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 & 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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.   Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.   Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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