Generalized extremal length of an infinite network

Nakamura, Tadashi; Yamasaki, Maretsugu

doi:10.32917/hmj/1206136452

Cited by 31 publications

(10 citation statements)

References 4 publications

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“…(Extremal length was first investigated by Duffin [16] in the discrete setting, and extremal length of order p was studied in Nakamura and Yamasaki [30]. By simple calculation, we verify that our definition is equivalent to theirs.…”

Section: Lemma 23 ([19 Proposition 4])mentioning

confidence: 54%

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Functions with finite Dirichlet sum of orderpand quasi-monomorphisms of infinite graphs

Hattori

Kasue

2012

Nagoya Mathematical Journal

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In this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of p-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension n and it is not p-parabolic for p > n - 1, then it admits a lot of p-harmonic functions with finite Dirichlet sum of order p.

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Section: Lemma 23 ([19 Proposition 4])mentioning

confidence: 54%

“…where o is a fixed point of V (see [30]). Let us denote by L 1,p 0 (G, r) the closure of the set of functions with finite supports.…”

Section: §2 Infinite Network and The Royden Compactificationmentioning

confidence: 99%

Functions with finite Dirichlet sum of orderpand quasi-monomorphisms of infinite graphs

Hattori

Kasue

2012

Nagoya Mathematical Journal

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“…Pick an a ∈ B 1 ∩ ∂D. By combining Theorem 2.1 and Theorem 2.4 of [4] we see that λ p (P a ) < ∞ if and only if cap p ({a}, ∞, D) > 0. Thus to finish the proof we need to show cap p ({a}, ∞, D) > 0, which we now proceed to do.…”

Section: Proof Of Theorem 32mentioning

confidence: 89%

Some results concerning the $p$-Royden and $p$-harmonic boundaries of a graph of bounded degree

Puls¹

2010

Preprint

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“…[8] for an introduction and survey on this topic). It was in this cultural climate that Nakamura and Yamasaki introduced in [9] on an infinite graph G with node set V the mapping 1…”

Section: Introductionmentioning

confidence: 99%

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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Generalized extremal length of an infinite network

Cited by 31 publications

References 4 publications

Functions with finite Dirichlet sum of orderpand quasi-monomorphisms of infinite graphs

Functions with finite Dirichlet sum of orderpand quasi-monomorphisms of infinite graphs

Some results concerning the $p$-Royden and $p$-harmonic boundaries of a graph of bounded degree

Parabolic theory of the discrete -Laplace operator

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