# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris, 1967.  Google Scholar [22] L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 649-675. doi: 10.1002/cpa.3160080414.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.  Google Scholar [26] N. N. Ural'ceva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184-222.  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), 115-135. doi: 10.1016/0022-247X(89)90098-X.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris, 1967.  Google Scholar [22] L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 649-675. doi: 10.1002/cpa.3160080414.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.  Google Scholar [26] N. N. Ural'ceva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184-222.  Google Scholar
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