# American Institute of Mathematical Sciences

March  2021, 41(3): 1071-1099. doi: 10.3934/dcds.2020310

## 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

Received  March 2020 Revised  July 2020 Published  March 2021 Early access  August 2020

Fund Project: 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

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.

Citation: Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310
##### References:
 [1] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., 15 (1962), 119-147.  doi: 10.1002/cpa.3160150203. [2] L. Börjeson, Estimates for the Bochner-Riesz operator with negative index, Indiana Univ. Math. J., 35 (1986), 225-233.  doi: 10.1512/iumj.1986.35.35013. [3] S.-S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310.  doi: 10.1002/cpa.20037. [4] S.-S. Byun and L. Wang, Parabolic equations in Reifenberg domains, Arch. Ration. Mech. Anal., 176 (2005), 271-301.  doi: 10.1007/s00205-005-0357-6. [5] L. A. Caffarelli and I. Peral, On $W^{1, p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1-21.  doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.0.CO;2-G. [6] F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2, p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ric. Mat., 40 (1991), 149-168. [7] J.-C. Cuenin, Eigenvalue bounds for Dirac and fractional Schrödinger operators with complex potentials, J. Funct. Anal., 272 (2017), 2987-3018.  doi: 10.1016/j.jfa.2016.12.008. [8] H. Dong and D. Kim, Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth, Comm. Partial Differential Equations, 36 (2011), 1750-1777.  doi: 10.1080/03605302.2011.571746. [9] H. Dong and D. Kim, Parabolic and elliptic systems in divergence form with variably partially BMO coefficients, SIAM J. Math. Anal., 43 (2011), 1075-1098.  doi: 10.1137/100794614. [10] H. Dong and D. Kim, Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces, Adv. Math., 274 (2015), 681-735.  doi: 10.1016/j.aim.2014.12.037. [11] H. Dong and N. V. Krylov, Fully nonlinear elliptic and parabolic equations in weighted and mixed-norm Sobolev spaces, Calc. Var. Partial Differential Equations, 58 (2019), Art. 145, 32 pp. doi: 10.1007/s00526-019-1591-3. [12] J. Földes and T. Phan, On higher integrability estimates for elliptic equations with singular coefficients, submitted, arXiv: 1804.03180. [13] R. L. Frank, A. Laptev, E. H. Lieb and R. Seiringer, Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials, Lett. Math. Phys., 77 (2006), 309-316.  doi: 10.1007/s11005-006-0095-1. [14] R. L. Frank and B. Simon, Eigenvalue bounds for Schrödinger operators with complex potentials Ⅱ, J. Spectr. Theory, 7 (2017), 633-658.  doi: 10.4171/JST/173. [15] R. L. Frank, Eigenvalue bounds for Schrödinger operators with complex potentials Ⅲ, Trans. Amer. Math. Soc., 370 (2018), 219-240.  doi: 10.1090/tran/6936. [16] Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Note, AMS, 1997. [17] L. T. Hoang, T. V. Nguyen and T. V. Phan, Gradient estimates and global existence of smooth solutions to a cross-diffusion system, SIAM J. Math. Anal., 47 (2015), 2122-2177.  doi: 10.1137/140981447. [18] E. Jeong, Y. Kwon and S. Lee, Uniform Sobolev inequalities for second order non-elliptic differential operators, Adv. Math., 302 (2016), 323-350.  doi: 10.1016/j.aim.2016.07.016. [19] B. Kang and H. Kim, On $L^p$-resolvent estimates for second-order elliptic equations in divergence form, Potential Anal., 50 (2019), 107-133.  doi: 10.1007/s11118-017-9675-1. [20] C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 55 (1987), 329-347.  doi: 10.1215/S0012-7094-87-05518-9. [21] D. Kim and N. V. Krylov, Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others, SIAM J. Math. Anal., 39 (2007), 489-506.  doi: 10.1137/050646913. [22] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations, 32 (2007), 453-475.  doi: 10.1080/03605300600781626. [23] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, 96. American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096. [24] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London, 1968. [25] A. Maugeri, D. K. Palagachev and L. G. Softova., Elliptic and Parabolic Equations with Discontinuous Coefficients, Mathematical Research, 109. Wiley-VCH Verlag Berlin GmbH, Berlin, 2000. doi: 10.1002/3527600868. [26] N. G. Meyers, An $L^{p}$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1963), 189-206. [27] T. Nguyen, Interior Calderón-Zygmund estimates for solutions to general parabolic equations of $p$-Laplacian type, Calc. Var. Partial Differential Equations, 56 (2017), Art. 173, 42 pp. doi: 10.1007/s00526-017-1265-y. [28] T. Nguyen and T. Phan, Interior gradient estimates for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 55 (2016), Art. 59, 33 pp. doi: 10.1007/s00526-016-0996-5. [29] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. [30] T. Phan, G. Todorova and B. Yordanov, $L^p$-diffusion phenomena for dissipative wave equations, preprint, (2020). [31] T. Phan, Regularity estimates for BMO-weak solutions of quasilinear elliptic equations with inhomogeneous boundary conditions, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Paper No. 8, 49 pp. doi: 10.1007/s00030-018-0501-2. [32] T. Phan, Lorentz estimates for weak solutions of quasi-linear parabolic equations with singular divergence-free drifts, Canad. J. Math., 71 (2019), 937-982.  doi: 10.4153/CJM-2017-049-3. [33] L. Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sin., 19 (2003), 381-396.  doi: 10.1007/s10114-003-0264-4. [34] W. Wei and Z. Zhang, $L^p$ resolvent estimates for constant coefficient elliptic systems on Lipschitz domains, J. Funct. Anal., 267 (2014), 3262-3293.  doi: 10.1016/j.jfa.2014.08.010.

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##### References:
 [1] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., 15 (1962), 119-147.  doi: 10.1002/cpa.3160150203. [2] L. Börjeson, Estimates for the Bochner-Riesz operator with negative index, Indiana Univ. Math. J., 35 (1986), 225-233.  doi: 10.1512/iumj.1986.35.35013. [3] S.-S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310.  doi: 10.1002/cpa.20037. [4] S.-S. Byun and L. Wang, Parabolic equations in Reifenberg domains, Arch. Ration. Mech. Anal., 176 (2005), 271-301.  doi: 10.1007/s00205-005-0357-6. [5] L. A. Caffarelli and I. Peral, On $W^{1, p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1-21.  doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.0.CO;2-G. [6] F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2, p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ric. Mat., 40 (1991), 149-168. [7] J.-C. Cuenin, Eigenvalue bounds for Dirac and fractional Schrödinger operators with complex potentials, J. Funct. Anal., 272 (2017), 2987-3018.  doi: 10.1016/j.jfa.2016.12.008. [8] H. Dong and D. Kim, Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth, Comm. Partial Differential Equations, 36 (2011), 1750-1777.  doi: 10.1080/03605302.2011.571746. [9] H. Dong and D. Kim, Parabolic and elliptic systems in divergence form with variably partially BMO coefficients, SIAM J. Math. Anal., 43 (2011), 1075-1098.  doi: 10.1137/100794614. [10] H. Dong and D. Kim, Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces, Adv. Math., 274 (2015), 681-735.  doi: 10.1016/j.aim.2014.12.037. [11] H. Dong and N. V. Krylov, Fully nonlinear elliptic and parabolic equations in weighted and mixed-norm Sobolev spaces, Calc. Var. Partial Differential Equations, 58 (2019), Art. 145, 32 pp. doi: 10.1007/s00526-019-1591-3. [12] J. Földes and T. Phan, On higher integrability estimates for elliptic equations with singular coefficients, submitted, arXiv: 1804.03180. [13] R. L. Frank, A. Laptev, E. H. Lieb and R. Seiringer, Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials, Lett. Math. Phys., 77 (2006), 309-316.  doi: 10.1007/s11005-006-0095-1. [14] R. L. Frank and B. Simon, Eigenvalue bounds for Schrödinger operators with complex potentials Ⅱ, J. Spectr. Theory, 7 (2017), 633-658.  doi: 10.4171/JST/173. [15] R. L. Frank, Eigenvalue bounds for Schrödinger operators with complex potentials Ⅲ, Trans. Amer. Math. Soc., 370 (2018), 219-240.  doi: 10.1090/tran/6936. [16] Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Note, AMS, 1997. [17] L. T. Hoang, T. V. Nguyen and T. V. Phan, Gradient estimates and global existence of smooth solutions to a cross-diffusion system, SIAM J. Math. Anal., 47 (2015), 2122-2177.  doi: 10.1137/140981447. [18] E. Jeong, Y. Kwon and S. Lee, Uniform Sobolev inequalities for second order non-elliptic differential operators, Adv. Math., 302 (2016), 323-350.  doi: 10.1016/j.aim.2016.07.016. [19] B. Kang and H. Kim, On $L^p$-resolvent estimates for second-order elliptic equations in divergence form, Potential Anal., 50 (2019), 107-133.  doi: 10.1007/s11118-017-9675-1. [20] C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 55 (1987), 329-347.  doi: 10.1215/S0012-7094-87-05518-9. [21] D. Kim and N. V. Krylov, Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others, SIAM J. Math. Anal., 39 (2007), 489-506.  doi: 10.1137/050646913. [22] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations, 32 (2007), 453-475.  doi: 10.1080/03605300600781626. [23] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, 96. American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096. [24] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London, 1968. [25] A. Maugeri, D. K. Palagachev and L. G. Softova., Elliptic and Parabolic Equations with Discontinuous Coefficients, Mathematical Research, 109. Wiley-VCH Verlag Berlin GmbH, Berlin, 2000. doi: 10.1002/3527600868. [26] N. G. Meyers, An $L^{p}$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1963), 189-206. [27] T. Nguyen, Interior Calderón-Zygmund estimates for solutions to general parabolic equations of $p$-Laplacian type, Calc. Var. Partial Differential Equations, 56 (2017), Art. 173, 42 pp. doi: 10.1007/s00526-017-1265-y. [28] T. Nguyen and T. Phan, Interior gradient estimates for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 55 (2016), Art. 59, 33 pp. doi: 10.1007/s00526-016-0996-5. [29] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. [30] T. Phan, G. Todorova and B. Yordanov, $L^p$-diffusion phenomena for dissipative wave equations, preprint, (2020). [31] T. Phan, Regularity estimates for BMO-weak solutions of quasilinear elliptic equations with inhomogeneous boundary conditions, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Paper No. 8, 49 pp. doi: 10.1007/s00030-018-0501-2. [32] T. Phan, Lorentz estimates for weak solutions of quasi-linear parabolic equations with singular divergence-free drifts, Canad. J. Math., 71 (2019), 937-982.  doi: 10.4153/CJM-2017-049-3. [33] L. Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sin., 19 (2003), 381-396.  doi: 10.1007/s10114-003-0264-4. [34] W. Wei and Z. Zhang, $L^p$ resolvent estimates for constant coefficient elliptic systems on Lipschitz domains, J. Funct. Anal., 267 (2014), 3262-3293.  doi: 10.1016/j.jfa.2014.08.010.
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