利普希兹区域上薛定谔方程Neumann问题的加权估计

Abstract

Abstract:

In this paper, we consider the weighted estimates in the weighted space H^p(∂Ω,ω_αdσ) or L^p(∂Ω,ω_αdσ)(1-ε < p≤2) for Schrödinger equation -Δu+Vu=0 on Lipschitz domains. Let Ω be a bounded Lipschitz domain with connected boundary in Rⁿ, n≥3. Let ω_α(Q)=|Q-Q₀|^α, where Q₀ is a fixed point on ∂Ω. For Schrödinger equation -Δu+Vu=0 in Ω, with singular non-negative potentials V belonging to the reverse Hölder class B_n, we study the Neumann problem with boundary date lies in the weighted space H^p(∂Ω,ω_αdσ) or L^p(∂Ω,ω_αdσ), where dσ denotes the surface measure on ∂Ω. We show that for certain ranges of α, there is a unique solution u, such that the non-tangential maximal function of ▽u is in H^p(∂Ω,ω_αdσ) or L^p(∂Ω,ω_αdσ). Moreover, the uniform estimates are founded.

Key words: Neumann Problem, Schrödinger Equation, Lipschitz Domains, Weighted Estimates

CLC Number:

O174.2

Huang Wenli, Tao Xiangxing. Weighted Estimates on the Neumann Problem for L² for Schrödinger Equations in Lipschitz Domains[J].Acta mathematica scientia,Series A, 2016, 36(6): 1165-1185.

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References

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Weighted Estimates on the Neumann Problem for L² for Schrödinger Equations in Lipschitz Domains

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Weighted Estimates on the Neumann Problem for L2 for Schrödinger Equations in Lipschitz Domains

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Weighted Estimates on the Neumann Problem for L² for Schrödinger Equations in Lipschitz Domains