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Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients

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Abstract

Using Duhamel’s formula, we prove sharp two-sided estimates for the spectral fractional Laplacian’s heat kernel with time-dependent gradient perturbation in bounded C1,1 domains. In addition, we obtain a gradient estimate as well as the Hölder continuity of the heat kernel’s gradient.

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References

  1. Abatangelo N, Dupaigne L. Nonhomogeneous boundary conditions for the spectral fractional Laplacian. Ann Inst H Poincaré Anal. Non Lineaire, 2017, 34: 439–467

    Article  MathSciNet  Google Scholar 

  2. Bogdan K, Jakubowski T. Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Comm Math Phys, 2007, 271: 179–198

    Article  MathSciNet  Google Scholar 

  3. Bonforte M, Sire Y, Vázquez J L. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete Contin Dyn Syst, 2015, 35: 5725–5767

    Article  MathSciNet  Google Scholar 

  4. Chen P, Song R, Xie L, et al. Heat kernel estimates for Dirichlet fractional Laplacian with gradient perturbation. J Korean Math Soc, 2019, 56: 91–111

    MathSciNet  MATH  Google Scholar 

  5. Chen Z, Kim P, Kumagai T. Global heat kernel estimates for symmetric jump processes. Trans Amer Math Soc, 2011, 363: 5021–5055

    MathSciNet  MATH  Google Scholar 

  6. Chen Z, Kim P, Song R. Heat kernel estimates for the Dirichlet fractional Laplacian. J Eur Math Soc (JEMS), 2010, 12: 1307–1329

    Article  MathSciNet  Google Scholar 

  7. Chen Z, Kim P, Song R. Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Ann Probab, 2012, 40: 2483–2538

    MathSciNet  MATH  Google Scholar 

  8. Chen Z, Kim P, Song R. Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation. Trans Amer Math Soc, 2015, 367: 5237–5270

    MathSciNet  MATH  Google Scholar 

  9. Chen Z, Kumagai T. Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab Theory Related Fields, 2008, 140: 277–317

    MathSciNet  MATH  Google Scholar 

  10. Dhifli A, Maagli H, Zribi M. On the subordinate killed B.M in bounded domains and existence results for nonlinear fractional Dirichlet problems. Math Ann, 2012, 352: 259–291

    Article  MathSciNet  Google Scholar 

  11. Garroni M, Menaldi J. Green Functions for Second Order Parabolic Integral-Differential Problems. Harlow: Longman, 1992

    MATH  Google Scholar 

  12. Glover J, Pop-Stojanovic Z, Rao M, et al. Harmonic functions of subordinate killed Brownian motion. J Funct Anal, 2004, 215: 399–426

    Article  MathSciNet  Google Scholar 

  13. Jakubowski T, Szczypkowski K. Time-dependent gradient perturbations of fractional Laplacian. J Evol Equ, 2010, 10: 319–339

    Article  MathSciNet  Google Scholar 

  14. Kim P, Song R. Stable process with singular drift. Stochastic Process Appl, 2014, 124: 2479–2516

    Article  MathSciNet  Google Scholar 

  15. Kim P, Song R. Dirichlet heat kernel estimates for stable processes with singular drift in unbounded C1,1 open sets. Potential Anal, 2014, 41: 555–581

    Article  MathSciNet  Google Scholar 

  16. Kulczycki T, Ryznar M. Gradient estimates of Dirichlet heat kernels for unimodal Levy processes. Math Nachr, 2018, 291: 374–397.

    Article  MathSciNet  Google Scholar 

  17. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983

    Book  Google Scholar 

  18. Song R. Sharp bounds on the density, Green function and jumping function of subordinate killed BM. Probab Theory Related Fields, 2004, 128: 606–628

    Article  MathSciNet  Google Scholar 

  19. Song R, Vondracek Z. Potential theory of subordinate killed Brownian motion in a domain. Probab Theory Related Fields, 2003, 135: 578–592

    Article  MathSciNet  Google Scholar 

  20. Song R, Vondracek Z. On the relationship between subordinate killed and killed subordinate processes. Electron Comm Probab, 2008, 13: 325–336

    MathSciNet  MATH  Google Scholar 

  21. Stinga P R, Zhang C. Harnack inequality for fractional non-local equations. Discrete Contin Dyn Syst, 2013, 33: 3153–3370

    MathSciNet  MATH  Google Scholar 

  22. Wang F, Zhang X. Heat kernel for fractional diffusion operators with perturbations. Forum Math, 2015, 27: 973–994

    MathSciNet  MATH  Google Scholar 

  23. Xie L, Zhang X. Heat kernel estimates for critical fractional diffusion operators. Studia Math, 2014, 224: 221–263

    MathSciNet  MATH  Google Scholar 

  24. Zhang Q. Gaussian bounds for the fundamental solutions of ∇(A∇u) + B∇u − ut = 0. Manuscripta Math, 1997, 93: 381–390

    MathSciNet  Google Scholar 

  25. Zhang Q. The boundary behavior of heat kernels of Dirichlet Laplacians. J Differential Equations, 2002, 182: 416–430

    MathSciNet  MATH  Google Scholar 

  26. Zhang Q. Some gradient estimates for the heat equation on domains and for an equation by Perelman. Int Math Res Not IMRN, 2006, 2006: 92314

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author was supported by the Simons Foundation (Grant No. #429343). The second author was supported by the Alexander-von-Humboldt Foundation, National Natural Science Foundation of China (Grant No. 11701233) and National Science Foundation of Jiangsu (Grant No. BK20170226). The third author was supported by National Natural Science Foundation of China (Grant No. 11771187). The Priority Academic Program Development of Jiangsu Higher Education Institutions is also gratefully acknowledged. The authors thank the referees for carefully reading the manuscript and providing many helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Renming Song

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221000, China

    Longjie Xie & Yingchao Xie

Authors
  1. Renming Song
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  2. Longjie Xie
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  3. Yingchao Xie
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Correspondence to Longjie Xie.

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Song, R., Xie, L. & Xie, Y. Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients. Sci. China Math. 63, 2343–2362 (2020). https://doi.org/10.1007/s11425-018-9472-x

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  • DOI: https://doi.org/10.1007/s11425-018-9472-x

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