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

A fractional Korn-type inequality

James Scott and  Tadele Mengesha ,

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.

Keywords: Korn's inequality, fractional Sobolev spaces, Poisson integral, coupled nonlocal equations, self-improving properties.
Mathematics Subject Classification: Primary: 46E35, 46E40, 45G15; Secondary: 35B65, 74B99.
Citation: James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137
References:
[1]

P. Auscher, S. Bortz, M. Egert and O. Saari, Non-local self-improving properties: A functional analytic approach, Tunisian Journal of Mathematics, 1 (2019), 151-183.   Google Scholar

[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.  Google Scholar

[3]

S. Blatt, P. 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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

J. M. Martell, D. Mitrea, I. 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.  Google Scholar

[12]

J. M. Martell, D. Mitrea, I. 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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

E. D. Nezza, G. 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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

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

[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.  Google Scholar

show all references

References:
[1]

P. Auscher, S. Bortz, M. Egert and O. Saari, Non-local self-improving properties: A functional analytic approach, Tunisian Journal of Mathematics, 1 (2019), 151-183.   Google Scholar

[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.  Google Scholar

[3]

S. Blatt, P. 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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

J. M. Martell, D. Mitrea, I. 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.  Google Scholar

[12]

J. M. Martell, D. Mitrea, I. 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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

E. D. Nezza, G. 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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

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

[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.  Google Scholar

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