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

Patrick Winkert; Rico Zacher

doi:10.3934/dcdss.2012.5.865

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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

Abstract
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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
[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

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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