Fractional Piola identity and polyconvexity in fractional spaces

Bellido, José C.; Cueto, Javier; Mora-Corral, Carlos

doi:10.1016/j.anihpc.2020.02.006

Bellido, José C.¹ ; Cueto, Javier ¹ ; Mora-Corral, Carlos ²

¹ E.T.S.I. Industriales, Department of Mathematics and INEI, Universidad de Castilla-La Mancha, 13.071-Ciudad Real, Spain
² Department of Mathematics, Universidad Autónoma de Madrid, Cantoblanco, 28.049-Madrid, Spain

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 955-981.

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Résumé

In this paper we address nonlocal vector variational principles obtained by substitution of the classical gradient by the Riesz fractional gradient. We show the existence of minimizers in Bessel fractional spaces under the main assumption of polyconvexity of the energy density, and, as a consequence, the existence of solutions to the associated Euler–Lagrange system of nonlinear fractional PDE. The main ingredient is the fractional Piola identity, which establishes that the fractional divergence of the cofactor matrix of the fractional gradient vanishes. This identity implies the weak convergence of the determinant of the fractional gradient, and, in turn, the existence of minimizers of the nonlocal energy. Contrary to local problems in nonlinear elasticity, this existence result is compatible with solutions presenting discontinuities at points and along hypersurfaces.

MR Zbl

DOI : 10.1016/j.anihpc.2020.02.006

Classification : 35Q74, 35R11, 49J45
Mots-clés : Nonlocal variational problems, Riesz fractional gradient, Fractional Piola identity, Polyconvexity

Affiliations des auteurs :

Bellido, José C. ¹ ; Cueto, Javier ¹ ; Mora-Corral, Carlos ²

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     title = {Fractional {Piola} identity and polyconvexity in fractional spaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {955--981},
     publisher = {Elsevier},
     volume = {37},
     number = {4},
     year = {2020},
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     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2020.02.006/}
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Bellido, José C.; Cueto, Javier; Mora-Corral, Carlos. Fractional Piola identity and polyconvexity in fractional spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 955-981. doi : 10.1016/j.anihpc.2020.02.006. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.02.006/

Bibliographie
Cité par

[1] Adams, R.A. Sobolev Spaces, Pure and Applied Mathematics, vol. 65, Academic Press, New York-London, 1975 | MR | Zbl

[2] Ball, J.M. Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 63 (1977) no. 4, pp. 337–403 | DOI | MR | Zbl

[3] Ball, J.M. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. R. Soc. Lond. Ser. A, Volume 306 (1982), pp. 557–611 | MR | Zbl

[4] Ball, J.M. Foundations of Computational Mathematics, Singularities and computation of minimizers for variational problems (London Math. Soc. Lecture Note Ser.), Volume vol. 284, Cambridge Univ. Press, Cambridge (2001), pp. 1–20 (Oxford, 1999) | MR | Zbl

[5] Ball, J.M. Some open problems in elasticity, Geometry, Mechanics, and Dynamics, Springer, New York, 2002, pp. 3–59 | DOI | MR | Zbl

[6] Ball, J.M.; Currie, J.C.; Olver, P.J. Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal., Volume 41 (1981) no. 2, pp. 135–174 | DOI | MR | Zbl

[7] Barchiesi, M.; Henao, D.; Mora-Corral, C. Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity, Arch. Ration. Mech. Anal., Volume 224 (2017) no. 2, pp. 743–816 | DOI | MR | Zbl

[8] Comi, G.E.; Stefani, G. A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up, J. Funct. Anal. (2019) http://www.sciencedirect.com/science/article/pii/S0022123619301016 | MR | Zbl

[9] Dacorogna, B. Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, vol. 78, Springer, New York, 2008 | MR | Zbl

[10] Dal Maso, G.; Francfort, G.A.; Toader, R. Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 176 (2005) no. 2, pp. 165–225 | MR | Zbl

[11] Du, X.; Tian, X. Mathematics of smoothed particle hydrodynamics, part I: a nonlocal Stokes equation, Found. Comput. Math. (2019) | DOI | MR | Zbl

[12] Edelen, D.G.B.; Green, A.; Laws, N. Nonlocal continuum mechanics, Arch. Ration. Mech. Anal., Volume 43 (1971), pp. 34–44 | MR | Zbl

[13] Edelen, D.G.B.; Laws, N. On the thermodynamics of systems with nonlocality, Arch. Ration. Mech. Anal., Volume 43 (1971), pp. 24–35 | DOI | MR | Zbl

[14] Evgrafov, A.; Bellido, J.C. From non-local Eringen's model to fractional elasticity, Math. Mech. Solids, Volume 24 (2019) no. 6, pp. 1935–1953 | DOI | MR | Zbl

[15] Faraco, D.; Rogers, K.M. The Sobolev norm of characteristic functions with applications to the Calderón inverse problem, Q. J. Math., Volume 64 (2013) no. 1, pp. 133–147 | DOI | MR | Zbl

[16] Felsinger, M.; Kassmann, M.; Voigt, P. The Dirichlet problem for nonlocal operators, Math. Z., Volume 279 (2015) no. 3–4, pp. 779–809 | DOI | MR | Zbl

[17] Fonseca, I.; Leoni, G. Modern Methods in the Calculus of Variations: $L^{p}$ Spaces, Springer Monographs in Mathematics, Springer, New York, 2007 | MR | Zbl

[18] Henao, D.; Mora-Corral, C. Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 197 (2010), pp. 619–655 | DOI | MR | Zbl

[19] Kassmann, M.; Mengesha, T.; Scott, J. Solvability of nonlocal systems related to peridynamics, Commun. Pure Appl. Anal., Volume 18 (2019) no. 3, pp. 1303–1332 | MR | Zbl

[20] Lang, S. Real Analysis, Addison-Wesley, Reading, MA, 1983 | MR | Zbl

[21] Maz'ya, V. Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 342, Springer, Heidelberg, 2011 | DOI | MR | Zbl

[22] Mengesha, T.; Du, Q. On the variational limit of a class of nonlocal functionals related to peridynamics, Nonlinearity, Volume 28 (2015) no. 11, pp. 3999–4035 | DOI | MR | Zbl

[23] Mengesha, T.; Du, Q. Characterization of function spaces of vector fields and an application in nonlinear peridynamics, Nonlinear Anal., Volume 140 (2016), pp. 82–111 | DOI | MR | Zbl

[24] Mengesha, T.; Spector, D. Localization of nonlocal gradients in various topologies, Calc. Var. Partial Differ. Equ., Volume 52 (2015) no. 1–2, pp. 253–279 | DOI | MR | Zbl

[25] Müller, S.; Spector, S.J. An existence theory for nonlinear elasticity that allows for cavitation, Arch. Ration. Mech. Anal., Volume 131 (1995) no. 1, pp. 1–66 | DOI | MR | Zbl

[26] Ponce, A.C. Elliptic PDEs, Measures and Capacities: From the Poisson Equations to Nonlinear Thomas-Fermi Problems, EMS Tracts in Mathematics, vol. 23, European Mathematical Society (EMS), Zürich, 2016 | DOI | MR

[27] Ros-Oton, X. Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., Volume 60 (2016) no. 1, pp. 3–26 http://projecteuclid.org/euclid.pm/1450818481 | MR | Zbl

[28] Runst, T.; Sickel, W. Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996 | DOI | MR | Zbl

[29] Schikorra, A.; Spector, D.; Van Schaftingen, J. An $L^{1}$ -type estimate for Riesz potentials, Rev. Mat. Iberoam., Volume 33 (2017) no. 1, pp. 291–303 | DOI | MR | Zbl

[30] Scott, J.; Mengesha, T. A fractional Korn-type inequality, Discrete Contin. Dyn. Syst., Ser. A, Volume 39 (2019) no. 6, pp. 3315–3343 http://aimsciences.org//article/id/fa5bdce1-5e95-404f-8caf-3ae336205c8d | DOI | MR | Zbl

[31] Scott, J.; Mengesha, T. A potential space estimate for solutions of systems of nonlocal equations in peridynamics, SIAM J. Math. Anal., Volume 51 (2019) no. 1, pp. 86–109 | DOI | MR | Zbl

[32] Shieh, T.-T.; Spector, D.E. On a new class of fractional partial differential equations, Adv. Calc. Var., Volume 8 (2015) no. 4, pp. 321–336 | DOI | MR | Zbl

[33] Shieh, T.-T.; Spector, D.E. On a new class of fractional partial differential equations II, Adv. Calc. Var., Volume 11 (2018) no. 3, pp. 289–307 | DOI | MR | Zbl

[34] Sickel, W. The Maz'ya Anniversary Collection, vol. 2, Pointwise multipliers of Lizorkin-Triebel spaces (Oper. Theory Adv. Appl.), Volume vol. 110, Birkhäuser, Basel (1999), pp. 295–321 (Rostock, 1998) | MR | Zbl

[35] Šilhavý, M. Fractional vector analysis based on invariance requirements (critique of coordinate approaches), Contin. Mech. Thermodyn. (Jun 2019) | DOI | MR | Zbl

[36] Silling, S.; Lehoucq, R.; Aref, H.; van der Giessen, E. Peridynamic theory of solid mechanics, Advances in Applied Mechanics, Advances in Applied Mechanics, vol. 44, Elsevier, 2010, pp. 73–168 http://www.sciencedirect.com/science/article/pii/S0065215610440028

[37] Silling, S.A. Linearized theory of peridynamic states, J. Elast., Volume 99 (2010) no. 1, pp. 85–111 | DOI | MR | Zbl

[38] Stein, E.M. Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, NJ, 1970 | MR | Zbl

[39] Triebel, H. Theory of Function Spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983 | DOI | MR | Zbl

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