September  2021, 20(9): 3193-3213. doi: 10.3934/cpaa.2021102

A biharmonic transmission problem in Lp-spaces

Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France

* Corresponding author

Received  March 2021 Revised  May 2021 Published  September 2021 Early access  June 2021

Fund Project: This research is supported by CIFRE contract 2014/1307 with Qualiom Eco company and partially by the LMAH and the european funding ERDF through grant project Xterm

In this work we study, by a semigroup approach, a transmission problem based on biharmonic equations with boundary and transmission conditions, in two juxtaposed habitats. We give a result of existence and uniqueness of the classical solution in $ L^p $-spaces, for $ p \in (1,+\infty) $, using analytic semigroups and operators sum theory in Banach spaces. To this end, we invert explicitly the determinant operator of the transmission system in $ L^p $-spaces using the $ \mathcal{E}_{\infty} $-calculus and the Dore-Venni sums theory.

Citation: Alexandre Thorel. A biharmonic transmission problem in Lp-spaces. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3193-3213. doi: 10.3934/cpaa.2021102
References:
[1]

B. Barraza MartínezR. DenkJ. Hernández MonzónF. Kammerlander and M. Nendel, Regularity and asymptotic behavior for a damped plate-membrane transmission problem, J. Math. Anal. Appl, 474 (2019), 1082-1103.  doi: 10.1016/j.jmaa.2019.02.005.

[2]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 163-168.  doi: 10.1007/BF02384306.

[3]

D. L. Burkholder, A geometrical characterisation of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011. 

[4]

F. CakoniG. C. Hsiao and W. L. Wendland, On the boundary integral equation methodfor a mixed boundary value problem of the biharmonic equation, Complex Var., 50 (2005), 681-696.  doi: 10.1080/02781070500087394.

[5]

D. S. Cohen and J. D. Murray, A generalized diffusion model for growth and dispersal in population, J. Math. Biol., 12 (1981), 237-249.  doi: 10.1007/BF00276132.

[6]

M. Costabel, E. Stephan and W. L. Wengland, On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4e serie, 10 (1983), 197-241.

[7]

G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. pures et appl., 54 (1975), 305-387. 

[8]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.  doi: 10.1007/BF01163654.

[9]

A. FaviniR. LabbasK. LemrabetS. Maingot and H. Sidibé, Transmission Problem for an Abstract Fourth-order Differential Equation of Elliptic Type in UMD Spaces, Adv. Differ. Equ., 15 (2010), 43-72. 

[10]

A. FaviniR. LabbasA. Medeghri and A. Menad, Analytic semigroups generated by the dispersal process in two habitats incorporating individual behavior at the interface, J. Math. Anal. Appl., 471 (2019), 448-480.  doi: 10.1016/j.jmaa.2018.10.085.

[11]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.

[12]

P. Grisvard, Spazi di tracce e applicazioni, Rendiconti di Mat., Serie VI, 5 (1972), 657-729.

[13]

Z. GuoB. Lai and D. Ye, Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions, Proc. Amer. Math. Soc., 142 (2014), 2027-2034.  doi: 10.1090/S0002-9939-2014-11895-8.

[14]

M. Haase, The Functional Calculus for Sectorial Operators, Birkhauser, 2006. doi: 10.1007/3-7643-7698-8.

[15]

F. Hassine, Logarithmic stabilization of the Euler-Bernoulli transmission plate equation with locally distributed Kelvin-Voigt damping, J. Math. Anal. Appl., 455 (2017), 1765-1782.  doi: 10.1016/j.jmaa.2017.06.068.

[16]

A. F. Hrustalev and B. I. Kogan, A boundary-value problem for the biharmonic equation in elasticity theory, (Russian) Izv. Vys$\check s$. U$\check cebn$. Zaved. Matematika, 4 (1958), 241-247. 

[17]

H. Komatsu, Fractional powers of operators, Pac. J. Math., 19 (1966), 285-346. 

[18]

M. Kotschote, Maximal $L^p$-regularity for a linear three-phase problem of para-bolic-elliptic type, J. Evol. Equ., 10 (2010), 293-318.  doi: 10.1007/s00028-009-0050-6.

[19]

R. LabbasS. MaingotD. Manceau and A. Thorel, On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain, J. Math. Anal. Appl., 450 (2017), 351-376.  doi: 10.1016/j.jmaa.2017.01.026.

[20]

R. LabbasK. LemrabetS. Maingot and A. Thorel, Generalized linear models for population dynamics in two juxtaposed habitats, Discrete Contin. Dyn. Syst. - A, 39 (2019), 2933-2960.  doi: 10.3934/dcds.2019122.

[21]

K. Limam, R. Labbas, K. Lemrabet, A. Medeghri and M. Meisner, On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution, J. Differ. Equ., 259 (2015), 2695-2731. doi: 10.1016/j.jde.2015.04.002.

[22]

F. Lin and Y. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proceedings of the Royal Society of London A, 463 (2007), 1323-1337.  doi: 10.1098/rspa.2007.1816.

[23]

J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Publications mathé-matiques de l'I.H.É.S., 19 (1964), 5-68. 

[24]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Basel, Boston, Berlin, 1995.

[25]

F. L. Ochoa, A generalized reaction-diffusion model for spatial structures formed by motile cells, BioSystems, 17 (1984), 35-50. 

[26]

K. M. Perfekt, The transmission problem on a three-dimensional wedge, Arch. Ration. Mech. Anal., 231 (2019), 1745-1780.  doi: 10.1007/s00205-018-1308-3.

[27]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Mathematische Zeitschrift, 203 (1990), 429-452.  doi: 10.1007/BF02570748.

[28]

J. Prüss and H. Sohr, Imaginary powers of elliptic second order differential operators in $L^p$-spaces, Hiroshima Math. J., 23 (1993), 161-192. 

[29]

J. L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, Probability and Banach Spaces: Lecture Notes in Math., 1221 (1986), 195-222.  doi: 10.1007/BFb0099115.

[30]

H. Saker and N. Bouselsal, On the bilaplacian problem with nonlinear boundary conditions, Indian J. Pure Appl. Math., 47 (2016), 425-435.  doi: 10.1007/s13226-016-0178-3.

[31]

A. Thorel, Operational approach for biharmonic equations in $L^p$-spaces, J. Evol. Equ., 20 (2020), 631-657.  doi: 10.1007/s00028-019-00536-2.

[32]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978.

[33]

J. WangJ. Lang and Y. Chen, Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission, Discrete Contin. Dyn. Syst. - B, 22 (2017), 3721-3747.  doi: 10.3934/dcdsb.2017186.

[34]

C. F. Yanga and S. Buterin, Uniqueness of the interior transmission problem with partial information on the potential and eigenvalues, J. Differ. Equ., 260 (2016), 4871-4887. doi: 10.1016/j.jde.2015.11.031.

show all references

References:
[1]

B. Barraza MartínezR. DenkJ. Hernández MonzónF. Kammerlander and M. Nendel, Regularity and asymptotic behavior for a damped plate-membrane transmission problem, J. Math. Anal. Appl, 474 (2019), 1082-1103.  doi: 10.1016/j.jmaa.2019.02.005.

[2]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 163-168.  doi: 10.1007/BF02384306.

[3]

D. L. Burkholder, A geometrical characterisation of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011. 

[4]

F. CakoniG. C. Hsiao and W. L. Wendland, On the boundary integral equation methodfor a mixed boundary value problem of the biharmonic equation, Complex Var., 50 (2005), 681-696.  doi: 10.1080/02781070500087394.

[5]

D. S. Cohen and J. D. Murray, A generalized diffusion model for growth and dispersal in population, J. Math. Biol., 12 (1981), 237-249.  doi: 10.1007/BF00276132.

[6]

M. Costabel, E. Stephan and W. L. Wengland, On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4e serie, 10 (1983), 197-241.

[7]

G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. pures et appl., 54 (1975), 305-387. 

[8]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.  doi: 10.1007/BF01163654.

[9]

A. FaviniR. LabbasK. LemrabetS. Maingot and H. Sidibé, Transmission Problem for an Abstract Fourth-order Differential Equation of Elliptic Type in UMD Spaces, Adv. Differ. Equ., 15 (2010), 43-72. 

[10]

A. FaviniR. LabbasA. Medeghri and A. Menad, Analytic semigroups generated by the dispersal process in two habitats incorporating individual behavior at the interface, J. Math. Anal. Appl., 471 (2019), 448-480.  doi: 10.1016/j.jmaa.2018.10.085.

[11]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.

[12]

P. Grisvard, Spazi di tracce e applicazioni, Rendiconti di Mat., Serie VI, 5 (1972), 657-729.

[13]

Z. GuoB. Lai and D. Ye, Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions, Proc. Amer. Math. Soc., 142 (2014), 2027-2034.  doi: 10.1090/S0002-9939-2014-11895-8.

[14]

M. Haase, The Functional Calculus for Sectorial Operators, Birkhauser, 2006. doi: 10.1007/3-7643-7698-8.

[15]

F. Hassine, Logarithmic stabilization of the Euler-Bernoulli transmission plate equation with locally distributed Kelvin-Voigt damping, J. Math. Anal. Appl., 455 (2017), 1765-1782.  doi: 10.1016/j.jmaa.2017.06.068.

[16]

A. F. Hrustalev and B. I. Kogan, A boundary-value problem for the biharmonic equation in elasticity theory, (Russian) Izv. Vys$\check s$. U$\check cebn$. Zaved. Matematika, 4 (1958), 241-247. 

[17]

H. Komatsu, Fractional powers of operators, Pac. J. Math., 19 (1966), 285-346. 

[18]

M. Kotschote, Maximal $L^p$-regularity for a linear three-phase problem of para-bolic-elliptic type, J. Evol. Equ., 10 (2010), 293-318.  doi: 10.1007/s00028-009-0050-6.

[19]

R. LabbasS. MaingotD. Manceau and A. Thorel, On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain, J. Math. Anal. Appl., 450 (2017), 351-376.  doi: 10.1016/j.jmaa.2017.01.026.

[20]

R. LabbasK. LemrabetS. Maingot and A. Thorel, Generalized linear models for population dynamics in two juxtaposed habitats, Discrete Contin. Dyn. Syst. - A, 39 (2019), 2933-2960.  doi: 10.3934/dcds.2019122.

[21]

K. Limam, R. Labbas, K. Lemrabet, A. Medeghri and M. Meisner, On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution, J. Differ. Equ., 259 (2015), 2695-2731. doi: 10.1016/j.jde.2015.04.002.

[22]

F. Lin and Y. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proceedings of the Royal Society of London A, 463 (2007), 1323-1337.  doi: 10.1098/rspa.2007.1816.

[23]

J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Publications mathé-matiques de l'I.H.É.S., 19 (1964), 5-68. 

[24]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Basel, Boston, Berlin, 1995.

[25]

F. L. Ochoa, A generalized reaction-diffusion model for spatial structures formed by motile cells, BioSystems, 17 (1984), 35-50. 

[26]

K. M. Perfekt, The transmission problem on a three-dimensional wedge, Arch. Ration. Mech. Anal., 231 (2019), 1745-1780.  doi: 10.1007/s00205-018-1308-3.

[27]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Mathematische Zeitschrift, 203 (1990), 429-452.  doi: 10.1007/BF02570748.

[28]

J. Prüss and H. Sohr, Imaginary powers of elliptic second order differential operators in $L^p$-spaces, Hiroshima Math. J., 23 (1993), 161-192. 

[29]

J. L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, Probability and Banach Spaces: Lecture Notes in Math., 1221 (1986), 195-222.  doi: 10.1007/BFb0099115.

[30]

H. Saker and N. Bouselsal, On the bilaplacian problem with nonlinear boundary conditions, Indian J. Pure Appl. Math., 47 (2016), 425-435.  doi: 10.1007/s13226-016-0178-3.

[31]

A. Thorel, Operational approach for biharmonic equations in $L^p$-spaces, J. Evol. Equ., 20 (2020), 631-657.  doi: 10.1007/s00028-019-00536-2.

[32]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978.

[33]

J. WangJ. Lang and Y. Chen, Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission, Discrete Contin. Dyn. Syst. - B, 22 (2017), 3721-3747.  doi: 10.3934/dcdsb.2017186.

[34]

C. F. Yanga and S. Buterin, Uniqueness of the interior transmission problem with partial information on the potential and eigenvalues, J. Differ. Equ., 260 (2016), 4871-4887. doi: 10.1016/j.jde.2015.11.031.

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