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 and 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), 115-135. doi: 10.1016/0022-247X(89)90098-X.

[2]

E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system, J. Reine Angew. Math., 584 (2005), 117-148. doi: 10.1515/crll.2005.2005.584.117.

[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), 552-577. doi: 10.1002/cpa.20036.

[4]

H. Beirão da Veiga, Navier-Stokes equations with shear thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 233-257. doi: 10.1007/s00021-008-0257-2.

[5]

H. Beirão da Veiga, Navier-Stokes equations with shear thinning viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 258-273. doi: 10.1007/s00021-008-0258-1.

[6]

H. Beirão da Veiga, On non-Newtonian $p$-fluids. The pseudo-plastic case, J. Math. Anal. Appl., 344 (2008), 175-185. doi: 10.1016/j.jmaa.2008.02.046.

[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), 127-167. doi: 10.4171/JEMS/144.

[8]

H. Beirão da Veiga, On the global regularity of shear thinning flows in smooth domains, J. Math. Anal. Appl., 349 (2009), 335-360. doi: 10.1016/j.jmaa.2008.09.009.

[9]

H. Beirão da Veiga and F. Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian typearXiv:1008.3262v1.

[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), 4346-4354. doi: 10.1016/j.na.2012.03.021.

[11]

H. Beirão da Veiga, P. Kaplický and M. Růžička, Boundary regularity of shear thickening flows, J. Math. Fluid Mech., 13 (2011), 387-404. doi: 10.1007/s00021-010-0025-y.

[12]

F. Crispo, A note on the global regularity of steady flows of generalized Newtonian fluids, Port. Math., 66 (2009), 211-223. doi: 10.4171/PM/1841.

[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), 455-487. doi: 10.1007/s00021-008-0282-1.

[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), 119-132. doi: 10.1016/j.jmaa.2009.02.013.

[15]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5.

[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), 1107-1134. doi: 10.2307/2375066.

[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), 457-472. doi: 10.1137/05064120X.

[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), 227-250. doi: 10.1007/s002291020227.

[19]

M. Fuchs and G. Seregin, "Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids," Lecture Notes in Mathematics, 1749, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103751.

[20]

M. Giaquinta and L. Martinazzi, "An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs," Appunti, Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes, Scuola Normale Superiore di Pisa (New Series)], 2, Edizioni della Normale, Pisa, 2005.

[21]

M. Giaquinta and G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math., 57 (1986), 55-99. doi: 10.1007/BF01172492.

[22]

E. Giusti, "Metodi Diretti nel Calcolo delle Variazioni," Unione Matematica Italiana, Bologna, 1994.

[23]

C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math., 431 (1992), 7-64. doi: 10.1515/crll.1992.431.7.

[24]

A. I. Košelev, On boundedness of $L^p$ of derivatives of solutions of elliptic differential equations, (Russian) Mat. Sbornik N.S., 38 (1956), 359-372.

[25]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Second English edition, Translated from the Russian by Richard A. Silverman Gordon and John Chu, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

[26]

O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York-London, 1968.

[27]

J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J., 32 (1983), 849-858. doi: 10.1512/iumj.1983.32.32058.

[28]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.

[29]

G. M. Lieberman, Gradient estimates for a new class of degenerate elliptic and parabolic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 21 (1994), 497-522.

[30]

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

[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), 470-487. doi: 10.1006/jmaa.1993.1319.

[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), 257-302.

[33]

P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth, J. Differ. Equations, 221 (2006), 412-443. doi: 10.1016/j.jde.2004.11.011.

[34]

G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math., 51 (2006), 355-426. doi: 10.1007/s10778-006-0110-3.

[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), 298-313. doi: 10.1007/s00021-004-0120-z.

[36]

C. Parés, Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids, Appl. Anal., 43 (1992), 245-296. doi: 10.1080/00036819208840063.

[37]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.

[38]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240.

[39]

N. N. Ural'ceva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184-222.

[40]

V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138 (1961), 805-808, English translation in Soviet Math. Doklady, 2 (1961), 746-749.

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), 115-135. doi: 10.1016/0022-247X(89)90098-X.

[2]

E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system, J. Reine Angew. Math., 584 (2005), 117-148. doi: 10.1515/crll.2005.2005.584.117.

[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), 552-577. doi: 10.1002/cpa.20036.

[4]

H. Beirão da Veiga, Navier-Stokes equations with shear thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 233-257. doi: 10.1007/s00021-008-0257-2.

[5]

H. Beirão da Veiga, Navier-Stokes equations with shear thinning viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 258-273. doi: 10.1007/s00021-008-0258-1.

[6]

H. Beirão da Veiga, On non-Newtonian $p$-fluids. The pseudo-plastic case, J. Math. Anal. Appl., 344 (2008), 175-185. doi: 10.1016/j.jmaa.2008.02.046.

[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), 127-167. doi: 10.4171/JEMS/144.

[8]

H. Beirão da Veiga, On the global regularity of shear thinning flows in smooth domains, J. Math. Anal. Appl., 349 (2009), 335-360. doi: 10.1016/j.jmaa.2008.09.009.

[9]

H. Beirão da Veiga and F. Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian typearXiv:1008.3262v1.

[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), 4346-4354. doi: 10.1016/j.na.2012.03.021.

[11]

H. Beirão da Veiga, P. Kaplický and M. Růžička, Boundary regularity of shear thickening flows, J. Math. Fluid Mech., 13 (2011), 387-404. doi: 10.1007/s00021-010-0025-y.

[12]

F. Crispo, A note on the global regularity of steady flows of generalized Newtonian fluids, Port. Math., 66 (2009), 211-223. doi: 10.4171/PM/1841.

[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), 455-487. doi: 10.1007/s00021-008-0282-1.

[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), 119-132. doi: 10.1016/j.jmaa.2009.02.013.

[15]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5.

[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), 1107-1134. doi: 10.2307/2375066.

[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), 457-472. doi: 10.1137/05064120X.

[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), 227-250. doi: 10.1007/s002291020227.

[19]

M. Fuchs and G. Seregin, "Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids," Lecture Notes in Mathematics, 1749, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103751.

[20]

M. Giaquinta and L. Martinazzi, "An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs," Appunti, Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes, Scuola Normale Superiore di Pisa (New Series)], 2, Edizioni della Normale, Pisa, 2005.

[21]

M. Giaquinta and G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math., 57 (1986), 55-99. doi: 10.1007/BF01172492.

[22]

E. Giusti, "Metodi Diretti nel Calcolo delle Variazioni," Unione Matematica Italiana, Bologna, 1994.

[23]

C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math., 431 (1992), 7-64. doi: 10.1515/crll.1992.431.7.

[24]

A. I. Košelev, On boundedness of $L^p$ of derivatives of solutions of elliptic differential equations, (Russian) Mat. Sbornik N.S., 38 (1956), 359-372.

[25]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Second English edition, Translated from the Russian by Richard A. Silverman Gordon and John Chu, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

[26]

O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York-London, 1968.

[27]

J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J., 32 (1983), 849-858. doi: 10.1512/iumj.1983.32.32058.

[28]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.

[29]

G. M. Lieberman, Gradient estimates for a new class of degenerate elliptic and parabolic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 21 (1994), 497-522.

[30]

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

[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), 470-487. doi: 10.1006/jmaa.1993.1319.

[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), 257-302.

[33]

P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth, J. Differ. Equations, 221 (2006), 412-443. doi: 10.1016/j.jde.2004.11.011.

[34]

G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math., 51 (2006), 355-426. doi: 10.1007/s10778-006-0110-3.

[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), 298-313. doi: 10.1007/s00021-004-0120-z.

[36]

C. Parés, Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids, Appl. Anal., 43 (1992), 245-296. doi: 10.1080/00036819208840063.

[37]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.

[38]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240.

[39]

N. N. Ural'ceva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184-222.

[40]

V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138 (1961), 805-808, English translation in Soviet Math. Doklady, 2 (1961), 746-749.

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