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August  2012, 5(4): 865-878. doi: 10.3934/dcdss.2012.5.865

A priori bounds for weak solutions to elliptic equations with nonstandard growth

Patrick Winkert 1, and  Rico Zacher 2,

1. 

Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany
2. 

Martin-Luther-Universität Halle-Wittenberg, Institut für Mathematik, Theodor-Lieser-Strasse 5, D-06120 Halle, Germany

Received  March 2011 Revised  July 2011 Published  November 2011

In this paper we study elliptic equations with a nonlinear conormal derivative boundary condition involving nonstandard growth terms. By means of the localization method and De Giorgi's iteration technique we derive global a priori bounds for weak solutions of such problems.
Keywords: A priori estimates, De Giorgi iteration, Nonstandard growth, Variable exponent spaces., Partition of unity, Elliptic equations.
Mathematics Subject Classification: Primary: 35J60, 35B45; Secondary: 35J2.
Citation: Patrick Winkert, Rico Zacher. A priori bounds for weak solutions to elliptic equations with nonstandard growth. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 865-878. doi: 10.3934/dcdss.2012.5.865
References:
[1]

E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140. doi: 10.1007/s002050100117.  Google Scholar

[2]

E. Acerbi and G. Mingione, Regularity results for electrorheological fluids: The stationary case, C. R. Math. Acad. Sci. Paris, 334 (2002), 817-822.  Google Scholar

[3]

S. N. Antontsev and L. Consiglieri, Elliptic boundary value problems with nonstandard growth conditions, Nonlinear Anal., 71 (2009), 891-902. doi: 10.1016/j.na.2008.10.109.  Google Scholar

[4]

S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 19-36. doi: 10.1007/s11565-006-0002-9.  Google Scholar

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V. Chiadò Piat and A. Coscia, Hölder continuity of minimizers of functionals with variable growth exponent, Manuscripta Math., 93 (1997), 283-299. doi: 10.1007/BF02677472.  Google Scholar

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E. DiBenedetto, "Degenerate Parabolic Equations," Universitext, Springer-Verlag, New York, 1993.  Google Scholar

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L. Diening, "Theoretical and Numerical Results for Electrorheological Fluids," Ph.D thesis, Univ. Freiburg in Breisgau, Mathematische Fakultät, (2002), 156 pp. Google Scholar

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L. Diening, F. Ettwein and M. Růžička, $C^{1,\alpha}$-regularity for electrorheological fluids in two dimensions, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 207-217. doi: 10.1007/s00030-007-5026-z.  Google Scholar

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L. Diening, P. Harjulehto, P. Hästö and M. Růžička, "Lebesgue and Sobolev spaces with variable exponents," Lecture Notes in Mathematics, 2017, Springer-Verlag, Heidelberg, 2011.  Google Scholar

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M. Eleuteri and J. Habermann, Regularity results for a class of obstacle problems under nonstandard growth conditions, J. Math. Anal. Appl., 344 (2008), 1120-1142. doi: 10.1016/j.jmaa.2008.03.068.  Google Scholar

[12]

X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339 (2008), 1395-1412. doi: 10.1016/j.jmaa.2007.08.003.  Google Scholar

[13]

X. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417. Google Scholar

[14]

X. Fan, Local boundedness of quasi-minimizers of integral functions with variable exponent anisotropic growth and applications, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 619-637. doi: 10.1007/s00030-010-0072-3.  Google Scholar

[15]

X. Fan and J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760.  Google Scholar

[16]

X. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36 (1999), 295-318. doi: 10.1016/S0362-546X(97)00628-7.  Google Scholar

[17]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.  Google Scholar

[18]

X. Fan and D. Zhao, The quasi-minimizer of integral functionals with $m(x)$ growth conditions, Nonlinear Anal., 39 (2000), 807-816. doi: 10.1016/S0362-546X(98)00239-9.  Google Scholar

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L. Gasiński and N. S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations, 42 (2011), 323-354. doi: 10.1007/s00526-011-0390-2.  Google Scholar

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J. Habermann and A. Zatorska-Goldstein, Regularity for minimizers of functionals with nonstandard growth by $\mathcalA$-harmonic approximation, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 169-194. doi: 10.1007/s00030-007-7007-7.  Google Scholar

[21]

P. Harjulehto, J. Kinnunen and T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl., (2007), Art. ID 48348, 20 pp.  Google Scholar

[22]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41(116) (1991), 592-618.  Google Scholar

[23]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1967.  Google Scholar

[24]

V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal., 71 (2009), 3305-3321. doi: 10.1016/j.na.2009.01.211.  Google Scholar

[25]

V. Liskevich and I. I. Skrypnik, Harnack inequality and continuity of solutions to elliptic equations with nonstandard growth conditions and lower order terms, Ann. Mat. Pura Appl. (4), 189 (2010), 333-356.  Google Scholar

[26]

T. Lukkari, Boundary continuity of solutions to elliptic equations with nonstandard growth, Manuscripta Math., 132 (2010), 463-482. doi: 10.1007/s00229-010-0355-3.  Google Scholar

[27]

T. Lukkari, Singular solutions of elliptic equations with nonstandard growth, Math. Nachr., 282 (2009), 1770-1787. doi: 10.1002/mana.200610822.  Google Scholar

[28]

P. Pucci and R. Servadei, Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations, Indiana Univ. Math. J., 57 (2008), 3329-3363. doi: 10.1512/iumj.2008.57.3525.  Google Scholar

[29]

K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials, Cont. Mech. and Thermodyn., 13 (2001), 59-78. doi: 10.1007/s001610100034.  Google Scholar

[30]

W. Rudin, "Functional Analysis," McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.  Google Scholar

[31]

M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory," Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000.  Google Scholar

[32]

V. Vergara and R. Zacher, A priori bounds for degenerate and singular evolutionary partial integro-differential equations, Nonlinear Anal., 73 (2010), 3572-3585. doi: 10.1016/j.na.2010.07.039.  Google Scholar

[33]

P. Winkert, Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values, Adv. Differential Equations, 15 (2010), 561-599.  Google Scholar

[34]

P. Winkert, $L^\infty$ -estimates for nonlinear elliptic Neumann boundary value problems, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 289-302. doi: 10.1007/s00030-009-0054-5.  Google Scholar

[35]

V. V. Zhikov, Meyer-type estimates for solving the nonlinear Stokes system, Differ. Equ., 33 (1997), 108-115.  Google Scholar

[36]

V. V. Zhikov, On some variational problems, Russian J. Math. Phys., 5 (1997), 105-116.  Google Scholar

show all references

References:
[1]

E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140. doi: 10.1007/s002050100117.  Google Scholar

[2]

E. Acerbi and G. Mingione, Regularity results for electrorheological fluids: The stationary case, C. R. Math. Acad. Sci. Paris, 334 (2002), 817-822.  Google Scholar

[3]

S. N. Antontsev and L. Consiglieri, Elliptic boundary value problems with nonstandard growth conditions, Nonlinear Anal., 71 (2009), 891-902. doi: 10.1016/j.na.2008.10.109.  Google Scholar

[4]

S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 19-36. doi: 10.1007/s11565-006-0002-9.  Google Scholar

[5]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522.  Google Scholar

[6]

V. Chiadò Piat and A. Coscia, Hölder continuity of minimizers of functionals with variable growth exponent, Manuscripta Math., 93 (1997), 283-299. doi: 10.1007/BF02677472.  Google Scholar

[7]

E. DiBenedetto, "Degenerate Parabolic Equations," Universitext, Springer-Verlag, New York, 1993.  Google Scholar

[8]

L. Diening, "Theoretical and Numerical Results for Electrorheological Fluids," Ph.D thesis, Univ. Freiburg in Breisgau, Mathematische Fakultät, (2002), 156 pp. Google Scholar

[9]

L. Diening, F. Ettwein and M. Růžička, $C^{1,\alpha}$-regularity for electrorheological fluids in two dimensions, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 207-217. doi: 10.1007/s00030-007-5026-z.  Google Scholar

[10]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, "Lebesgue and Sobolev spaces with variable exponents," Lecture Notes in Mathematics, 2017, Springer-Verlag, Heidelberg, 2011.  Google Scholar

[11]

M. Eleuteri and J. Habermann, Regularity results for a class of obstacle problems under nonstandard growth conditions, J. Math. Anal. Appl., 344 (2008), 1120-1142. doi: 10.1016/j.jmaa.2008.03.068.  Google Scholar

[12]

X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339 (2008), 1395-1412. doi: 10.1016/j.jmaa.2007.08.003.  Google Scholar

[13]

X. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417. Google Scholar

[14]

X. Fan, Local boundedness of quasi-minimizers of integral functions with variable exponent anisotropic growth and applications, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 619-637. doi: 10.1007/s00030-010-0072-3.  Google Scholar

[15]

X. Fan and J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760.  Google Scholar

[16]

X. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36 (1999), 295-318. doi: 10.1016/S0362-546X(97)00628-7.  Google Scholar

[17]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.  Google Scholar

[18]

X. Fan and D. Zhao, The quasi-minimizer of integral functionals with $m(x)$ growth conditions, Nonlinear Anal., 39 (2000), 807-816. doi: 10.1016/S0362-546X(98)00239-9.  Google Scholar

[19]

L. Gasiński and N. S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations, 42 (2011), 323-354. doi: 10.1007/s00526-011-0390-2.  Google Scholar

[20]

J. Habermann and A. Zatorska-Goldstein, Regularity for minimizers of functionals with nonstandard growth by $\mathcalA$-harmonic approximation, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 169-194. doi: 10.1007/s00030-007-7007-7.  Google Scholar

[21]

P. Harjulehto, J. Kinnunen and T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl., (2007), Art. ID 48348, 20 pp.  Google Scholar

[22]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41(116) (1991), 592-618.  Google Scholar

[23]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1967.  Google Scholar

[24]

V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal., 71 (2009), 3305-3321. doi: 10.1016/j.na.2009.01.211.  Google Scholar

[25]

V. Liskevich and I. I. Skrypnik, Harnack inequality and continuity of solutions to elliptic equations with nonstandard growth conditions and lower order terms, Ann. Mat. Pura Appl. (4), 189 (2010), 333-356.  Google Scholar

[26]

T. Lukkari, Boundary continuity of solutions to elliptic equations with nonstandard growth, Manuscripta Math., 132 (2010), 463-482. doi: 10.1007/s00229-010-0355-3.  Google Scholar

[27]

T. Lukkari, Singular solutions of elliptic equations with nonstandard growth, Math. Nachr., 282 (2009), 1770-1787. doi: 10.1002/mana.200610822.  Google Scholar

[28]

P. Pucci and R. Servadei, Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations, Indiana Univ. Math. J., 57 (2008), 3329-3363. doi: 10.1512/iumj.2008.57.3525.  Google Scholar

[29]

K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials, Cont. Mech. and Thermodyn., 13 (2001), 59-78. doi: 10.1007/s001610100034.  Google Scholar

[30]

W. Rudin, "Functional Analysis," McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.  Google Scholar

[31]

M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory," Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000.  Google Scholar

[32]

V. Vergara and R. Zacher, A priori bounds for degenerate and singular evolutionary partial integro-differential equations, Nonlinear Anal., 73 (2010), 3572-3585. doi: 10.1016/j.na.2010.07.039.  Google Scholar

[33]

P. Winkert, Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values, Adv. Differential Equations, 15 (2010), 561-599.  Google Scholar

[34]

P. Winkert, $L^\infty$ -estimates for nonlinear elliptic Neumann boundary value problems, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 289-302. doi: 10.1007/s00030-009-0054-5.  Google Scholar

[35]

V. V. Zhikov, Meyer-type estimates for solving the nonlinear Stokes system, Differ. Equ., 33 (1997), 108-115.  Google Scholar

[36]

V. V. Zhikov, On some variational problems, Russian J. Math. Phys., 5 (1997), 105-116.  Google Scholar

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