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Parabolic equations with unbounded lower-order coefficients in Sobolev spaces with mixed norms

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Abstract

We prove the \(L_{p,q}\)-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the coefficients for the lower-order terms are not necessarily bounded. We study both the Dirichlet and conormal derivative boundary value problems on irregular domains. We also prove embedding results for parabolic Sobolev spaces, the proof of which is of independent interest.

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References

  1. H. Aikawa. Martin boundary and boundary Harnack principle for non-smooth domains [mr1962228]. In Selected papers on differential equations and analysis, volume 215 of Amer. Math. Soc. Transl. Ser. 2, pages 33–55. Amer. Math. Soc., Providence, RI, 2005.

  2. Y. A. Alkhutov, A. N. Gordeev. \(L_p\)-estimates for solutions to second order parabolic equations. In Proceedings of the St. Petersburg Mathematical Society. Vol. XIII, volume 222 of Amer. Math. Soc. Transl. Ser. 2, pages 1–21. Amer. Math. Soc., Providence, RI, 2008.

  3. O. V. Besov, V. P. Il’in, S. M. Nikol’skiĭ. Integral representations of functions and imbedding theorems. Vol. I. V. H. Winston & Sons, Washington, D.C., 1978. Translated from the Russian, Scripta Series in Mathematics, Edited by Mitchell H. Taibleson.

  4. M. Bramanti, M. Cristina Cerutti. \(W_p^{1,2}\) solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients. Commun. Partial Differ. Equ., 18(9-10):1735–1763, 1993.

  5. S.-S. Byun, D. K. Palagachev, L. Wang. Parabolic systems with measurable coefficients in Reifenberg domains. Int. Math. Res. Not. IMRN, 13:3053–3086, 2013.

    Article  MathSciNet  Google Scholar 

  6. S.-S. Byun, L. Wang. Elliptic equations with measurable coefficients in Reifenberg domains. Adv. Math., 225(5):2648–2673, 2010.

    Article  MathSciNet  Google Scholar 

  7. F. Chiarenza, M. Frasca, P. Longo. Interior \(W^{2,p}\) estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche Mat., 40(1):149–168, 1991.

  8. F. Chiarenza, M. Frasca, P. Longo. \(W^{2,p}\)-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc., 336(2):841–853, 1993.

  9. J. Choi, H. Dong, D. Kim. Conormal derivative problems for stationary Stokes system in Sobolev spaces. Discrete Contin. Dyn. Syst., 38(5):2349–2374, 2018.

    Article  MathSciNet  Google Scholar 

  10. G. Di Fazio. \(L^p\) estimates for divergence form elliptic equations with discontinuous coefficients. Boll. Un. Mat. Ital. A (7), 10(2):409–420, 1996.

  11. H. Dong, D. Kim. Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains. J. Funct. Anal., 261(11):3279–3327, 2011.

    Article  MathSciNet  Google Scholar 

  12. H. Dong, D. Kim. \(L_p\) solvability of divergence type parabolic and elliptic systems with partially BMO coefficients. Calc. Var. Partial Differ. Equ., 40(3-4):357–389, 2011.

    Article  Google Scholar 

  13. H. Dong, D. Kim. On the impossibility of \(W_p^2\) estimates for elliptic equations with piecewise constant coefficients. J. Funct. Anal., 267(10):3963–3974, 2014.

    Article  MathSciNet  Google Scholar 

  14. J. Droniou. Non-coercive linear elliptic problems. Potential Anal., 17(2):181–203, 2002.

    Article  MathSciNet  Google Scholar 

  15. J. Droniou, J.-L. Vázquez. Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions. Calc. Var. Partial Differ. Equ., 34(4):413–434, 2009.

    Article  MathSciNet  Google Scholar 

  16. D. Jerison, C. E. Kenig. The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal., 130(1):161–219, 1995.

    Article  MathSciNet  Google Scholar 

  17. P. W. Jones. Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math., 147(1-2):71–88, 1981.

    Article  MathSciNet  Google Scholar 

  18. B. Kang, H. Kim. \(W^{1,p}\)-estimates for elliptic equations with lower order terms. Commun. Pure Appl. Anal., 16(3):799–821, 2017.

    Article  MathSciNet  Google Scholar 

  19. B. Kang, H. Kim. On \(L^p\)-Resolvent Estimates for Second-Order Elliptic Equations in Divergence Form. Potential Anal., 50(1):107–133, 2019.

    Article  MathSciNet  Google Scholar 

  20. C. E. Kenig, T. Toro. Harmonic measure on locally flat domains. Duke Math. J., 87(3):509–551, 1997.

    Article  MathSciNet  Google Scholar 

  21. D. Kim, S. Ryu. The weak maximum principle for second-order elliptic and parabolic conormal derivative problems. Commun. Pure Appl. Anal., 19(1):493–510, 2020.

    Article  MathSciNet  Google Scholar 

  22. H. Kim, Y.-H. Kim. On weak solutions of elliptic equations with singular drifts. SIAM J. Math. Anal., 47(2):1271–1290, 2015.

    Article  MathSciNet  Google Scholar 

  23. N. V. Krylov. Parabolic and elliptic equations with VMO coefficients. Commum. Partial Differ. Equ., 32(1-3):453–475, 2007.

    Article  MathSciNet  Google Scholar 

  24. N. V. Krylov. Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms. J. Funct. Anal., 250(2):521–558, 2007.

    Article  MathSciNet  Google Scholar 

  25. N. V. Krylov. Lectures on elliptic and parabolic equations in Sobolev spaces, volume 96 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008.

  26. N. V. Krylov. On parabolic equations in one space dimension. Commum. Partial Differ. Equ., 41(4):644–664, 2016.

    Article  MathSciNet  Google Scholar 

  27. N. V. Krylov. Elliptic equations with VMO \(a, b \in L_d\), and \(c \in L_{d/2}\). Trans. Am. Math. Soc., 374(4):2805–2822, 2021.

    Article  Google Scholar 

  28. O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural’ceva. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1967.

  29. A. Lemenant, E. Milakis, L. V. Spinolo. On the extension property of Reifenberg-flat domains. Ann. Acad. Sci. Fenn. Math., 39(1):51–71, 2014.

    Article  MathSciNet  Google Scholar 

  30. G. Leoni. A first course in Sobolev spaces, volume 181 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2017.

  31. A. I. Nazarov, N. N. Ural’ tseva. The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients. Algebra i Analiz, 23(1):136–168, 2011.

  32. E. M. Stein. Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J. 1970.

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Acknowledgements

The authors would like to thank Hongjie Dong for helpful suggestions and discussion including the full ranges of \((p_k,q_k)\) in Theorem 5.2 and an example supporting the assertion in Remark 2.3. The authors also thank Jongkeun Choi for discussion on an early version of the paper and the referee for his/her careful review and helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea

    Doyoon Kim & Kwan Woo

  2. Department of Mathematics, University of Seoul, 163 Seoulsiripdaero, Dongdaemun-gu, Seoul, 02504, Republic of Korea

    Seungjin Ryu

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  1. Doyoon Kim
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Correspondence to Kwan Woo.

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D. Kim and K. Woo were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1A2C1084683).

S. Ryu was supported by NRF-2017R1C1B1010966 and NRF-2020R1C1C1A01014310.

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Kim, D., Ryu, S. & Woo, K. Parabolic equations with unbounded lower-order coefficients in Sobolev spaces with mixed norms. J. Evol. Equ. 22, 9 (2022). https://doi.org/10.1007/s00028-022-00761-2

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