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

Previous Article
DCDS-S Home
This Issue
Next Article
Noncoercive elliptic equations with subcritical growth

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

Abstract
Related Papers

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 and 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.
[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.

[1]	Tomasz Adamowicz, Przemysław Górka. The Liouville theorems for elliptic equations with nonstandard growth. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2377-2392. doi: 10.3934/cpaa.2015.14.2377
[2]	Sun-Sig Byun, Yumi Cho, Shuang Liang. Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3843-3855. doi: 10.3934/dcdsb.2020038
[3]	SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure and Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026
[4]	Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
[5]	Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128
[6]	Luis A. Caffarelli, Alexis F. Vasseur. The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 409-427. doi: 10.3934/dcdss.2010.3.409
[7]	D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure and Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499
[8]	Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095
[9]	Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
[10]	Svetlana Pastukhova, Valeria Chiadò Piat. Homogenization of multivalued monotone operators with variable growth exponent. Networks and Heterogeneous Media, 2020, 15 (2) : 281-305. doi: 10.3934/nhm.2020013
[11]	Prashanta Garain, Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5059-5075. doi: 10.3934/cpaa.2020226
[12]	Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017
[13]	Changfeng Gui. On some problems related to de Giorgi’s conjecture. Communications on Pure and Applied Analysis, 2003, 2 (1) : 101-106. doi: 10.3934/cpaa.2003.2.101
[14]	Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145
[15]	Laura Baldelli, Roberta Filippucci. A priori estimates for elliptic problems via Liouville type theorems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1883-1898. doi: 10.3934/dcdss.2020148
[16]	Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107
[17]	Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131
[18]	Carla Baroncini, Julián Fernández Bonder. An extension of a Theorem of V. Šverák to variable exponent spaces. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1987-2007. doi: 10.3934/cpaa.2015.14.1987
[19]	Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013
[20]	Paolo Baroni, Agnese Di Castro, Giampiero Palatucci. Intrinsic geometry and De Giorgi classes for certain anisotropic problems. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 647-659. doi: 10.3934/dcdss.2017032

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Patrick Winkert Rico Zacher
on Google Scholar
Patrick Winkert Rico Zacher

[Back to Top]