# American Institute of Mathematical Sciences

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
 [1] Luigi C. Berselli, Carlo R. Grisanti. On the regularity up to the boundary for certain nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 53-71. doi: 10.3934/dcdss.2016.9.53 [2] Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063 [3] Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194 [4] Shenzhou Zheng, Laping Zhang, Zhaosheng Feng. Everywhere regularity for P-harmonic type systems under the subcritical growth. Communications on Pure & Applied Analysis, 2008, 7 (1) : 107-117. doi: 10.3934/cpaa.2008.7.107 [5] Jan Burczak, P. Kaplický. Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2401-2445. doi: 10.3934/cpaa.2016042 [6] Petru Jebelean. Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (2) : 267-275. doi: 10.3934/cpaa.2008.7.267 [7] Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51. [8] Yoshikazu Giga, Jürgen Saal. $L^1$ maximal regularity for the laplacian and applications. Conference Publications, 2011, 2011 (Special) : 495-504. doi: 10.3934/proc.2011.2011.495 [9] De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431 [10] Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $p$-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034 [11] Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 [12] Daniela De Silva, Ovidiu Savin. A note on higher regularity boundary Harnack inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6155-6163. doi: 10.3934/dcds.2015.35.6155 [13] Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1751-1759. doi: 10.3934/dcdsb.2012.17.1751 [14] H. Beirão da Veiga. Vorticity and regularity for flows under the Navier boundary condition. Communications on Pure & Applied Analysis, 2006, 5 (4) : 907-918. doi: 10.3934/cpaa.2006.5.907 [15] Kangsheng Liu, Xu Liu, Bopeng Rao. Eventual regularity of a wave equation with boundary dissipation. Mathematical Control & Related Fields, 2012, 2 (1) : 17-28. doi: 10.3934/mcrf.2012.2.17 [16] Carlos E. Kenig, Tatiana Toro. On the free boundary regularity theorem of Alt and Caffarelli. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 397-422. doi: 10.3934/dcds.2004.10.397 [17] Jae-Myoung  Kim. Local regularity of the magnetohydrodynamics equations near the curved boundary. Communications on Pure & Applied Analysis, 2016, 15 (2) : 507-517. doi: 10.3934/cpaa.2016.15.507 [18] Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425 [19] Jaeho Choi, Nitin Krishna, Nicole Magill, Alejandro Sarria. On the $L^p$ regularity of solutions to the generalized Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6349-6365. doi: 10.3934/dcdsb.2019142 [20] Jishan Fan, Fucai Li, Gen Nakamura. A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1757-1766. doi: 10.3934/dcdsb.2018079

2019 Impact Factor: 1.233