June  2019, 39(6): 3315-3343. doi: 10.3934/dcds.2019137

A fractional Korn-type inequality

The University of Tennessee, Knoxville, TN 37996, USA

* Corresponding author

Received  July 2018 Revised  December 2018 Published  February 2019

Fund Project: The second author is supported by NSF grant DMS-1615726.

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of "directional" difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to understand better the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical space embeddings in proving that weak solutions to the nonlocal system enjoy both improved differentiability and improved integrability.

Citation: James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137
References:
[1]

P. AuscherS. BortzM. Egert and O. Saari, Non-local self-improving properties: A functional analytic approach, Tunisian Journal of Mathematics, 1 (2019), 151-183. 

[2]

R. F. Bass and H. Ren, Meyers inequality and strong stability for stable-like operators, J. of Func. Anal, 265 (2013), 28-48.  doi: 10.1016/j.jfa.2013.03.008.

[3]

S. BlattP. Reiter and A. Schikorra, Harmonic analysis meets critical knots (stationary points of the moebius energy are smooth, Trans. Amer. Math. Soc., 368 (2016), 6391-6438.  doi: 10.1090/tran/6603.

[4]

J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in: Optimal Control and Partial Differential, Equations. A Volume in Honour of A. Bensoussans 60th Birthday, IOS Press, 2001, 439–455.

[5]

K. de Leeuw and H. Mirkil, A priori estimates for differential operators in $L^{\infty}$ norm, Illinois J. Math., 8 (1964), 112-124. 

[6]

F. Demengel and G. Demengel, Function Spaces for the Theory of Elliptic Partial Differential Equations, Springer, 2012. doi: 10.1007/978-1-4471-2807-6.

[7]

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

[8]

T. KuusiG. Mingione and Y. Sire, Nonlocal self-improving properties, Analysis and PDE, 8 (2015), 57-114.  doi: 10.2140/apde.2015.8.57.

[9]

R. Lipton, Dynamic brittle fracture as a small horizon limit of peridynamics, Journal of Elasticity, 117 (2014), 21-50.  doi: 10.1007/s10659-013-9463-0.

[10]

R. Lipton, Cohesive dynamics and fracture, Journal of Elasticity, 124 (2016), 143-191.  doi: 10.1007/s10659-015-9564-z.

[11]

J. M. MartellD. MitreaI. Mitrea and M. Mitrea, The higher order regularity Dirichlet problem for elliptic systems in the upper-half space, Harmonic Analysis and Partial Differential Equations. Proceedings of the 9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 11-15, 2012, Contemporary Mathematics, 612 (2014), 123-141.  doi: 10.1090/conm/612/12228.

[12]

J. M. MartellD. MitreaI. Mitrea and M. Mitrea, The Dirichlet problem for elliptic systems with data in Köthe function spaces, Revista Matemática Iberoamericana, 268 (2016), 913-970.  doi: 10.4171/RMI/903.

[13]

T. Mengesha, Nonlocal Korn-type characterization of Sobolev vector fields, Commun. Contemp. Math., 14 (2012), 1250028, 28pp. doi: 10.1142/S0219199712500289.

[14]

T. Mengesha, Fractional Korn and Hardy-type inequalities for vector fields in half space, To appear in Communications in Contemporary Mathematics, 2018, URL https://arXiv.org/abs/1805.06434. doi: 10.1142/S0219199718500554.

[15]

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

[16]

T. Mengesha and Q. Du, On the variational limit of a class of nonlocal functionals related to peridynamics, Nonlinearity, 28 (2015), 3999-4035.  doi: 10.1088/0951-7715/28/11/3999.

[17]

D. Mitrea, Distributions, Partial Differential Equations, and Harmonic Analysis, Springer Universitext, 2013. doi: 10.1007/978-1-4614-8208-6.

[18]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[19]

J. A. Nitsche, On Korn's second inequality, ESAIM: M2AN, 15 (1981), 237-248.  doi: 10.1051/m2an/1981150302371.

[20]

D. Ornstein, A non-inequality for differential operators in the $L^{1}$ norm, Arch. Rational Mech. Anal., 11 (1962), 40-49.  doi: 10.1007/BF00253928.

[21]

A. Schikorra, Nonlinear commutators for the fractional $p-$Laplacian and applications, Math. Ann., 366 (2016), 695-720.  doi: 10.1007/s00208-015-1347-0.

[22]

J. Scott and T. Mengesha, A potential space estimate for solutions of systems of nonlocal equations in peridynamics, SIAM Journal of Mathematical Analysis, 51 (2019), 86-109.  doi: 10.1137/18M1189294.

[23]

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.

[24]

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

[25]

S. A. SillingM. EptonO. WecknerJ. Xu and E. Askari, Peridynamic states and constitutive modeling, J. Elast., 88 (2007), 151-184.  doi: 10.1007/s10659-007-9125-1.

[26] E. N. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. 
[27]

M. H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space: I. principal properties, Journal of Mathematics and Mechanics, 13 (1964), 407-479. 

[28]

H. B. Veiga and F. Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian type, Discrete & Continuous Dynamical Systems, 6 (2013), 1173-1191.  doi: 10.3934/dcdss.2013.6.1173.

show all references

References:
[1]

P. AuscherS. BortzM. Egert and O. Saari, Non-local self-improving properties: A functional analytic approach, Tunisian Journal of Mathematics, 1 (2019), 151-183. 

[2]

R. F. Bass and H. Ren, Meyers inequality and strong stability for stable-like operators, J. of Func. Anal, 265 (2013), 28-48.  doi: 10.1016/j.jfa.2013.03.008.

[3]

S. BlattP. Reiter and A. Schikorra, Harmonic analysis meets critical knots (stationary points of the moebius energy are smooth, Trans. Amer. Math. Soc., 368 (2016), 6391-6438.  doi: 10.1090/tran/6603.

[4]

J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in: Optimal Control and Partial Differential, Equations. A Volume in Honour of A. Bensoussans 60th Birthday, IOS Press, 2001, 439–455.

[5]

K. de Leeuw and H. Mirkil, A priori estimates for differential operators in $L^{\infty}$ norm, Illinois J. Math., 8 (1964), 112-124. 

[6]

F. Demengel and G. Demengel, Function Spaces for the Theory of Elliptic Partial Differential Equations, Springer, 2012. doi: 10.1007/978-1-4471-2807-6.

[7]

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

[8]

T. KuusiG. Mingione and Y. Sire, Nonlocal self-improving properties, Analysis and PDE, 8 (2015), 57-114.  doi: 10.2140/apde.2015.8.57.

[9]

R. Lipton, Dynamic brittle fracture as a small horizon limit of peridynamics, Journal of Elasticity, 117 (2014), 21-50.  doi: 10.1007/s10659-013-9463-0.

[10]

R. Lipton, Cohesive dynamics and fracture, Journal of Elasticity, 124 (2016), 143-191.  doi: 10.1007/s10659-015-9564-z.

[11]

J. M. MartellD. MitreaI. Mitrea and M. Mitrea, The higher order regularity Dirichlet problem for elliptic systems in the upper-half space, Harmonic Analysis and Partial Differential Equations. Proceedings of the 9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 11-15, 2012, Contemporary Mathematics, 612 (2014), 123-141.  doi: 10.1090/conm/612/12228.

[12]

J. M. MartellD. MitreaI. Mitrea and M. Mitrea, The Dirichlet problem for elliptic systems with data in Köthe function spaces, Revista Matemática Iberoamericana, 268 (2016), 913-970.  doi: 10.4171/RMI/903.

[13]

T. Mengesha, Nonlocal Korn-type characterization of Sobolev vector fields, Commun. Contemp. Math., 14 (2012), 1250028, 28pp. doi: 10.1142/S0219199712500289.

[14]

T. Mengesha, Fractional Korn and Hardy-type inequalities for vector fields in half space, To appear in Communications in Contemporary Mathematics, 2018, URL https://arXiv.org/abs/1805.06434. doi: 10.1142/S0219199718500554.

[15]

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

[16]

T. Mengesha and Q. Du, On the variational limit of a class of nonlocal functionals related to peridynamics, Nonlinearity, 28 (2015), 3999-4035.  doi: 10.1088/0951-7715/28/11/3999.

[17]

D. Mitrea, Distributions, Partial Differential Equations, and Harmonic Analysis, Springer Universitext, 2013. doi: 10.1007/978-1-4614-8208-6.

[18]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[19]

J. A. Nitsche, On Korn's second inequality, ESAIM: M2AN, 15 (1981), 237-248.  doi: 10.1051/m2an/1981150302371.

[20]

D. Ornstein, A non-inequality for differential operators in the $L^{1}$ norm, Arch. Rational Mech. Anal., 11 (1962), 40-49.  doi: 10.1007/BF00253928.

[21]

A. Schikorra, Nonlinear commutators for the fractional $p-$Laplacian and applications, Math. Ann., 366 (2016), 695-720.  doi: 10.1007/s00208-015-1347-0.

[22]

J. Scott and T. Mengesha, A potential space estimate for solutions of systems of nonlocal equations in peridynamics, SIAM Journal of Mathematical Analysis, 51 (2019), 86-109.  doi: 10.1137/18M1189294.

[23]

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.

[24]

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

[25]

S. A. SillingM. EptonO. WecknerJ. Xu and E. Askari, Peridynamic states and constitutive modeling, J. Elast., 88 (2007), 151-184.  doi: 10.1007/s10659-007-9125-1.

[26] E. N. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. 
[27]

M. H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space: I. principal properties, Journal of Mathematics and Mechanics, 13 (1964), 407-479. 

[28]

H. B. Veiga and F. Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian type, Discrete & Continuous Dynamical Systems, 6 (2013), 1173-1191.  doi: 10.3934/dcdss.2013.6.1173.

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