February  2016, 9(1): 53-71. doi: 10.3934/dcdss.2016.9.53

On the regularity up to the boundary for certain nonlinear elliptic systems

1. 

Dipartimento di Matematica, Via F. Buonarroti 1/c, Pisa, I-56127, Italy

Received  September 2014 Revised  February 2015 Published  December 2015

We consider a class of nonlinear elliptic systems and we prove regularity up to the boundary for second order derivatives. In the proof we trace carefully the dependence on the various parameters of the problem, in order to establish, in a further work, results for more general systems.
Citation: Luigi C. Berselli, Carlo R. Grisanti. On the regularity up to the boundary for certain nonlinear elliptic systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 53-71. doi: 10.3934/dcdss.2016.9.53
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]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[3]

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.

[4]

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.

[5]

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.

[6]

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 Anal., 75 (2012), 4346-4354. doi: 10.1016/j.na.2012.03.021.

[7]

H. Beirão da Veiga and F. Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian type, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1173-1191. doi: 10.3934/dcdss.2013.6.1173.

[8]

L. C. Berselli, L. Diening and M. Růžička, Existence of strong solutions for incompressible fluids with shear dependent viscosities, J. Math. Fluid Mech., 12 (2010), 101-132. doi: 10.1007/s00021-008-0277-y.

[9]

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.

[10]

F. Crispo and P. Maremonti, On the higher regularity of solutions to the p-Laplacean system in the subquadratic case, Riv. Math. Univ. Parma (N.S.), 5 (2014), 39-63.

[11]

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.

[12]

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 (electronic). doi: 10.1137/05064120X.

[13]

L. C. Evans, Partial Differential Equations, 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[14]

L. C. Evans, A new proof of local $C^{1,\alpha}$ regularity for solutions of certain degenerate elliptic p.d.e, J. Differential Equations, 45 (1982), 356-373. doi: 10.1016/0022-0396(82)90033-X.

[15]

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.

[16]

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.

[17]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.

[18]

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.

[19]

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. Available from: http://projecteuclid.org/euclid.ade/1357141212.

[20]

J. Naumann and J. Wolf, On the interior regularity of weak solutions of degenerate elliptic systems (the case $1< p <2$), Rend. Sem. Mat. Univ. Padova, 88 (1992), 55-81. Available from: http://www.numdam.org/item?id=RSMUP_1992__88__55_0.

[21]

J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris, 1967.

[22]

L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 649-675. doi: 10.1002/cpa.3160080414.

[23]

P. Tolksdorf, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl. (4), 134 (1983), 241-266. doi: 10.1007/BF01773507.

[24]

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.

[25]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.

[26]

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

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]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[3]

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.

[4]

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.

[5]

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.

[6]

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 Anal., 75 (2012), 4346-4354. doi: 10.1016/j.na.2012.03.021.

[7]

H. Beirão da Veiga and F. Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian type, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1173-1191. doi: 10.3934/dcdss.2013.6.1173.

[8]

L. C. Berselli, L. Diening and M. Růžička, Existence of strong solutions for incompressible fluids with shear dependent viscosities, J. Math. Fluid Mech., 12 (2010), 101-132. doi: 10.1007/s00021-008-0277-y.

[9]

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.

[10]

F. Crispo and P. Maremonti, On the higher regularity of solutions to the p-Laplacean system in the subquadratic case, Riv. Math. Univ. Parma (N.S.), 5 (2014), 39-63.

[11]

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.

[12]

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 (electronic). doi: 10.1137/05064120X.

[13]

L. C. Evans, Partial Differential Equations, 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[14]

L. C. Evans, A new proof of local $C^{1,\alpha}$ regularity for solutions of certain degenerate elliptic p.d.e, J. Differential Equations, 45 (1982), 356-373. doi: 10.1016/0022-0396(82)90033-X.

[15]

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.

[16]

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.

[17]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.

[18]

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.

[19]

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. Available from: http://projecteuclid.org/euclid.ade/1357141212.

[20]

J. Naumann and J. Wolf, On the interior regularity of weak solutions of degenerate elliptic systems (the case $1< p <2$), Rend. Sem. Mat. Univ. Padova, 88 (1992), 55-81. Available from: http://www.numdam.org/item?id=RSMUP_1992__88__55_0.

[21]

J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris, 1967.

[22]

L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 649-675. doi: 10.1002/cpa.3160080414.

[23]

P. Tolksdorf, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl. (4), 134 (1983), 241-266. doi: 10.1007/BF01773507.

[24]

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.

[25]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.

[26]

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

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