Hopf's lemma for a class of singular/degenerate PDE-s

Mikayelyan, Hayk; Shahgholian, Henrik

doi:10.5186/aasfm.2015.4033

Cited by 13 publications

(9 citation statements)

References 12 publications

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“…However, if for instance u 0 > 0 in Ω or if u 0 ≥ 0 and Ω is regular enough in order to have a Hopf's lemma (for instance C 1,α , cf. [10]), then it is straightforward to verify that ψ is indeed non-zero. The iteration scheme (1.2) was introduced by R. Biezuner, G. Ercole, and E. Martins in [1] who conjectured the limit…”

Section: Introductionmentioning

confidence: 99%

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Inverse iteration for $p$-ground states

Hynd

Lindgren

2015

Proc. Amer. Math. Soc.

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We adapt the inverse iteration method for symmetric matrices to some nonlinear PDE eigenvalue problems. In particular, for p ∈ (1, ∞) and a given domain Ω ⊂ R n , we analyze a scheme that allows us to approximate the smallest value the ratio Ω |Dψ| p dx/ Ω |ψ| p dx can assume for functions ψ that vanish on ∂Ω. The scheme in question also provides a natural way to approximate minimizing ψ. Our analysis also extends in the limit as p → ∞ and thereby fashions a new approximation method for ground states of the infinity Laplacian.

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Section: Introductionmentioning

confidence: 99%

“…It may not be obvious how to verify that the limiting function ψ is not identically zero. However, if for instance u 0 > 0 in Ω or if u 0 ≥ 0 and Ω is regular enough in order to have a Hopf's lemma (for instance C 1,α , cf [10]…”

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confidence: 99%

Inverse iteration for $p$-ground states

Hynd

Lindgren

2015

Proc. Amer. Math. Soc.

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“…Such a minimal Green function G provides us with a positive p-harmonic function defined in a relative neighbourhood of ∂Ω which will be used in the sequel. Importantly, if ∂Ω is of class C 1,γ with 0 < γ < 1, then G is C 1,α up to the boundary, G(x) = 0 and ∇G(x) = 0 for all x ∈ ∂Ω since the Hopf lemma holds, see [26].…”

Section: ω{X0}mentioning

confidence: 99%

“…We believe that Lemma 3.2 is of independent interest since it is proved without the use of the tubular neighbourhood theorem and actually allows to bypass it. We note that the Hopf lemma holds also if ∂Ω is of class C 1,Dini (see, [26]) and this allows to gain some generality as it is explained in Remark 4.2.…”

Section: Introductionmentioning

confidence: 96%

$L^p$ Hardy inequality on $C^{1,\gamma}$ domains

Lamberti¹,

Pinchover²

2019

Annali Scuola Normale Superiore - Classe Di Scienze

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We consider the L p Hardy inequality involving the distance to the boundary of a domain in the n-dimensional Euclidean space with nonempty compact boundary. We extend the validity of known existence and nonexistence results, as well as the appropriate tight decay estimates for the corresponding minimizers, from the case of domains of class C 2 to the case of domains of class C 1,γ with γ ∈ (0, 1]. We consider both bounded and exterior domains. The upper and lower estimates for the minimizers in the case of exterior domains and the corresponding related non-existence result seem to be new even for C 2 -domains.2010 Mathematics Subject Classification. 49R05, 35B09, 35J92.

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“…The widening of the class of operators to singular/degenerate ones was made in the papers [KH75], [KH77] and [ABM + 11], while the uniform elliptic operators with unbounded lower order coefficients were studied in [Saf10] and [Naz12] (see also [NU09]). We mention also the publications [Tol83] and [MS15] where the Boundary Point Principle was established for a class of degenerate quasilinear operators including the p-Laplacian.…”

Section: Introductionmentioning

confidence: 99%