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Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials

Tuoc Phan, ,Department of Mathematics, University of Tennessee - Knoxville, 227 Ayress Hall, 1403 Circle Drive, Knoxville, TN 37996, USA, Grozdena Todorova, Borislav Yordanov, ,Integrated Science Program, Office of International Affairs, Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Hokkaido, Japan
2019 Discrete and Continuous Dynamical Systems. Series A  

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This paper studies second order elliptic equations in both divergence and non-divergence forms with measurable complex valued principle coefficients and measurable complex valued potentials. The PDE operators can be considered as generalized Schrödinger operators. Under some sufficient conditions, we prove existence, uniqueness, and regularity estimates in Sobolev spaces for solutions to the equations. We particularly show that the non-zero imaginary parts of the potentials are the main
more » ... ms that control the solutions. Our results can be considered as limiting absorption principle for Schrödinger operators with measurable coefficients and they could be useful in applications. The approach is based on the perturbation technique that freezes the potentials. The results of the paper not only generalize known results but also provide a key ingredient for the study of L p -diffusion phenomena for dissipative wave equations. 2020 Mathematics Subject Classification. Primary: 35J10, 35J15; Secondary: 35B45.
doi:10.3934/dcds.2020310 fatcat:4bmsy2bcmrhbpevsojkvois2b4
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Tuoc Phan, ,Department of Mathematics, University of Tennessee - Knoxville, 227 Ayress Hall, 1403 Circle Drive, Knoxville, TN 37996, USA, Grozdena Todorova, Borislav Yordanov, ,Integrated Science Program, Office of International Affairs, Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Hokkaido, Japan. "Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials." Discrete and Continuous Dynamical Systems. Series A 0.0 (2019) 0-0