Global Comparison Principles for the p-Laplace Operator on Riemannian Manifolds

Holopainen, Ilkka; Pigola, Stefano; Veronelli, Giona

doi:10.1007/s11118-010-9199-4

Cited by 14 publications

(20 citation statements)

References 13 publications

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“…In the theorem below we demonstrate a global comparison principle for complete Riemannian manifolds with infinite volume whose geodesic flow is recurrent. Similar results can be found in [18], [14] and [6].…”

Section: Applicationssupporting

confidence: 89%

A divergence theorem for non-compact Riemannian manifolds: a dynamical approach

Melo¹,

Pujals

2019

Nonlinearity

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In this paper we use a dynamical approach to prove some new divergence theorems on complete non-compact Riemannian manifolds. 10ÍTALO MELO AND ENRIQUE PUJALS Proof of Theorem 1.3Observe that SM is the disjoint union SM = D ∪ C + ∪ (C − \C + ). Since C + is φ tinvariant and for almost all θ ∈ C + , θ is recurrent, by the proof of Theorem 1.1 we haveNote that C − \C + is also φ t -invariant and for almost all θ ∈ C − , θ is recurrent with respect to the inverse geodesic flow. By the proof of Theorem 1.1 we haveLet d,d be the distances on M and SM respectively, for almost all θ ∈ D we havẽ d(θ, φ t (θ)) → ∞ as |t| → ∞. On the other hand, we haved(θ, φ t (θ)) ≤ d(p, γ θ (t)) + 2π, where θ = (p, v). Thus, for almost θ ∈ SM we get d(p, γ θ (t)) → ∞ as |t| → ∞.On the other hand,Since f X is integrable and |X| → 0 at infinity in M , for almost all θ ∈ D it follows that ∞ −∞ f X (φ t (θ)) dt = 0.By Proposition 2.13 follows that

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Section: Applicationssupporting

confidence: 89%

A divergence theorem for non-compact Riemannian manifolds: a dynamical approach

Melo¹,

Pujals

2019

Nonlinearity

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“…In [17], the authors take a first step in this direction by proving that a map u : M → N with finite p-energy and homotopic to a constant is constant provided M is p-parabolic and N Sect ≤ 0. Using Theorem 6 in the proof of their result, we easily obtain the following In [12], the authors apply the Kelvin-Nevanlinna-Royden criterion to obtain a vector valued version of their comparison theorem. In some sense this result is a further step in treating the problem of the uniqueness of p-harmonic representative.…”

Section: Applicationsmentioning

confidence: 99%

“…As a matter of fact, in their proof X is defined in such a way that the smoothness of u and v seems to be strictly necessary to do computations. In order to generalize their result, in the direction of Proposition 7, apparently assumption (12) can not be dropped. Nevertheless, in case |du|, |dv| ∈ L p and their L p -norms decay fast with respect to the volume in the A M,p sense, we obtain a similar result for non p-parabolic manifolds and for maps with low regularity.…”

Section: Applicationsmentioning

confidence: 99%

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Stokes' theorem, volume growth and parabolicity

Valtorta¹,

Veronelli²

2011

Tohoku Math. J. (2)

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“…[47] 6.3). In [15], I. Holopainen, S. Pigola and G. Veronelli showed that if u, v ∈ W 1,p loc (M) ∩ C 0 (M) satisfy ∆ p u ≥ ∆ p v weakly and |∇u| , |∇v| ∈ L p (M) , for p > 1, then u − v is constant provided M is connected, possibly incomplete, p-parabolic Riemannian manifold. They also discussed L q comparison principles in the non-parabolic setting.…”

Section: Introductionmentioning

confidence: 99%

Liouville properties for 𝑝-harmonic maps with finite 𝑞-energy

Chang

Chen

Wei

2015

Trans. Amer. Math. Soc.

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Abstract. We introduce and study an approximate solution of the p-Laplace equation, and a linearlization L ǫ of a perturbed p-Laplace operator. By deriving an L ǫ -type Bochner's formula and Kato type inequalities, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type Theorem for p-harmonic morphisms.

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Global Comparison Principles for the p-Laplace Operator on Riemannian Manifolds

Abstract: Abstract. We prove global comparison results for the p-Laplacian on a p-parabolic manifold. These involve both real-valued and vector-valued maps with finite p-energy.

Cited by 14 publications

References 13 publications

A divergence theorem for non-compact Riemannian manifolds: a dynamical approach

A divergence theorem for non-compact Riemannian manifolds: a dynamical approach

Stokes' theorem, volume growth and parabolicity

Liouville properties for 𝑝-harmonic maps with finite 𝑞-energy

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