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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Abatangelo N, Dupaigne L. Nonhomogeneous boundary conditions for the spectral fractional Laplacian. Ann Inst H Poincaré Anal. Non Lineaire, 2017, 34: 439–467
Bogdan K, Jakubowski T. Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Comm Math Phys, 2007, 271: 179–198
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
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
Chen Z, Kim P, Kumagai T. Global heat kernel estimates for symmetric jump processes. Trans Amer Math Soc, 2011, 363: 5021–5055
Chen Z, Kim P, Song R. Heat kernel estimates for the Dirichlet fractional Laplacian. J Eur Math Soc (JEMS), 2010, 12: 1307–1329
Chen Z, Kim P, Song R. Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Ann Probab, 2012, 40: 2483–2538
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
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
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
Garroni M, Menaldi J. Green Functions for Second Order Parabolic Integral-Differential Problems. Harlow: Longman, 1992
Glover J, Pop-Stojanovic Z, Rao M, et al. Harmonic functions of subordinate killed Brownian motion. J Funct Anal, 2004, 215: 399–426
Jakubowski T, Szczypkowski K. Time-dependent gradient perturbations of fractional Laplacian. J Evol Equ, 2010, 10: 319–339
Kim P, Song R. Stable process with singular drift. Stochastic Process Appl, 2014, 124: 2479–2516
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
Kulczycki T, Ryznar M. Gradient estimates of Dirichlet heat kernels for unimodal Levy processes. Math Nachr, 2018, 291: 374–397.
Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983
Song R. Sharp bounds on the density, Green function and jumping function of subordinate killed BM. Probab Theory Related Fields, 2004, 128: 606–628
Song R, Vondracek Z. Potential theory of subordinate killed Brownian motion in a domain. Probab Theory Related Fields, 2003, 135: 578–592
Song R, Vondracek Z. On the relationship between subordinate killed and killed subordinate processes. Electron Comm Probab, 2008, 13: 325–336
Stinga P R, Zhang C. Harnack inequality for fractional non-local equations. Discrete Contin Dyn Syst, 2013, 33: 3153–3370
Wang F, Zhang X. Heat kernel for fractional diffusion operators with perturbations. Forum Math, 2015, 27: 973–994
Xie L, Zhang X. Heat kernel estimates for critical fractional diffusion operators. Studia Math, 2014, 224: 221–263
Zhang Q. Gaussian bounds for the fundamental solutions of ∇(A∇u) + B∇u − ut = 0. Manuscripta Math, 1997, 93: 381–390
Zhang Q. The boundary behavior of heat kernels of Dirichlet Laplacians. J Differential Equations, 2002, 182: 416–430
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
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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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