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. doi: 10.1016/0022-247X(89)90098-X.

[2]

R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (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. 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. 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. 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. 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. 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. 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. 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.

[11]

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.

[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. doi: 10.1137/05064120X.

[13]

L. C. Evans, Partial Differential Equations,, 2nd edition, (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. 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. doi: 10.1512/iumj.1983.32.32058.

[16]

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

[17]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires,, Dunod, (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. 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.

[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.

[21]

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

[22]

L. Nirenberg, Remarks on strongly elliptic partial differential equations,, Comm. Pure Appl. Math., 8 (1955), 649. 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. doi: 10.1007/BF01773507.

[24]

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.

[25]

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

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.

[2]

R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (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. 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. 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. 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. 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. 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. 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. 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.

[11]

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.

[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. doi: 10.1137/05064120X.

[13]

L. C. Evans, Partial Differential Equations,, 2nd edition, (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. 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. doi: 10.1512/iumj.1983.32.32058.

[16]

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

[17]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires,, Dunod, (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. 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.

[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.

[21]

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

[22]

L. Nirenberg, Remarks on strongly elliptic partial differential equations,, Comm. Pure Appl. Math., 8 (1955), 649. 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. doi: 10.1007/BF01773507.

[24]

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.

[25]

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

[1]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184

[2]

Guji Tian, Xu-Jia Wang. Partial regularity for elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 899-913. doi: 10.3934/dcds.2010.28.899

[3]

Alassane Niang. Boundary regularity for a degenerate elliptic equation with mixed boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 107-128. doi: 10.3934/cpaa.2019007

[4]

Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357

[5]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[6]

Matteo Novaga, Diego Pallara, Yannick Sire. A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 815-831. doi: 10.3934/dcdss.2016030

[7]

Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363

[8]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307

[9]

Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149

[10]

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

[11]

Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213

[12]

Maria Colombo, Alessio Figalli. An excess-decay result for a class of degenerate elliptic equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 631-652. doi: 10.3934/dcdss.2014.7.631

[13]

Inbo Sim, Yun-Ho Kim. Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents. Conference Publications, 2013, 2013 (special) : 695-707. doi: 10.3934/proc.2013.2013.695

[14]

Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709

[15]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473

[16]

Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble. The cost of controlling weakly degenerate parabolic equations by boundary controls. Mathematical Control & Related Fields, 2017, 7 (2) : 171-211. doi: 10.3934/mcrf.2017006

[17]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177

[18]

Chunlai Mu, Zhaoyin Xiang. Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux. Communications on Pure & Applied Analysis, 2007, 6 (2) : 487-503. doi: 10.3934/cpaa.2007.6.487

[19]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621

[20]

Raluca Clendenen, Gisèle Ruiz Goldstein, Jerome A. Goldstein. Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 651-660. doi: 10.3934/dcdss.2016019

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]