October  2013, 6(5): 1173-1191. doi: 10.3934/dcdss.2013.6.1173

On the global regularity for nonlinear systems of the $p$-Laplacian type

1. 

Dipartimento di Matematica Applicata, Università di Pisa, Via Buonarroti 1/C, 56127 Pisa, Italy

2. 

Dipartimento di Matematica, Seconda Università degli Studi di Napoli, Via Vivaldi 43, 81100 Caserta, Italy

Received  October 2011 Published  March 2013

We are interested in regularity results, up to the boundary, for the second derivatives of the solutions of some nonlinear systems of partial differential equations with $p$-growth. We choose two representative cases: the ''full gradient case'', corresponding to a $p$-Laplacian, and the ''symmetric gradient case'', arising from mathematical physics. The domain is either the ''cubic domain'' or a bounded open subset of $\mathbb{R}^3$ with a smooth boundary. Depending on the model and on the range of $p$, $p<2$ or $p>2$, we prove different regularity results. It is worth noting that in the full gradient case with $p<2$ we cover the singular case and obtain $W^{2,q}$-global regularity results, for arbitrarily large values of $q$. In turn, the regularity achieved implies the Hölder continuity of the gradient of the solution.
Citation: Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173
References:
[1]

E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: The case $1 < p < 2$,, J. Math. Anal. Appl., 140 (1989), 115.  doi: 10.1016/0022-247X(89)90098-X.  Google Scholar

[2]

E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system,, J. Reine Angew. Math., 584 (2005), 117.  doi: 10.1515/crll.2005.2005.584.117.  Google Scholar

[3]

H. Beirão da Veiga, On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip or non-slip boundary conditions,, Comm. Pure Appl. Math., 58 (2005), 552.  doi: 10.1002/cpa.20036.  Google Scholar

[4]

H. Beirão da Veiga, Navier-Stokes equations with shear thickening viscosity. Regularity up to the boundary,, J. Math. Fluid Mech., 11 (2009), 233.  doi: 10.1007/s00021-008-0257-2.  Google Scholar

[5]

H. Beirão da Veiga, Navier-Stokes equations with shear thinning viscosity. Regularity up to the boundary,, J. Math. Fluid Mech., 11 (2009), 258.  doi: 10.1007/s00021-008-0258-1.  Google Scholar

[6]

H. Beirão da Veiga, On non-Newtonian $p$-fluids. The pseudo-plastic case,, J. Math. Anal. Appl., 344 (2008), 175.  doi: 10.1016/j.jmaa.2008.02.046.  Google Scholar

[7]

H. Beirão da Veiga, On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem,, J. Eur. Math. Soc. (JEMS), 11 (2009), 127.  doi: 10.4171/JEMS/144.  Google Scholar

[8]

H. Beirão da Veiga, On the global regularity of shear thinning flows in smooth domains,, J. Math. Anal. Appl., 349 (2009), 335.  doi: 10.1016/j.jmaa.2008.09.009.  Google Scholar

[9]

H. Beirão da Veiga and F. Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian type,, , ().   Google Scholar

[10]

H. Beirão da Veiga and F. Crispo, On the global $W^{2,q}$ regularity for nonlinear $N-$systems of the $p$-Laplacian type in $n$ space variables,, Nonlinear Analysis, 75 (2012), 4346.  doi: 10.1016/j.na.2012.03.021.  Google Scholar

[11]

H. Beirão da Veiga, P. Kaplický and M. Růžička, Boundary regularity of shear thickening flows,, J. Math. Fluid Mech., 13 (2011), 387.  doi: 10.1007/s00021-010-0025-y.  Google Scholar

[12]

F. Crispo, A note on the global regularity of steady flows of generalized Newtonian fluids,, Port. Math., 66 (2009), 211.  doi: 10.4171/PM/1841.  Google Scholar

[13]

F. Crispo and C. R. Grisanti, On the existence, uniqueness and $C^{1,\gamma}(\overline\Omega)\cap W^{2,2}(\Omega)$ regularity for a class of shear-thinning fluids,, J. Math. Fluid Mech., 10 (2008), 455.  doi: 10.1007/s00021-008-0282-1.  Google Scholar

[14]

F. Crispo and C. R. Grisanti, On the $C^{1,\gamma}(\overline\Omega)\cap W^{2,2}(\Omega)$ regularity for a class of electro-rheological fluids,, J. Math. Anal. Appl., 356 (2009), 119.  doi: 10.1016/j.jmaa.2009.02.013.  Google Scholar

[15]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[16]

E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems,, Amer. J. Math., 115 (1993), 1107.  doi: 10.2307/2375066.  Google Scholar

[17]

L. Diening, C. Ebmeyer and M. Růžička, Optimal convergence for the implicit space-time discretization of parabolic systems with $p$-structure,, SIAM J. Numer. Anal., 45 (2007), 457.  doi: 10.1137/05064120X.  Google Scholar

[18]

M. Fuchs and G. Mingione, Full $C^{1,\alpha}$-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth,, Manuscripta Math., 102 (2000), 227.  doi: 10.1007/s002291020227.  Google Scholar

[19]

M. Fuchs and G. Seregin, "Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids,", Lecture Notes in Mathematics, 1749 (2000).  doi: 10.1007/BFb0103751.  Google Scholar

[20]

M. Giaquinta and L. Martinazzi, "An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs,", Appunti, 2 (2005).   Google Scholar

[21]

M. Giaquinta and G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals,, Manuscripta Math., 57 (1986), 55.  doi: 10.1007/BF01172492.  Google Scholar

[22]

E. Giusti, "Metodi Diretti nel Calcolo delle Variazioni,", Unione Matematica Italiana, (1994).   Google Scholar

[23]

C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals,, J. Reine Angew. Math., 431 (1992), 7.  doi: 10.1515/crll.1992.431.7.  Google Scholar

[24]

A. I. Košelev, On boundedness of $L^p$ of derivatives of solutions of elliptic differential equations,, (Russian) Mat. Sbornik N.S., 38 (1956), 359.   Google Scholar

[25]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Second English edition, (1969).   Google Scholar

[26]

O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,", Academic Press, (1968).   Google Scholar

[27]

J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations,, Indiana Univ. Math. J., 32 (1983), 849.  doi: 10.1512/iumj.1983.32.32058.  Google Scholar

[28]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[29]

G. M. Lieberman, Gradient estimates for a new class of degenerate elliptic and parabolic equations,, Ann. Sc. Norm. Super. Pisa, 21 (1994), 497.   Google Scholar

[30]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod; Gauthier-Villars, (1969).   Google Scholar

[31]

W. B. Liu and J. W. Barrett, A remark on the regularity of the solutions of the $p$-Laplacian and its application to their finite element approximation,, J. Math. Anal. Appl., 178 (1993), 470.  doi: 10.1006/jmaa.1993.1319.  Google Scholar

[32]

J. Málek, J. Nečas and M. Růžička, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: The case $p\geq2$,, Adv. Differential Equations, 6 (2001), 257.   Google Scholar

[33]

P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth,, J. Differ. Equations, 221 (2006), 412.  doi: 10.1016/j.jde.2004.11.011.  Google Scholar

[34]

G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations,, Appl. Math., 51 (2006), 355.  doi: 10.1007/s10778-006-0110-3.  Google Scholar

[35]

J. Naumann and J. Wolf, Interior differentiability of weak solutions to the equations of stationary motion of a class of non-Newtonian fluids,, J. Math. Fluid Mech., 7 (2005), 298.  doi: 10.1007/s00021-004-0120-z.  Google Scholar

[36]

C. Parés, Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids,, Appl. Anal., 43 (1992), 245.  doi: 10.1080/00036819208840063.  Google Scholar

[37]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

[38]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems,, Acta Math., 138 (1977), 219.   Google Scholar

[39]

N. N. Ural'ceva, Degenerate quasilinear elliptic systems,, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184.   Google Scholar

[40]

V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations,, Dokl. Akad. Nauk SSSR, 138 (1961), 805.   Google Scholar

show all references

References:
[1]

E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: The case $1 < p < 2$,, J. Math. Anal. Appl., 140 (1989), 115.  doi: 10.1016/0022-247X(89)90098-X.  Google Scholar

[2]

E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system,, J. Reine Angew. Math., 584 (2005), 117.  doi: 10.1515/crll.2005.2005.584.117.  Google Scholar

[3]

H. Beirão da Veiga, On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip or non-slip boundary conditions,, Comm. Pure Appl. Math., 58 (2005), 552.  doi: 10.1002/cpa.20036.  Google Scholar

[4]

H. Beirão da Veiga, Navier-Stokes equations with shear thickening viscosity. Regularity up to the boundary,, J. Math. Fluid Mech., 11 (2009), 233.  doi: 10.1007/s00021-008-0257-2.  Google Scholar

[5]

H. Beirão da Veiga, Navier-Stokes equations with shear thinning viscosity. Regularity up to the boundary,, J. Math. Fluid Mech., 11 (2009), 258.  doi: 10.1007/s00021-008-0258-1.  Google Scholar

[6]

H. Beirão da Veiga, On non-Newtonian $p$-fluids. The pseudo-plastic case,, J. Math. Anal. Appl., 344 (2008), 175.  doi: 10.1016/j.jmaa.2008.02.046.  Google Scholar

[7]

H. Beirão da Veiga, On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem,, J. Eur. Math. Soc. (JEMS), 11 (2009), 127.  doi: 10.4171/JEMS/144.  Google Scholar

[8]

H. Beirão da Veiga, On the global regularity of shear thinning flows in smooth domains,, J. Math. Anal. Appl., 349 (2009), 335.  doi: 10.1016/j.jmaa.2008.09.009.  Google Scholar

[9]

H. Beirão da Veiga and F. Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian type,, , ().   Google Scholar

[10]

H. Beirão da Veiga and F. Crispo, On the global $W^{2,q}$ regularity for nonlinear $N-$systems of the $p$-Laplacian type in $n$ space variables,, Nonlinear Analysis, 75 (2012), 4346.  doi: 10.1016/j.na.2012.03.021.  Google Scholar

[11]

H. Beirão da Veiga, P. Kaplický and M. Růžička, Boundary regularity of shear thickening flows,, J. Math. Fluid Mech., 13 (2011), 387.  doi: 10.1007/s00021-010-0025-y.  Google Scholar

[12]

F. Crispo, A note on the global regularity of steady flows of generalized Newtonian fluids,, Port. Math., 66 (2009), 211.  doi: 10.4171/PM/1841.  Google Scholar

[13]

F. Crispo and C. R. Grisanti, On the existence, uniqueness and $C^{1,\gamma}(\overline\Omega)\cap W^{2,2}(\Omega)$ regularity for a class of shear-thinning fluids,, J. Math. Fluid Mech., 10 (2008), 455.  doi: 10.1007/s00021-008-0282-1.  Google Scholar

[14]

F. Crispo and C. R. Grisanti, On the $C^{1,\gamma}(\overline\Omega)\cap W^{2,2}(\Omega)$ regularity for a class of electro-rheological fluids,, J. Math. Anal. Appl., 356 (2009), 119.  doi: 10.1016/j.jmaa.2009.02.013.  Google Scholar

[15]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[16]

E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems,, Amer. J. Math., 115 (1993), 1107.  doi: 10.2307/2375066.  Google Scholar

[17]

L. Diening, C. Ebmeyer and M. Růžička, Optimal convergence for the implicit space-time discretization of parabolic systems with $p$-structure,, SIAM J. Numer. Anal., 45 (2007), 457.  doi: 10.1137/05064120X.  Google Scholar

[18]

M. Fuchs and G. Mingione, Full $C^{1,\alpha}$-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth,, Manuscripta Math., 102 (2000), 227.  doi: 10.1007/s002291020227.  Google Scholar

[19]

M. Fuchs and G. Seregin, "Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids,", Lecture Notes in Mathematics, 1749 (2000).  doi: 10.1007/BFb0103751.  Google Scholar

[20]

M. Giaquinta and L. Martinazzi, "An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs,", Appunti, 2 (2005).   Google Scholar

[21]

M. Giaquinta and G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals,, Manuscripta Math., 57 (1986), 55.  doi: 10.1007/BF01172492.  Google Scholar

[22]

E. Giusti, "Metodi Diretti nel Calcolo delle Variazioni,", Unione Matematica Italiana, (1994).   Google Scholar

[23]

C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals,, J. Reine Angew. Math., 431 (1992), 7.  doi: 10.1515/crll.1992.431.7.  Google Scholar

[24]

A. I. Košelev, On boundedness of $L^p$ of derivatives of solutions of elliptic differential equations,, (Russian) Mat. Sbornik N.S., 38 (1956), 359.   Google Scholar

[25]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Second English edition, (1969).   Google Scholar

[26]

O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,", Academic Press, (1968).   Google Scholar

[27]

J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations,, Indiana Univ. Math. J., 32 (1983), 849.  doi: 10.1512/iumj.1983.32.32058.  Google Scholar

[28]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[29]

G. M. Lieberman, Gradient estimates for a new class of degenerate elliptic and parabolic equations,, Ann. Sc. Norm. Super. Pisa, 21 (1994), 497.   Google Scholar

[30]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod; Gauthier-Villars, (1969).   Google Scholar

[31]

W. B. Liu and J. W. Barrett, A remark on the regularity of the solutions of the $p$-Laplacian and its application to their finite element approximation,, J. Math. Anal. Appl., 178 (1993), 470.  doi: 10.1006/jmaa.1993.1319.  Google Scholar

[32]

J. Málek, J. Nečas and M. Růžička, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: The case $p\geq2$,, Adv. Differential Equations, 6 (2001), 257.   Google Scholar

[33]

P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth,, J. Differ. Equations, 221 (2006), 412.  doi: 10.1016/j.jde.2004.11.011.  Google Scholar

[34]

G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations,, Appl. Math., 51 (2006), 355.  doi: 10.1007/s10778-006-0110-3.  Google Scholar

[35]

J. Naumann and J. Wolf, Interior differentiability of weak solutions to the equations of stationary motion of a class of non-Newtonian fluids,, J. Math. Fluid Mech., 7 (2005), 298.  doi: 10.1007/s00021-004-0120-z.  Google Scholar

[36]

C. Parés, Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids,, Appl. Anal., 43 (1992), 245.  doi: 10.1080/00036819208840063.  Google Scholar

[37]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

[38]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems,, Acta Math., 138 (1977), 219.   Google Scholar

[39]

N. N. Ural'ceva, Degenerate quasilinear elliptic systems,, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184.   Google Scholar

[40]

V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations,, Dokl. Akad. Nauk SSSR, 138 (1961), 805.   Google Scholar

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