Optimal L p Hardy-type inequalities

Devyver, Baptiste; Pinchover, Yehuda

doi:10.1016/j.anihpc.2014.08.005

Optimal L ^p Hardy-type inequalities

Devyver, Baptiste ; Pinchover, Yehuda

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 93-118

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Résumé

Let Ω be a domain in $R^{n}$ or a noncompact Riemannian manifold of dimension $n \geq 2$ , and $1 < p < \infty$ . Consider the functional $Q (φ) : = \int_{Ω} ({| \nabla φ |}^{p} + V {| φ |}^{p}) d ν$ defined on $C_{0}^{\infty} (Ω)$ , and assume that $Q \geq 0$ . The aim of the paper is to generalize to the quasilinear case ( $p \neq 2$ ) some of the results obtained in [6] for the linear case ( $p = 2$ ), and in particular, to obtain “as large as possible” nonnegative (optimal) Hardy-type weight W satisfying

Q (φ) \geq \int_{Ω} W {| φ |}^{p} d ν \forall φ \in C_{0}^{\infty} (Ω) .

Our main results deal with the case where $V = 0$ , and Ω is a general punctured domain (for $V \neq 0$ we obtain only some partial results). In the case $1 < p \leq n$ , an optimal Hardy-weight is given by

W : = {(\frac{p - 1}{p})}^{p} {| \frac{\nabla G}{G} |}^{p},

where G is the associated positive minimal Green function with a pole at 0. On the other hand, for

p > n

, several cases should be considered, depending on the behavior of G at infinity in Ω. The results are extended to annular and exterior domains.

MR Zbl

DOI : 10.1016/j.anihpc.2014.08.005

Keywords: Hardy inequality, Optimal, p-Laplacian

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     author = {Devyver, Baptiste and Pinchover, Yehuda},
     title = {Optimal {\protect\emph{L}}         \protect\textsuperscript{            \protect\emph{p}         } {Hardy-type} inequalities},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {93--118},
     publisher = {Elsevier},
     volume = {33},
     number = {1},
     year = {2016},
     doi = {10.1016/j.anihpc.2014.08.005},
     mrnumber = {3436428},
     zbl = {1331.35013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.08.005/}
}

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Devyver, Baptiste; Pinchover, Yehuda. Optimal L         ^p Hardy-type inequalities. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 93-118. doi: 10.1016/j.anihpc.2014.08.005

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