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February  2019, 39(2): 1101-1115. doi: 10.3934/dcds.2019046

On fractional Leibniz rule for Dirichlet Laplacian in exterior domain

1. 

Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5 Pisa, 56127, Italy

2. 

IMI–BAS, Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria

3. 

Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

4. 

Department of Mathematics, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

Received  April 2018 Revised  August 2018 Published  November 2018

Fund Project: The first author was supported in part by Project 2017 "Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari" of INDAM, GNAMPA - Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni, by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, by Top Global University Project, Waseda University and the Project PRA 2018 49 of University of Pisa

The goal of the work is to verify the fractional Leibniz rule for Dirichlet Laplacian in the exterior domain of a compact set. The key point is the proof of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem.

Citation: Vladimir Georgiev, Koichi Taniguchi. On fractional Leibniz rule for Dirichlet Laplacian in exterior domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1101-1115. doi: 10.3934/dcds.2019046
References:
[1]

B. Cassano and P. D'Ancona, Scattering in the energy space for the NLS with variable coefficients, Math. Ann., 366 (2016), 479-543. doi: 10.1007/s00208-015-1335-4. Google Scholar

[2]

W. Dan and Y. Shibata, Remarks on the LqLp estimate of Stokes semigroup in a 2-dimensional exterior domain, Pacific Journal of Math., 189 (1999), 223-239. doi: 10.2140/pjm.1999.189.223. Google Scholar

[3]

P. D'Ancona, A short proof of commutator estimates, J. Fourier Anal. Appl., (2018) to appear, https://doi.org/10.1007/s00041-018-9612-8.Google Scholar

[4]

P. D'Ancona and V. Pierfelice, On the wave equation with a large rough potential, J. Functional Analysis, 227 (2005), 30-77. doi: 10.1016/j.jfa.2005.05.013. Google Scholar

[5]

S. FornaroG. Metafune and E. Priola, Gradient estimates for Dirichlet parabolic problems in unbounded domains, J. Differential Equations, 205 (2004), 329-353. doi: 10.1016/j.jde.2004.06.019. Google Scholar

[6]

K. FujiwaraV. Georgiev and T. Ozawa, Higher order fractional Leibniz rule, J. Fourier Anal. Appl., 24 (2018), 650-665. doi: 10.1007/s00041-017-9541-y. Google Scholar

[7]

E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 8 (1959), 24-51. Google Scholar

[8]

M. GeissertH. HeckM. Hieber and I. Wood, The Ornstein-Uhlenbeck semigroup in exterior domains, Arch. Math. (Basel), 85 (2005), 554-562. doi: 10.1007/s00013-005-1400-4. Google Scholar

[9]

V. Georgiev and N. Visciglia, Decay estimates for the wave equation with potential, Comm. Partial Differential Equations, 28 (2003), 1325-1369. doi: 10.1081/PDE-120024371. Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Reprint of the 1998 edition, 2001. Google Scholar

[11]

L. GrafakosD. Maldonado and V. Naibo, A remark on an endpoint Kato-Ponce inequality, Differential Integral Equations, 27 (2014), 415-424. Google Scholar

[12]

L. Grafakos and Z. Si, The Hörmander multiplier theorem for multilinear operators, J. Reine Angew. Math., 668 (2012), 133-147. Google Scholar

[13]

P. Han, Large time behavior for the incompressible Navier-Stokes flows in 2D exterior domains, Manuscripta Math., 138 (2012), 347-370. doi: 10.1007/s00229-011-0495-0. Google Scholar

[14]

T. Hishida, Lq - Lr estimate of the Oseen flow in plane exterior domains, J. Math. Soc. Japan, 68 (2016), 295-346. doi: 10.2969/jmsj/06810295. Google Scholar

[15]

T. Hishida and Y. Shibata, Lp - Lq estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 193 (2009), 339-421. doi: 10.1007/s00205-008-0130-8. Google Scholar

[16]

D. IftimieG. Karch and C. Lacave, Asymptotics of solutions to the Navier-Stokes system in exterior domains, J. Lond. Math. Soc., 90 (2014), 785-806. doi: 10.1112/jlms/jdu052. Google Scholar

[17]

K. Ishige, Gradient estimates for the heat equation in the exterior domains under the Neumann boundary condition, Differential Integral Equations, 22 (2009), 401-410. Google Scholar

[18]

K. Ishige and Y. Kabeya, Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball, J. Math. Soc. Japan, 59 (2007), 861-898. Google Scholar

[19]

O. Ivanovici and F. Planchon, Square function and heat flow estimates on domains, Comm. Partial Differential Equations, 42 (2017), 1447-1466. doi: 10.1080/03605302.2017.1365267. Google Scholar

[20]

T. Iwabuchi, T. Matsuyama and K. Taniguchi, Besov spaces on open sets, Bull. Sci. Math., (2018) to appear.Google Scholar

[21]

T. Iwabuchi, T. Matsuyama and K. Taniguchi, Bilinear estimates in Besov spaces generated by the Dirichlet Laplacian, arXiv: 1705.08595v2.Google Scholar

[22]

T. IwabuchiT. Matsuyama and K. Taniguchi, Boundedness of spectral multipliers for Schrödinger operators on open sets, Rev. Mat. Iberoam., 34 (2018), 1277-1322. doi: 10.4171/RMI/1024. Google Scholar

[23]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1998), 891-907. doi: 10.1002/cpa.3160410704. Google Scholar

[24]

R. KillipM. Visan and X. Zhang, Riesz transforms outside a convex obstacle, Int. Math. Res. Not. IMRN, (2016), 5875-5921. doi: 10.1093/imrn/rnv338. Google Scholar

[25]

T. Kobayashi and T. Kubo, Weighted LpLq estimates of the Stokes semigroup in some unbounded domains, Tsukuba J. Math., 37 (2013), 179-205. Google Scholar

[26]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, Vol. 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. Google Scholar

[27]

P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 395-449. Google Scholar

[28]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. Google Scholar

[29]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, Vol. 31, Princeton University Press, Princeton, NJ, 2005. Google Scholar

[30]

K. Taniguchi, Besov spaces generated by the Neumann Laplacian, Eur. J. Math., (2018) to appear, https://doi.org/10.1007/s40879-018-0224-2.Google Scholar

[31]

Q. S. Zhang, The global behavior of heat kernels in exterior domains, J. Functional Analysis, 200 (2003), 160-176. doi: 10.1016/S0022-1236(02)00074-5. Google Scholar

show all references

References:
[1]

B. Cassano and P. D'Ancona, Scattering in the energy space for the NLS with variable coefficients, Math. Ann., 366 (2016), 479-543. doi: 10.1007/s00208-015-1335-4. Google Scholar

[2]

W. Dan and Y. Shibata, Remarks on the LqLp estimate of Stokes semigroup in a 2-dimensional exterior domain, Pacific Journal of Math., 189 (1999), 223-239. doi: 10.2140/pjm.1999.189.223. Google Scholar

[3]

P. D'Ancona, A short proof of commutator estimates, J. Fourier Anal. Appl., (2018) to appear, https://doi.org/10.1007/s00041-018-9612-8.Google Scholar

[4]

P. D'Ancona and V. Pierfelice, On the wave equation with a large rough potential, J. Functional Analysis, 227 (2005), 30-77. doi: 10.1016/j.jfa.2005.05.013. Google Scholar

[5]

S. FornaroG. Metafune and E. Priola, Gradient estimates for Dirichlet parabolic problems in unbounded domains, J. Differential Equations, 205 (2004), 329-353. doi: 10.1016/j.jde.2004.06.019. Google Scholar

[6]

K. FujiwaraV. Georgiev and T. Ozawa, Higher order fractional Leibniz rule, J. Fourier Anal. Appl., 24 (2018), 650-665. doi: 10.1007/s00041-017-9541-y. Google Scholar

[7]

E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 8 (1959), 24-51. Google Scholar

[8]

M. GeissertH. HeckM. Hieber and I. Wood, The Ornstein-Uhlenbeck semigroup in exterior domains, Arch. Math. (Basel), 85 (2005), 554-562. doi: 10.1007/s00013-005-1400-4. Google Scholar

[9]

V. Georgiev and N. Visciglia, Decay estimates for the wave equation with potential, Comm. Partial Differential Equations, 28 (2003), 1325-1369. doi: 10.1081/PDE-120024371. Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Reprint of the 1998 edition, 2001. Google Scholar

[11]

L. GrafakosD. Maldonado and V. Naibo, A remark on an endpoint Kato-Ponce inequality, Differential Integral Equations, 27 (2014), 415-424. Google Scholar

[12]

L. Grafakos and Z. Si, The Hörmander multiplier theorem for multilinear operators, J. Reine Angew. Math., 668 (2012), 133-147. Google Scholar

[13]

P. Han, Large time behavior for the incompressible Navier-Stokes flows in 2D exterior domains, Manuscripta Math., 138 (2012), 347-370. doi: 10.1007/s00229-011-0495-0. Google Scholar

[14]

T. Hishida, Lq - Lr estimate of the Oseen flow in plane exterior domains, J. Math. Soc. Japan, 68 (2016), 295-346. doi: 10.2969/jmsj/06810295. Google Scholar

[15]

T. Hishida and Y. Shibata, Lp - Lq estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 193 (2009), 339-421. doi: 10.1007/s00205-008-0130-8. Google Scholar

[16]

D. IftimieG. Karch and C. Lacave, Asymptotics of solutions to the Navier-Stokes system in exterior domains, J. Lond. Math. Soc., 90 (2014), 785-806. doi: 10.1112/jlms/jdu052. Google Scholar

[17]

K. Ishige, Gradient estimates for the heat equation in the exterior domains under the Neumann boundary condition, Differential Integral Equations, 22 (2009), 401-410. Google Scholar

[18]

K. Ishige and Y. Kabeya, Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball, J. Math. Soc. Japan, 59 (2007), 861-898. Google Scholar

[19]

O. Ivanovici and F. Planchon, Square function and heat flow estimates on domains, Comm. Partial Differential Equations, 42 (2017), 1447-1466. doi: 10.1080/03605302.2017.1365267. Google Scholar

[20]

T. Iwabuchi, T. Matsuyama and K. Taniguchi, Besov spaces on open sets, Bull. Sci. Math., (2018) to appear.Google Scholar

[21]

T. Iwabuchi, T. Matsuyama and K. Taniguchi, Bilinear estimates in Besov spaces generated by the Dirichlet Laplacian, arXiv: 1705.08595v2.Google Scholar

[22]

T. IwabuchiT. Matsuyama and K. Taniguchi, Boundedness of spectral multipliers for Schrödinger operators on open sets, Rev. Mat. Iberoam., 34 (2018), 1277-1322. doi: 10.4171/RMI/1024. Google Scholar

[23]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1998), 891-907. doi: 10.1002/cpa.3160410704. Google Scholar

[24]

R. KillipM. Visan and X. Zhang, Riesz transforms outside a convex obstacle, Int. Math. Res. Not. IMRN, (2016), 5875-5921. doi: 10.1093/imrn/rnv338. Google Scholar

[25]

T. Kobayashi and T. Kubo, Weighted LpLq estimates of the Stokes semigroup in some unbounded domains, Tsukuba J. Math., 37 (2013), 179-205. Google Scholar

[26]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, Vol. 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. Google Scholar

[27]

P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 395-449. Google Scholar

[28]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. Google Scholar

[29]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, Vol. 31, Princeton University Press, Princeton, NJ, 2005. Google Scholar

[30]

K. Taniguchi, Besov spaces generated by the Neumann Laplacian, Eur. J. Math., (2018) to appear, https://doi.org/10.1007/s40879-018-0224-2.Google Scholar

[31]

Q. S. Zhang, The global behavior of heat kernels in exterior domains, J. Functional Analysis, 200 (2003), 160-176. doi: 10.1016/S0022-1236(02)00074-5. Google Scholar

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