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Dirichlet problem for Schrödinger equation with the boundary value in the BMO space

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Abstract

Let (X, d, μ) be a metric measure space satisfying a Q-doubling condition (Q > 1) and an L2-Poincaré inequality. Let \({\cal L} = {\cal L} + V\) be a Schrödinger operator on X, where \({\cal L}\) is a non-negative operator generalized by a Dirichlet form, and V is a non-negative Muckenhoupt weight that satisfies a reverse Hölder condition RHq for some q ⩾ (Q + 1)/2. We show that a solution to \(({\cal L} - \partial _t^2)u = 0\) on X × ℝ+ satisfies the Carleson condition

$$\mathop {{\rm{sub}}}\limits_{B({x_B},{r_B})} {1 \over {\mu (B({x_B},{r_B}))}}\int_0^{{r_B}} {\int_{B({x_B},{r_B})} {{{\left| {t\nabla u(x,t)} \right|}^2}{{d\mu dt} \over t} < \infty } } $$

if and only if u can be represented as the Poisson integral of the Schrödinger operator ℒ with the trace in the BMO (bounded mean oscillation) space associated with ℒ.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11922114, 11671039 and 11771043). The first author thanks Professor Lixin Yan for many helpful discussions during the preparation of this work. The authors are grateful for the referees’ comments on the original version of this paper.

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  1. Bo Li

    Present address: Current address: College of Data Science, Jiaxing University, Jiaxing, 314001, China

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  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Renjin Jiang & Bo Li

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Jiang, R., Li, B. Dirichlet problem for Schrödinger equation with the boundary value in the BMO space. Sci. China Math. 65, 1431–1468 (2022). https://doi.org/10.1007/s11425-020-1834-1

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