February  2022, 42(2): 1011-1037. doi: 10.3934/dcds.2021145

On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle

Department of Mathematics and Institute for Scientific Computing and Applied Mathematics, Indiana University, Rawles Hall, 831 East 3rd St, Bloomington, IN 47405-5701, USA, ORCID: 0000-0001-8087-7811

*Corresponding author: Paolo Piersanti

The author is greatly indebted to Professor Philippe G. Ciarlet for his encouragement and guidance.
The author would like to express his sincere gratitude to the Anonymous Referee for the proposed suggestions and improvements

Received  February 2021 Revised  August 2021 Published  February 2022 Early access  October 2021

In this paper we show that the solution of an obstacle problem for linearly elastic elliptic membrane shells enjoys higher differentiability properties in the interior of the domain where it is defined.

Citation: Paolo Piersanti. On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 1011-1037. doi: 10.3934/dcds.2021145
References:
[1]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.

[2]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.

[3] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511804441.
[4]

L. A. Caffarelli and A. Friedman, The obstacle problem for the biharmonic operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 6 (1979), 151-184. 

[5]

L. A. CaffarelliA. Friedman and A. Torelli, The two-obstacle problem for the biharmonic operator, Pacific J. Math., 103 (1982), 325-335.  doi: 10.2140/pjm.1982.103.325.

[6]

P. G. Ciarlet, Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988.

[7]

P. G. Ciarlet, Mathematical Elasticity. Vol. III: Theory of Shells., North-Holland, Amsterdam, 2000.

[8]

P. G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, 2005.

[9]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2013.

[10]

P. G. Ciarlet and P. Destuynder, A justification of the two-dimensional linear plate model, J. Mécanique, 18 (1979), 315-344. 

[11]

P. G. Ciarlet and V. Lods, Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations, Arch. Rational Mech. Anal., 136 (1996), 119-161.  doi: 10.1007/BF02316975.

[12]

P. G. Ciarlet and V. Lods, On the ellipticity of linear membrane shell equations, J. Math. Pures Appl., 75 (1996), 107-124. 

[13]

P. G. CiarletC. Mardare and P. Piersanti, Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique, C. R. Math. Acad. Sci. Paris, 356 (2018), 1040-1051.  doi: 10.1016/j.crma.2018.08.002.

[14]

P. G. CiarletC. Mardare and P. Piersanti, An obstacle problem for elliptic membrane shells, Math. Mech. Solids, 24 (2019), 1503-1529.  doi: 10.1177/1081286518800164.

[15]

P. G. Ciarlet and P. Piersanti, A confinement problem for a linearly elastic Koiter's shell, C. R. Math. Acad. Sci. Paris, 357 (2019), 221-230.  doi: 10.1016/j.crma.2019.01.004.

[16]

P. G. Ciarlet and P. Piersanti, Obstacle problems for Koiter's shells, Math. Mech. Solids, 24 (2019), 3061-3079.  doi: 10.1177/1081286519825979.

[17]

P. G. Ciarlet and E. Sanchez-Palencia, An existence and uniqueness theorem for the two-dimensional linear membrane shell equations, J. Math. Pures Appl., 75 (1996), 51-67. 

[18]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[19]

J. Frehse, Zum Differenzierbarkeitsproblem bei Variationsungleichungen höherer Ordnung. (German), Abh. Math. Sem. Univ. Hamburg, 36 (1971), 140-149.  doi: 10.1007/BF02995917.

[20]

J. Frehse, On the regularity of the solution of the biharmonic variational inequality, Manuscripta Math., 9 (1973), 91-103.  doi: 10.1007/BF01320669.

[21]

A. Léger and B. Miara, Mathematical justification of the obstacle problem in the case of a shallow shell, J. Elasticity, 90 (2008), 241-257.  doi: 10.1007/s10659-007-9141-1.

[22]

A. Léger and B. Miara, A linearly elastic shell over an obstacle: The flexural case, J. Elasticity, 131 (2018), 19-38.  doi: 10.1007/s10659-017-9643-4.

[23]

M. E. Mezabia, D. A. Chacha and A. Bensayah, Modelling of frictionless Signorini problem for a linear elastic membrane shell, Applicable Analysis, 2020. doi: 10.1080/00036811.2020.1807008.

[24]

P. Piersanti, On the improved interior regularity of the solution of a fourth order elliptic problem modelling the displacement of a linearly elastic shallow shell subject to an obstacle, Asymptot. Anal..

[25]

R. Piersanti, P. C. Africa, M. Fedele, C. Vergara, L. Dedè, A. F. Corno and A. Quarteroni, Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations, Comput. Methods Appl. Mech. Engrg., 373 (2021), 33pp. doi: 10.1016/j.cma.2020.113468.

[26]

F. RegazzoniL. Dedè and A. Quarteroni, Active force generation in cardiac muscle cells: Mathematical modeling and numerical simulation of the actin-myosin interaction, Vietnam J. Math., 49 (2021), 87-118.  doi: 10.1007/s10013-020-00433-z.

[27]

A. Rodríguez-Arós, Mathematical justification of the obstacle problem for elastic elliptic membrane shells, Appl. Anal., 97 (2018), 1261-1280.  doi: 10.1080/00036811.2017.1337894.

[28]

A. ZingaroL. DedèF. Menghini and A. Quarteroni, Hemodynamics of the heart's left atrium based on a Variational Multiscale-LES numerical method, Eur. J. Mech. B Fluids, 89 (2021), 380-400.  doi: 10.1016/j.euromechflu.2021.06.014.

show all references

References:
[1]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.

[2]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.

[3] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511804441.
[4]

L. A. Caffarelli and A. Friedman, The obstacle problem for the biharmonic operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 6 (1979), 151-184. 

[5]

L. A. CaffarelliA. Friedman and A. Torelli, The two-obstacle problem for the biharmonic operator, Pacific J. Math., 103 (1982), 325-335.  doi: 10.2140/pjm.1982.103.325.

[6]

P. G. Ciarlet, Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988.

[7]

P. G. Ciarlet, Mathematical Elasticity. Vol. III: Theory of Shells., North-Holland, Amsterdam, 2000.

[8]

P. G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, 2005.

[9]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2013.

[10]

P. G. Ciarlet and P. Destuynder, A justification of the two-dimensional linear plate model, J. Mécanique, 18 (1979), 315-344. 

[11]

P. G. Ciarlet and V. Lods, Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations, Arch. Rational Mech. Anal., 136 (1996), 119-161.  doi: 10.1007/BF02316975.

[12]

P. G. Ciarlet and V. Lods, On the ellipticity of linear membrane shell equations, J. Math. Pures Appl., 75 (1996), 107-124. 

[13]

P. G. CiarletC. Mardare and P. Piersanti, Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique, C. R. Math. Acad. Sci. Paris, 356 (2018), 1040-1051.  doi: 10.1016/j.crma.2018.08.002.

[14]

P. G. CiarletC. Mardare and P. Piersanti, An obstacle problem for elliptic membrane shells, Math. Mech. Solids, 24 (2019), 1503-1529.  doi: 10.1177/1081286518800164.

[15]

P. G. Ciarlet and P. Piersanti, A confinement problem for a linearly elastic Koiter's shell, C. R. Math. Acad. Sci. Paris, 357 (2019), 221-230.  doi: 10.1016/j.crma.2019.01.004.

[16]

P. G. Ciarlet and P. Piersanti, Obstacle problems for Koiter's shells, Math. Mech. Solids, 24 (2019), 3061-3079.  doi: 10.1177/1081286519825979.

[17]

P. G. Ciarlet and E. Sanchez-Palencia, An existence and uniqueness theorem for the two-dimensional linear membrane shell equations, J. Math. Pures Appl., 75 (1996), 51-67. 

[18]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[19]

J. Frehse, Zum Differenzierbarkeitsproblem bei Variationsungleichungen höherer Ordnung. (German), Abh. Math. Sem. Univ. Hamburg, 36 (1971), 140-149.  doi: 10.1007/BF02995917.

[20]

J. Frehse, On the regularity of the solution of the biharmonic variational inequality, Manuscripta Math., 9 (1973), 91-103.  doi: 10.1007/BF01320669.

[21]

A. Léger and B. Miara, Mathematical justification of the obstacle problem in the case of a shallow shell, J. Elasticity, 90 (2008), 241-257.  doi: 10.1007/s10659-007-9141-1.

[22]

A. Léger and B. Miara, A linearly elastic shell over an obstacle: The flexural case, J. Elasticity, 131 (2018), 19-38.  doi: 10.1007/s10659-017-9643-4.

[23]

M. E. Mezabia, D. A. Chacha and A. Bensayah, Modelling of frictionless Signorini problem for a linear elastic membrane shell, Applicable Analysis, 2020. doi: 10.1080/00036811.2020.1807008.

[24]

P. Piersanti, On the improved interior regularity of the solution of a fourth order elliptic problem modelling the displacement of a linearly elastic shallow shell subject to an obstacle, Asymptot. Anal..

[25]

R. Piersanti, P. C. Africa, M. Fedele, C. Vergara, L. Dedè, A. F. Corno and A. Quarteroni, Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations, Comput. Methods Appl. Mech. Engrg., 373 (2021), 33pp. doi: 10.1016/j.cma.2020.113468.

[26]

F. RegazzoniL. Dedè and A. Quarteroni, Active force generation in cardiac muscle cells: Mathematical modeling and numerical simulation of the actin-myosin interaction, Vietnam J. Math., 49 (2021), 87-118.  doi: 10.1007/s10013-020-00433-z.

[27]

A. Rodríguez-Arós, Mathematical justification of the obstacle problem for elastic elliptic membrane shells, Appl. Anal., 97 (2018), 1261-1280.  doi: 10.1080/00036811.2017.1337894.

[28]

A. ZingaroL. DedèF. Menghini and A. Quarteroni, Hemodynamics of the heart's left atrium based on a Variational Multiscale-LES numerical method, Eur. J. Mech. B Fluids, 89 (2021), 380-400.  doi: 10.1016/j.euromechflu.2021.06.014.

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