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

Tuoc Phan; Grozdena Todorova; Borislav Yordanov

doi:10.3934/dcds.2020310

Article Contents

2021, Volume 41, Issue 3: 1071-1099. Doi: 10.3934/dcds.2020310

This issue Previous Article Pomeau-Manneville maps are global-local mixing Next Article Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere

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

1.
Department of Mathematics, University of Tennessee - Knoxville, 227 Ayress Hall, 1403 Circle Drive, Knoxville, TN 37996, USA
2.
Integrated Science Program, Office of International Affairs, Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Hokkaido, Japan

^* Corresponding author: phan@utk.edu
^* Corresponding author: phan@utk.edu

Received: March 2020

Revised: July 2020

Early access: August 2020

Published: March 2021

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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Abstract

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.

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

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