindelöf condition (SRPL) is defined. It means that a radial analogue of the classical Phragmén-Lindelöf Theorem holds on V. Here we derive a sufficient condition for V to satisfy (SRPL), which is formulated in terms of local hyperbolicity at infinite points of V. The latter condition as well as the extension of local hyperbolicity to varieties of arbitrary codimension are introduced here for the first time. The proof of the main result...
We prove a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for harmonic functions with respect to the other graph. As part of the proof, we give a characterization of the elliptic Harnack inequality.