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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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[29]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[29]

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

[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.

[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.

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