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Noncoercive elliptic equations with subcritical growth
A priori bounds for weak solutions to elliptic equations with nonstandard growth
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 |
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. |
[2] |
E. Acerbi and G. Mingione, Regularity results for electrorheological fluids: The stationary case, C. R. Math. Acad. Sci. Paris, 334 (2002), 817-822. |
[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. |
[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. |
[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. |
[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. |
[7] |
E. DiBenedetto, "Degenerate Parabolic Equations," Universitext, Springer-Verlag, New York, 1993. |
[8] |
L. Diening, "Theoretical and Numerical Results for Electrorheological Fluids," Ph.D thesis, Univ. Freiburg in Breisgau, Mathematische Fakultät, (2002), 156 pp. |
[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. |
[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. |
[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. |
[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. |
[13] |
X. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
J. Habermann and A. Zatorska-Goldstein, Regularity for minimizers of functionals with nonstandard growth by $\mathcal{A}$-harmonic approximation, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 169-194.
doi: 10.1007/s00030-007-7007-7. |
[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. |
[22] |
O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41(116) (1991), 592-618. |
[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. |
[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. |
[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. |
[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. |
[27] |
T. Lukkari, Singular solutions of elliptic equations with nonstandard growth, Math. Nachr., 282 (2009), 1770-1787.
doi: 10.1002/mana.200610822. |
[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. |
[29] |
K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials, Cont. Mech. and Thermodyn., 13 (2001), 59-78.
doi: 10.1007/s001610100034. |
[30] |
W. Rudin, "Functional Analysis," McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. |
[31] |
M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory," Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000. |
[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. |
[33] |
P. Winkert, Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values, Adv. Differential Equations, 15 (2010), 561-599. |
[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. |
[35] |
V. V. Zhikov, Meyer-type estimates for solving the nonlinear Stokes system, Differ. Equ., 33 (1997), 108-115. |
[36] |
V. V. Zhikov, On some variational problems, Russian J. Math. Phys., 5 (1997), 105-116. |
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. |
[2] |
E. Acerbi and G. Mingione, Regularity results for electrorheological fluids: The stationary case, C. R. Math. Acad. Sci. Paris, 334 (2002), 817-822. |
[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. |
[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. |
[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. |
[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. |
[7] |
E. DiBenedetto, "Degenerate Parabolic Equations," Universitext, Springer-Verlag, New York, 1993. |
[8] |
L. Diening, "Theoretical and Numerical Results for Electrorheological Fluids," Ph.D thesis, Univ. Freiburg in Breisgau, Mathematische Fakultät, (2002), 156 pp. |
[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. |
[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. |
[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. |
[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. |
[13] |
X. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
J. Habermann and A. Zatorska-Goldstein, Regularity for minimizers of functionals with nonstandard growth by $\mathcal{A}$-harmonic approximation, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 169-194.
doi: 10.1007/s00030-007-7007-7. |
[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. |
[22] |
O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41(116) (1991), 592-618. |
[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. |
[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. |
[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. |
[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. |
[27] |
T. Lukkari, Singular solutions of elliptic equations with nonstandard growth, Math. Nachr., 282 (2009), 1770-1787.
doi: 10.1002/mana.200610822. |
[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. |
[29] |
K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials, Cont. Mech. and Thermodyn., 13 (2001), 59-78.
doi: 10.1007/s001610100034. |
[30] |
W. Rudin, "Functional Analysis," McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. |
[31] |
M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory," Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000. |
[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. |
[33] |
P. Winkert, Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values, Adv. Differential Equations, 15 (2010), 561-599. |
[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. |
[35] |
V. V. Zhikov, Meyer-type estimates for solving the nonlinear Stokes system, Differ. Equ., 33 (1997), 108-115. |
[36] |
V. V. Zhikov, On some variational problems, Russian J. Math. Phys., 5 (1997), 105-116. |
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