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

  • * Corresponding author: phan@utk.edu

    * Corresponding author: phan@utk.edu 

T. Phan's research is partly supported by the Simons Foundation, grant # 354889. The third author would like to thank the Department of Mathematics at the University of Tennessee, Knoxville for its hospitality from which part of this work was done

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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 mechanisms 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.

    Mathematics Subject Classification: Primary: 35J10, 35J15; Secondary: 35B45.


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