Applications of Boundary Harnack Inequalities for p Harmonic Functions and Related Topics

Lewis, John L.

doi:10.1007/978-3-642-27145-8_1

Cited by 4 publications

(5 citation statements)

References 52 publications

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“…Note that the set S u∞ (respectively, S u 0 ) of critical points of u ∞ (respectively, u 0 ) is empty. Moreover, by Theorem 2.15 of [18], the boundary point lemma is valid for the p -Laplacian on ∂X \ {0}. It follows as in the proof of Theorem 1.2 that ∞ (respectively, 0) is a regular point with respect to u ∞ and u 0 , and that u ∞ (respectively, u 0 if p ≥ d) is the unique positive p -harmonic function in X of minimal growth in ∂ X \ {∞} (respectively, ∂ X \ {0}).…”

Section: Uniform Harnack Inequality and Behavior Near Regular Pointsmentioning

confidence: 88%

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POSITIVE LIOUVILLE THEOREMS AND ASYMPTOTIC BEHAVIOR FOR p-LAPLACIAN TYPE ELLIPTIC EQUATIONS WITH A FUCHSIAN POTENTIAL

Fraas

Pinchover

2011

Confluentes Math.

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We study positive Liouville theorems and the asymptotic behavior of positive solutions of p-Laplacian type elliptic equations of the formwhere X is a domain in R d , d ≥ 2, and 1 < p < ∞. We assume that the potential V has a Fuchsian type singularity at a point ζ, where either ζ = ∞ and X is a truncated C 2 -cone, or ζ = 0 and ζ is either an isolated point of ∂X or belongs to a C 2 -portion of ∂X.

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Section: Uniform Harnack Inequality and Behavior Near Regular Pointsmentioning

confidence: 88%

“…In particular, under some restrictions, Poisson's principle for a Fuchsian type p -Laplace equation of the form (1.1) in a bounded smooth domain is proved in [4]. For other Liouville theorems for quasilinear equations see for example [5,12,18,30], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

POSITIVE LIOUVILLE THEOREMS AND ASYMPTOTIC BEHAVIOR FOR p-LAPLACIAN TYPE ELLIPTIC EQUATIONS WITH A FUCHSIAN POTENTIAL

Fraas

Pinchover

2011

Confluentes Math.

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“…Results on NTA domains. For convenience of the reader, we recall here the notion of non-tangencially accessible domain (NTA), introduced in [28] (the version we present here is taken from [31]). (2) R N \ Ω satisfies the corkscrew condition (3) If w ∈ ∂Ω and w 1 , w 2 ∈ B r0 (w) ∩ Ω, then there is a rectifiable curve γ : [0, 1] → Ω with γ(0) = w 1 , γ(1) = w 2 , and: (a) H 1 (γ) M |w 1 − w 2 |, (b) min{H 1 (γ([0, t])), H 1 (γ([t, 1]))} M dist(γ(t), ∂Ω).…”

Section: 3mentioning

confidence: 99%

Section: Appendix a Boundary Behavior On Nta And Reifenberg Flat Domainsmentioning

confidence: 99%

Extremality Conditions and Regularity of Solutions to Optimal Partition Problems Involving Laplacian Eigenvalues

Ramos

Tavares

Terracini

2015

Arch Rational Mech Anal

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Abstract. Let Ω ⊂ R N be an open bounded domain and m ∈ N. Given k 1 , . . . , km ∈ N, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, of the following formwhere λ k i (ω i ) denotes the k i -th eigenvalue of (−∆, H 1 0 (ω i )) counting multiplicities, and Pm(Ω) is the set of all open partitions of Ω, namelyWhile existence of a quasi-open optimal partition (ω 1 , . . . , ωm) follows from a general result by Bucur, Buttazzo and Henrot [Adv. Math. Sci. Appl. 8, 1998], the aim of this paper is to associate with such minimal partitions and their eigenfunctions some suitable extremality conditions and to exploit them, proving as well the Lipschitz continuity of some eigenfunctions, and regularity of the partition in the sense that the free boundary ∪ m i=1 ∂ω i ∩ Ω is, up to a residual set, locally a C 1,α hypersurface. This last result extend the ones in the paper by Caffarelli and Lin [J. Sci. Comput. 31, 2007] to the case of higher eigenvalues.

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“…This is of course an old exploit used for a variety of theoretical and practical pursuits, particularly underlying the standard theory of p-harmonic functions and the p-Laplacian (see e.g. [16,23,11,1,19]). In the theory of conformal dimension, it offers an alternative to the methods based on the p-modulus or p-extremal length.…”

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confidence: 99%

Conformal dimension via p-resistance: Sierpinski carpet

Kwapisz¹

2020

Ann. Acad. Sci. Fenn. Math.

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We put forth the notion of p-resistance as a proxy for the combinatorial p-modulus and demonstrate its effectiveness by studying the (Ahlfors regular) conformal dimension of the Sierpiński carpet. Specifically, we construct large resistor network approximating the carpet, establish weak-sup and sub-multiplicativity of their p-resistances, identify the conformal dimension as the associated critical exponent, and provide numerical approximations and rigorous two-sided bounds. In particular, we prove that the conformal dimension of the carpet exceeds 1 + ln 2/ ln 3, the Hausdorff dimension of the Cantor comb contained therein. A conjectural construction (and a numerical picture) of the quasi-symmetric uniformization of the carpet emerges as a byproduct.

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Applications of Boundary Harnack Inequalities for p Harmonic Functions and Related Topics

Cited by 4 publications

References 52 publications

POSITIVE LIOUVILLE THEOREMS AND ASYMPTOTIC BEHAVIOR FOR p-LAPLACIAN TYPE ELLIPTIC EQUATIONS WITH A FUCHSIAN POTENTIAL

POSITIVE LIOUVILLE THEOREMS AND ASYMPTOTIC BEHAVIOR FOR p-LAPLACIAN TYPE ELLIPTIC EQUATIONS WITH A FUCHSIAN POTENTIAL

Extremality Conditions and Regularity of Solutions to Optimal Partition Problems Involving Laplacian Eigenvalues

Conformal dimension via p-resistance: Sierpinski carpet

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