A three-dimensional discrete model for approximating the deformation of a viral capsid subjected to lying over a flat surface in the static and time-dependent case. (English) Zbl 1501.35224
Summary: In this paper, we present a three-dimensional discrete model governing the deformation of a viral capsid, modelled as a regular icosahedron and subjected not to cross a given flat rigid surface on which it initially lies in correspondence of one vertex only. First, we set up the model in the form of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable space. Second, we show the existence and uniqueness of the solution for the proposed model. Third, we numerically test this model and we observe that the outputs of the numerical experiments comply with physics. Finally, we establish the existence of solutions for the corresponding time-dependent version of the obstacle problem under consideration.
MSC:
| 35J86 | Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators |
| 35J87 | Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators |
References:
| [1] | Abate, M. and Tovena, F., Curves and Surfaces, , Vol. 55 (Springer, Milan, 2012). · Zbl 1238.53001 |
| [2] | Bock, I., Jarušek, J. and Šilhavý, M., On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate, Nonlinear Anal. Real World Appl.32 (2016) 111-135. · Zbl 1346.35193 |
| [3] | Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer, New York, 2011). · Zbl 1220.46002 |
| [4] | Ciarlet, P. G., Linear and Nonlinear Functional Analysis with Applications, (Philadelphia, PA, 2013). · Zbl 1293.46001 |
| [5] | Ciarlet, P. G. and Piersanti, P., Obstacle problems for Koiter’s shells, Math. Mech. Solids24 (2019) 3061-3079. · Zbl 07273353 |
| [6] | Ciarlet, P. G. and Piersanti, P., A confinement problem for a linearly elastic Koiter’s shell, C. R. Acad. Sci. Paris, Sér. I357 (2019) 221-230. · Zbl 1409.74029 |
| [7] | Ciarlet, P. G., Mardare, C. and Piersanti, P., Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique, C. R. Math. Acad. Sci. Paris356(10) (2018) 1040-1051. · Zbl 1402.74067 |
| [8] | Ciarlet, P. G., Mardare, C. and Piersanti, P., An obstacle problem for elliptic membrane shells. Math. Mech. Solids24(5) (2019) 1503-1529. · Zbl 1440.74205 |
| [9] | Courant, R., Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc.49 (1943) 1-23. · Zbl 0063.00985 |
| [10] | Diestel, J. and Uhl, J. J., Vector Measures (American Mathematical Society, Providence, R.I., 1977). · Zbl 0369.46039 |
| [11] | Dinculeanu, N., Vector Measures (Pergamon Press, Oxford, New York, Toronto, Ont.; VEB Deutscher Verlag der Wissenschaften, Berlin, 1967). |
| [12] | Duvaut, G. and Lions, J.-L., Inequalities in Mechanics and Physics, , Vol. 219 (Springer-Verlag, Berlin, New York, 1976). · Zbl 0331.35002 |
| [13] | I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, Vol. 28 Society for Industrial and Applied Mathematics (Philadelphia, PA, 1999). · Zbl 0939.49002 |
| [14] | Filippucci, R., Existence of global solutions of elliptic systems, J. Math. Anal. Appl.293(2) (2004) 677-692. · Zbl 1160.35403 |
| [15] | Filippucci, R., Entire radial solutions of elliptic systems and inequalities of the mean curvature type, J. Math. Anal. Appl.334(1) (2007) 604-620]]. · Zbl 1194.35136 |
| [16] | Gronwall, T. H., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math.20(4) (1919) 292-296. · JFM 47.0399.02 |
| [17] | Hartman, P., Ordinary Differential Equations, 2nd edn., (Philadelphia, 1982). · Zbl 0476.34002 |
| [18] | Helfrich, W., Elastic properties of lipid bilayers: Theory and possible experiments. Zeitschrift für Naturforschung C28(11-12) (1973) 693-703. |
| [19] | D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Classics in Applied Mathematics, Vol. 31, Society for Industrial and Applied Mathematics (Philadelphia, PA, 2000). · Zbl 0988.49003 |
| [20] | Kyritsi-Yiallourou, S. and Papageorgiou, N. S., Handbook of Applied Analysis (Springer, New York, 2009). · Zbl 1189.49003 |
| [21] | Levine, H. A. and Serrin, J., Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal.137(4) (1997) 341-361. · Zbl 0886.35096 |
| [22] | Lions, J.-L., Quelques Méthodes de Résolution des Problèmes Aux Limites Nonlinéaires (Dunod, Paris; Gauthier-Villars, Paris, 1969). · Zbl 0189.40603 |
| [23] | Lions, J.-L. and Magenes, E., Non-homogeneous Boundary Value Problems and Applications, Vol. 1, (Springer-Verlag, New York, Heidelberg, 1972). · Zbl 0227.35001 |
| [24] | Lions, J.-L. and Magenes, E., Non-homogeneous Boundary Value Problems and Applications, Vol. 2, (Springer-Verlag, New York, Heidelberg, 1972). · Zbl 0223.35039 |
| [25] | Lions, J.-L. and Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, Vol. III, (Springer-Verlag, New York, Heidelberg, 1973). · Zbl 0251.35001 |
| [26] | Piersanti, P., 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.127(1-2) (2022) 35-55. · Zbl 1509.35309 |
| [27] | Piersanti, P., On the improved interior regularity of the solution of a second order elliptic boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle, Discrete Contin. Dyn. Syst.42(2) (2022) 1011-1032. · Zbl 1481.74516 |
| [28] | Piersanti, P. and Shen, X., Numerical methods for static shallow shells lying over an obstacle, Numer. Algorithms85(2) (2020) 623-652. · Zbl 1465.65144 |
| [29] | Piersanti, P., White, K., Dragnea, B. and Temam, R., A simplified model of virus deformation in contact with a surface, Appl. Anal.101(11) (2022) 3947-3957. · Zbl 1495.74046 |
| [30] | Prasolov, V. V. and Tikhomirov, V. M., Geometry, , Vol. 200 (American Mathematical Society, Providence, RI, 2001). · Zbl 0977.51001 |
| [31] | Roubíček, T.. Nonlinear Partial Differential Equations with Applications, , Vol. 153 (Birkhäuser Verlag, Basel, 2005). · Zbl 1087.35002 |
| [32] | Stampacchia, G.. Équations Elliptiques du Second Ordre à Coefficients Discontinus, , Vol. 16 (Été. Les Presses de l’Université de Montréal, Montreal, Que., 1966). · Zbl 0151.15501 |
| [33] | Sun, W. and Yuan, Y.-X., Optimization Theory and Methods, , Vol. 1 (Springer, New York, 2006). · Zbl 1129.90002 |
| [34] | Vliegenthart, G. A. and Gompper, G., Mechanical deformation of spherical viruses with icosahedral symmetry, Biophys. J.91(3) (2006) 834-841. https://doi.org/10.1529/biophysj.106.081422, http://www.ncbi.nlm.nih.gov/pubmed/16679375, http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC1563762. |
| [35] | Young, W. H., On classes of summable functions and their Fourier series, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.87 (1912) 225-229. · JFM 43.1114.12 |
| [36] | Zandi, R., Dragnea, B., Travesset, A. and Podgornik, R., On virus growth and form, Phys. Rep.847 (2020) 1-102. |
| [37] | Zeng, C., Hernando-Pérez, M., Dragnea, B., Ma, X., van der Schoot, P. and Zandi, R., Contact mechanics of a small icosahedral virus, Phys. Rev. Lett.119(3) (2017) 038102. https://doi.org/10.1103/PhysRevLett.119.038102, http://link.aps.org/doi/10.1103/PhysRevLett.119.038102. |
| [38] | Zinger, I., Linear functionals on the space of continuous mappings of a compact Hausdorff space into a Banach space, Rev. Math. Pures Appl.2 (1957) 301-315. |
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