Nonlinear potential theory on metric spaces

Kinnunen, Juha; Мартио, Олли

doi:10.1215/ijm/1258130989

Cited by 90 publications

(130 citation statements)

References 3 publications

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“…In the last decade, there has been a lot of development in the theory of p-harmonic functions on metric spaces and the Dirichlet problem for p-harmonic functions has been solved for rather general boundary data (including Sobolev and continuous functions) in e.g., Cheeger [11], Shanmugalingam [30], Kinnunen-Martio [23] and Björn et al [5,6]. A recent survey of calculus on metric spaces is in Heinonen [16].…”

Section: Introductionmentioning

confidence: 99%

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

Björn

2008

Calc. Var.

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We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincaré inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear elliptic equations on R n . Mathematics Subject Classification (2000)Primary 35J67; Secondary 49N60 · 35B45 · 35B65 · 31C45

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Section: Introductionmentioning

confidence: 99%

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

Björn

2008

Calc. Var.

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“…It is clear that u ≤ 0, that u is bounded from below and that u(x 0 ) = 0. By Proposition 5.5 in Kinnunen-Martio [21], u| A = P A h, where A = {y ∈ Ω 2 ∪ 2B : u(y) > h(y)}. Let G be a component of A, then u| G = P G h. Since C p (X \ G) ≥ C p (X \ (Ω 2 ∪ 2B)) > 0, Lemma 8.6 in Björn-Björn-Shanmugalingam [8] shows that C p (∂G) > 0.…”

Section: ¬(D) ⇒ ¬(C)mentioning

confidence: 85%

“…For a function f ∈ N 1,p (Ω) and an arbitrary function ψ : Ω → [−∞, ∞] we let, following KinnunenMartio [21], …”

Section: Regularity For Quasiminimizersmentioning

confidence: 99%

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Pasting Lemmas and Characterizations of Boundary Regularity for Quasiminimizers

Björn

Мартио

2009

Results. Math.

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Quasiharmonic functions correspond to p-harmonic functions when minimizers of the p-Dirichlet integral are replaced by quasiminimizers. In this paper, boundary regularity for quasiminimizers is characterized in several ways; in particular it is shown that regularity is a local property of the boundary. For these characterizations we employ a version of the so called pasting lemma; this is a useful tool in the theory of superharmonic functions and our version extends the classical pasting lemma to quasisuperharmonic functions and quasisuperminimizers.The results are obtained for metric measure spaces, but they are new also in the Euclidean spaces. Mathematics Subject Classification (2000). Primary: 31C45; Secondary: 31B25, 35J60, 35J65.

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“…As mentioned before, this problem has been extensively studied also on measure metric spaces with a doubling property and a Poincaré inequality (for example, bounded domains in Riemannian manifolds with respect to the measure induced by the metric), and Proposition 1.2 holds even in this more general setting (see [11]). …”

Section: Proposition 12 If Is a Bounded Domain In Amentioning

confidence: 99%

Reverse Khas’minskii condition

Valtorta

2010

Math. Z.

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The aim of this paper is to present and discuss some equivalent characterizations of p-parabolicity for complete Riemannian manifolds in terms of existence of special exhaustion functions. In particular, Khas'minskii in Ergodic properties of recurrent diffusion prossesses and stabilization of solution to the Cauchy problem for parabolic equations (Theor Prob Appl 5(2), 1960) proved that if there exists a 2-superharmonic function K defined outside a compact set on a complete Riemannian manifold R such that lim x→∞ K(x) = ∞, then R is 2-parabolic, and Sario and Nakai in Classification theory of Riemann surfaces (Springer, Berlin, 1970) were able to improve this result by showing that R is 2-parabolic if and only if there exists an Evans potential, i.e. a 2-harmonic function E : R\K → R + with lim x→∞ E(x) = ∞. In this paper, we will prove a reverse Khas'minskii condition valid for any p > 1 and discuss the existence of Evans potentials in the nonlinear case.

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Nonlinear potential theory on metric spaces

Cited by 90 publications

References 3 publications

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

Pasting Lemmas and Characterizations of Boundary Regularity for Quasiminimizers

Reverse Khas’minskii condition

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