Estimates of Green and Martin kernels for Schrödinger operators with singular potential in Lipschitz domains

Marcus, Moshe

doi:10.1016/j.anihpc.2018.09.003

Marcus, Moshe

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1183-1200

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Résumé

Consider operators of the form $L^{γ V} : = Δ + γ V$ in a bounded Lipschitz domain $Ω \subset R^{N}$ . Assume that $V \in C^{1} (Ω)$ satisfies $| V (x) | \leq \bar{a} dist {(x, \partial Ω)}^{- 2}$ for every $x \in Ω$ and γ is a number in a range $(γ_{-}, γ_{+})$ described in the introduction. The model case is $V (x) = dist {(x, F)}^{- 2}$ where F is a closed subset of ∂Ω and $γ < c_{H} (V) =$ Hardy constant for V. We provide sharp two sided estimates of the Green and Martin kernel for $L^{γ V}$ in Ω. In addition we establish a pointwise version of the 3G inequality.

MR Zbl

DOI : 10.1016/j.anihpc.2018.09.003

Keywords: Hardy constant, First eigenfunction, Boundary Harnack principle, 3G inequality

@article{AIHPC_2019__36_5_1183_0,
     author = {Marcus, Moshe},
     title = {Estimates of {Green} and {Martin} kernels for {Schr\"odinger} operators with singular potential in {Lipschitz} domains},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1183--1200},
     year = {2019},
     publisher = {Elsevier},
     volume = {36},
     number = {5},
     doi = {10.1016/j.anihpc.2018.09.003},
     mrnumber = {3985541},
     zbl = {1426.35098},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2018.09.003/}
}

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Marcus, Moshe. Estimates of Green and Martin kernels for Schrödinger operators with singular potential in Lipschitz domains. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1183-1200. doi: 10.1016/j.anihpc.2018.09.003

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