Parabolic and hyperbolic infinite networks

Yamasaki, Maretsugu

doi:10.32917/hmj/1206135953

Cited by 72 publications

(36 citation statements)

References 6 publications

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“…: E 2 (f ) < ∞}/R-in accordance with the setting used in all Yamasaki's papers, beginning with [25].…”

Section: Introductionmentioning

confidence: 70%

Parabolic theory of the discrete -Laplace operator

Mugnolo

2013

Nonlinear Analysis: Theory, Methods & Applications

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a b s t r a c tWe study the discrete version of the p-Laplace operator. Based on its variational properties we discuss some features of the associated parabolic problem. We prove well-posedness of the problem and obtain information about positivity and comparison principles as well as compatibility with the symmetries of the underlying graph. Our methods consist in an interplay of the theory of subdifferentials and of combinatorial methods. We conclude briefly discussing the variational properties of a handful of nonlinear generalized Laplacians appearing in different parabolic equations.

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“…: E 2 (f ) < ∞}/R-in accordance with the setting used in all Yamasaki's papers, beginning with [25].…”

Section: Introductionmentioning

confidence: 70%

Parabolic theory of the discrete -Laplace operator

Mugnolo

2013

Nonlinear Analysis: Theory, Methods & Applications

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“…Other known consequences of the triviality of the reduced ℓ p -cohomology include the triviality of the p-capacity between finite sets and ∞ (see Yamasaki [49] and Puls [39, Corollary 2.3]) and existence of continuous translation invariant linear functionals on D p (Γ)/K (see [39, §8]). It also has implication on the possibility of realising the graph as a packing of spheres in R k (see Benjamini & Schramm [6]).…”

Section: Discussionmentioning

confidence: 99%

Boundary values, random walks, and $\ell^p$-cohomology in degree one

Gournay¹

2015

Groups Geom. Dyn.

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The vanishing of reduced ℓ 2 -cohomology for amenable groups can be traced to the work of Cheeger & Gromov in [10]. The subject matter here is reduced ℓ p -cohomology for p ∈]1, ∞[, particularly its vanishing. Results for the triviality of ℓ p H 1 (G) are obtained, for example: when p ∈]1, 2] and G is amenable; when p ∈]1, ∞[ and G is Liouville (e.g. of intermediate growth). This is done by answering a question of Pansu in [34, §1.9] for graphs satisfying certain isoperimetric profile. Namely, the triviality of the reduced ℓ p -cohomology is equivalent to the absence of non-constant harmonic functions with gradient in ℓ q (q depends on the profile). In particular, one reduces questions of non-linear analysis (p-harmonic functions) to linear ones (harmonic functions with a very restrictive growth condition).

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“…For the equivalence between 1, 3 and 4, see [Tr2, Proposition 1 and Theorem 3], and also [GoT2,Theorem 3.1]. For the equivalence between 1 and 2, see [Y,Theorem 3.2]. The equivalence between p-parabolicity and 4 is proved in [Ho1,Theorem 5.2], as well as in [K] in the Riemannian setting.…”

Section: Volume Growth and P-parabolicitymentioning

confidence: 99%