Picard principle for negative planar potentials

Nakai, Mitsuru; Tada, Toshimasa

doi:10.1007/s002080050077

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Remark 551

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2009

2011

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Cited by 2 publications

(2 citation statements)

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“…As we already stated in the introduction, it is proven in [19] that + μ satisfies the Picard principle, if d = 2. Since the proof is quite involved, let us observe that Lemma 5.4 leads to a simple proof in the case, where g 0 = ∞.…”

Section: Remark 55mentioning

confidence: 82%

See 1 more Smart Citation

On the Picard principle for Δ + μ

Hansen

Netuka

2011

Math. Z.

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Given a (local) Kato measure μ on R d \ {0}, d ≥ 2, let H +μ 0 (U ) be the convex cone of all continuous real solutions u ≥ 0 to the equation u + uμ = 0 on the punctured unit ball U satisfying lim |x|→1 uwhere G denotes the Green function on U , is bounded on L 2 (U, μ) and has a norm which is at most one. Moreover, extremal rays in H +μ 0 (U ) are characterized and it is proven that + μ satisfies the Picard principle on U , that is, that H +μ 0 (U ) consists of one ray, provided there exists a suitable sequence of shells in U such that, on these shells, μ is either small or not too far from being radial. Further, it is shown that the verification of the Picard principle can be localized. Several results on L 2 -(sub)eigenfunctions and 3G-inequalities which are used in the paper, but may be of independent interest, are proved at the end of the paper.

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Section: Remark 55mentioning

confidence: 82%

“…In [19] it is shown that (1.2) holds, if d = 2. Moreover, it is satisfied if μ has a rotationally invariant density which is locally Hölder continuous ( [18]).…”

Section: Introductionmentioning

confidence: 99%