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May 2019, 18(3): 1303-1332. doi: 10.3934/cpaa.2019063

Solvability of nonlocal systems related to peridynamics

1. 

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

2. 

Department of Mathematics, The University of Tennessee Knoxville, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996, USA

* Corresponding author

Received  May 2018 Revised  May 2018 Published  November 2018

Fund Project: M. Kassmann acknowledges the support of the German Science Foundation through CRC 1283. T. Mengesha and J. Scott acknowledge the support of the U.S. NSF under grant DMS-1615726

In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lamé system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces.

Citation: Moritz Kassmann, Tadele Mengesha, James Scott. Solvability of nonlocal systems related to peridynamics. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1303-1332. doi: 10.3934/cpaa.2019063
References:
[1]

U. BiccariM. Warma and E. Zuazua, Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 837-839. doi: 10.1515/ans-2017-6020.

[2]

U. BiccariM. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409. doi: 10.1515/ans-2017-0014.

[3]

H. Brezis, Functional analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[4]

Z.-Q. Chen, On notions of harmonicity, Proc. Amer. Math. Soc., 137 (2009), 3497-3510. doi: 10.1090/S0002-9939-09-09945-6.

[5]

M. Cozzi, Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces, Ann. Mat. Pura Appl., 196 (2017), 555-578. doi: 10.1007/s10231-016-0586-3.

[6]

F. Da Lio, Fractional harmonic maps into manifolds in odd dimension $ n>1$, Calc. Var. Partial Differential Equations, 48 (2013), 421-445. doi: 10.1007/s00526-012-0556-6.

[7]

F. Da Lio and T. Rivière, Three-term commutator estimates and the regularity of $ \frac{1}{2}$-harmonic maps into spheres, Anal. PDE, 4 (2011), 149-190. doi: 10.2140/apde.2011.4.149.

[8]

F. Da Lio and A. Schikorra, On regularity theory for n/p-harmonic maps into manifolds, arXiv e-prints, (2017). doi: 10.1016/j.na.2017.10.001.

[9]

H. Dong and D. Kim, On $ L_p$-estimates for a class of non-local elliptic equations, J. Funct. Anal., 262 (2012), 1166-1199. doi: 10.1016/j.jfa.2011.11.002.

[10]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, J. Elasticity, 113 (2013), 193-217. doi: 10.1007/s10659-012-9418-x.

[11]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540. doi: 10.1142/S0218202512500546.

[12]

Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, ESAIM Math. Model. Numer. Anal., 45 (2011), 217-234. doi: 10.1051/m2an/2010040.

[13]

E. Emmrich and O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity, Commun. Math. Sci., 5 (2007), 851-864.

[14]

M. FelsingerM. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809. doi: 10.1007/s00209-014-1394-3.

[15]

M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric MArkov Processes, vol. 19 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, extended ed., 2011.

[16]

M. Fukushima and T. Uemura, Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms, Ann. Probab., 40 (2012), 858-889. doi: 10.1214/10-AOP633.

[17]

M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs vol. 11 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, second ed., 2012. doi: 10.1007/978-88-7642-443-4.

[18]

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018.

[19]

M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598. doi: 10.1137/090766607.

[20]

Z.-C. HuZ.-M. Ma and W. Sun, On representations of non-symmetric Dirichlet forms, Potential Anal., 32 (2010), 101-131. doi: 10.1007/s11118-009-9145-5.

[21]

M. Kassmann and B. Dyda, Function spaces and extension results for nonlocal Dirichlet problems, arXiv e-prints, (2016).

[22]

T. LeonoriI. PeralA. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068. doi: 10.3934/dcds.2015.35.6031.

[23]

Z. MaR. Zhu and X. Zhu, On notions of harmonicity for non-symmetric Dirichlet form, Sci. China Math., 53 (2010), 1407-1420. doi: 10.1007/s11425-010-4001-z.

[24]

T. Mengesha, Fractional Korn and Hardy-type inequalities for vector fields in half space, Communications in Contemporary Mathematics, to appear. doi: 10.1142/S0219199718500554.

[25]

T. Mengesha and Q. Du, The bond-based peridynamic system with Dirichlet-type volume constraint, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 161-186. doi: 10.1017/S0308210512001436.

[26]

T. Mengesha and Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation, J. Elasticity, 116 (2014), 27-51. doi: 10.1007/s10659-013-9456-z.

[27]

V. Millot and Y. Sire, On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Ration. Mech. Anal., 215 (2015), 125-210. doi: 10.1007/s00205-014-0776-3.

[28]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.

[29]

A. Rutkowski, The Dirichlet problem for nonlocal Lévy-type operators, Publ. Mat., 62 (2018), 213-251. doi: 10.5565/PUBLMAT6211811.

[30]

A. Schikorra, Regularity of n/2-harmonic maps into spheres, J. Differential Equations, 252 (2012), 1862-1911. doi: 10.1016/j.jde.2011.08.021.

[31]

A. Schikorra, Lp-gradient harmonic maps into spheres and SO(N), Differential Integral Equations, 28 (2015), 383-408.

[32]

R. L. Schilling and J. Wang, Lower bounded semi-Dirichlet forms associated with Lévy type operators, in Festschrift Masatoshi Fukushima, vol. 17 of Interdiscip. Math. Sci., World Sci. Publ., Hackensack, NJ, 2015,507-526. doi: 10.1142/9789814596534_0025.

[33]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105.

[34]

S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), 175-209. doi: 10.1016/S0022-5096(99)00029-0.

[35]

S. A. Silling, Linearized theory of peridynamic states, J. Elasticity, 99 (2010), 85-111. doi: 10.1007/s10659-009-9234-0.

[36]

S. A. Silling, M. Epton, O. Weckner, J. Xu, and E. Askari, Peridynamic states and constitutive modeling, J. Elasticity, 88 (2007), 151-184. doi: 10.1007/s10659-007-9125-1.

[37]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.

[38]

R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, second ed., 2005. doi: 10.1017/CBO9780511755422.

show all references

References:
[1]

U. BiccariM. Warma and E. Zuazua, Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 837-839. doi: 10.1515/ans-2017-6020.

[2]

U. BiccariM. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409. doi: 10.1515/ans-2017-0014.

[3]

H. Brezis, Functional analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[4]

Z.-Q. Chen, On notions of harmonicity, Proc. Amer. Math. Soc., 137 (2009), 3497-3510. doi: 10.1090/S0002-9939-09-09945-6.

[5]

M. Cozzi, Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces, Ann. Mat. Pura Appl., 196 (2017), 555-578. doi: 10.1007/s10231-016-0586-3.

[6]

F. Da Lio, Fractional harmonic maps into manifolds in odd dimension $ n>1$, Calc. Var. Partial Differential Equations, 48 (2013), 421-445. doi: 10.1007/s00526-012-0556-6.

[7]

F. Da Lio and T. Rivière, Three-term commutator estimates and the regularity of $ \frac{1}{2}$-harmonic maps into spheres, Anal. PDE, 4 (2011), 149-190. doi: 10.2140/apde.2011.4.149.

[8]

F. Da Lio and A. Schikorra, On regularity theory for n/p-harmonic maps into manifolds, arXiv e-prints, (2017). doi: 10.1016/j.na.2017.10.001.

[9]

H. Dong and D. Kim, On $ L_p$-estimates for a class of non-local elliptic equations, J. Funct. Anal., 262 (2012), 1166-1199. doi: 10.1016/j.jfa.2011.11.002.

[10]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, J. Elasticity, 113 (2013), 193-217. doi: 10.1007/s10659-012-9418-x.

[11]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540. doi: 10.1142/S0218202512500546.

[12]

Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, ESAIM Math. Model. Numer. Anal., 45 (2011), 217-234. doi: 10.1051/m2an/2010040.

[13]

E. Emmrich and O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity, Commun. Math. Sci., 5 (2007), 851-864.

[14]

M. FelsingerM. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809. doi: 10.1007/s00209-014-1394-3.

[15]

M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric MArkov Processes, vol. 19 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, extended ed., 2011.

[16]

M. Fukushima and T. Uemura, Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms, Ann. Probab., 40 (2012), 858-889. doi: 10.1214/10-AOP633.

[17]

M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs vol. 11 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, second ed., 2012. doi: 10.1007/978-88-7642-443-4.

[18]

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018.

[19]

M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598. doi: 10.1137/090766607.

[20]

Z.-C. HuZ.-M. Ma and W. Sun, On representations of non-symmetric Dirichlet forms, Potential Anal., 32 (2010), 101-131. doi: 10.1007/s11118-009-9145-5.

[21]

M. Kassmann and B. Dyda, Function spaces and extension results for nonlocal Dirichlet problems, arXiv e-prints, (2016).

[22]

T. LeonoriI. PeralA. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068. doi: 10.3934/dcds.2015.35.6031.

[23]

Z. MaR. Zhu and X. Zhu, On notions of harmonicity for non-symmetric Dirichlet form, Sci. China Math., 53 (2010), 1407-1420. doi: 10.1007/s11425-010-4001-z.

[24]

T. Mengesha, Fractional Korn and Hardy-type inequalities for vector fields in half space, Communications in Contemporary Mathematics, to appear. doi: 10.1142/S0219199718500554.

[25]

T. Mengesha and Q. Du, The bond-based peridynamic system with Dirichlet-type volume constraint, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 161-186. doi: 10.1017/S0308210512001436.

[26]

T. Mengesha and Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation, J. Elasticity, 116 (2014), 27-51. doi: 10.1007/s10659-013-9456-z.

[27]

V. Millot and Y. Sire, On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Ration. Mech. Anal., 215 (2015), 125-210. doi: 10.1007/s00205-014-0776-3.

[28]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.

[29]

A. Rutkowski, The Dirichlet problem for nonlocal Lévy-type operators, Publ. Mat., 62 (2018), 213-251. doi: 10.5565/PUBLMAT6211811.

[30]

A. Schikorra, Regularity of n/2-harmonic maps into spheres, J. Differential Equations, 252 (2012), 1862-1911. doi: 10.1016/j.jde.2011.08.021.

[31]

A. Schikorra, Lp-gradient harmonic maps into spheres and SO(N), Differential Integral Equations, 28 (2015), 383-408.

[32]

R. L. Schilling and J. Wang, Lower bounded semi-Dirichlet forms associated with Lévy type operators, in Festschrift Masatoshi Fukushima, vol. 17 of Interdiscip. Math. Sci., World Sci. Publ., Hackensack, NJ, 2015,507-526. doi: 10.1142/9789814596534_0025.

[33]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105.

[34]

S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), 175-209. doi: 10.1016/S0022-5096(99)00029-0.

[35]

S. A. Silling, Linearized theory of peridynamic states, J. Elasticity, 99 (2010), 85-111. doi: 10.1007/s10659-009-9234-0.

[36]

S. A. Silling, M. Epton, O. Weckner, J. Xu, and E. Askari, Peridynamic states and constitutive modeling, J. Elasticity, 88 (2007), 151-184. doi: 10.1007/s10659-007-9125-1.

[37]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.

[38]

R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, second ed., 2005. doi: 10.1017/CBO9780511755422.

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