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Homogenization of a locally periodic time-dependent domain
Global higher integrability of weak solutions of porous medium systems
1. | Department of Mathematics and Systems Analysis, Aalto University, P. O. Box 11100, FI-00076 Aalto, Finland |
2. | Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany |
3. | Katedra matematické analyźy, Matematicko-fyzikální fakulta Univerzity Karlovy, Sokolovská 83,186 75 Praha 8, Czech Republic |
4. | Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany |
$ \begin{equation*} \partial_t u-\Delta(|u|^{m-1}u) = \mathrm{div}\,F\,, \end{equation*} $ |
$ m>1 $ |
$ D(|u|^{m-1}u) $ |
$ 2 $ |
References:
[1] |
K. Adimurthi and S. Byun,
Boundary Higher Integrability for Very Weak Solutions of Quasilinear Parabolic Equations, J. Math. Pures Appl., 121 (2019), 244-285.
doi: 10.1016/j.matpur.2018.06.005. |
[2] |
V. Bögelein,
Higher integrability for weak solutions of higher order degenerate parabolic systems, Ann. Acad. Sci. Fenn. Math., 33 (2008), 387-412.
|
[3] |
V. Bögelein, F. Duzaar, J. Kinnunen and C. Scheven, Higher integrability for doubly nonlinear parabolic systems, Preprint., |
[4] |
V. Bögelein, F. Duzaar, R. Korte and C. Scheven,
The higher integrability of weak solutions of porous medium systems, Adv. Nonlinear Anal., 8 (2019), 1004-1034.
doi: 10.1515/anona-2017-0270. |
[5] |
V. Bögelein, F. Duzaar and P. Marcellini,
Parabolic systems with $p, q$-growth: a variational approach, Arch. Ration. Mech. Anal., 210 (2013), 219-267.
doi: 10.1007/s00205-013-0646-4. |
[6] |
V. Bögelein, F. Duzaar and C. Scheven, Higher integrability for the singular porous medium system, Preprint, 2018. |
[7] |
V. Bögelein and M. Parviainen,
Self-improving property of nonlinear higher order parabolic systems near the boundary, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 21-54.
doi: 10.1007/s00030-009-0038-5. |
[8] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Universitytext xv, 387, New York, NY, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[9] |
E. DiBenedetto and A. Friedman, Hölder estimates for non-linear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1–22. 1985.
doi: 10.1515/crll.1985.357.1. |
[10] |
E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, 2011.
doi: 10.1007/978-1-4614-1584-8. |
[11] |
L. Diening, P. Kaplický and S. Schwarzacher,
BMO estimates for the $p$-Laplacian, Nonlinear Anal., 75 (2012), 637-650.
doi: 10.1016/j.na.2011.08.065. |
[12] |
L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
doi: 10.1007/978-1-4614-1584-8. |
[13] |
H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969. |
[14] |
F. W. Gehring,
The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277.
doi: 10.1007/BF02392268. |
[15] |
U. Gianazza and S. Schwarzacher,
Self-improving property of degenerate parabolic equations of porous medium-type, Amer. J. Math., 141 (2019), 399-446.
doi: 10.1353/ajm.2019.0009. |
[16] |
U. Gianazza and S. Schwarzacher, Self-improving property of the fast diffusion equation, J. Funct. Anal., 277 (2019), 108291.
doi: 10.1016/j.jfa.2019.108291. |
[17] |
M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, 1983.
![]() ![]() |
[18] |
M. Giaquinta and G. Modica,
Regularity results for some classes of higher order nonlinear elliptic systems, J. Reine Ang. Math., 311/312 (1979), 145-169.
|
[19] |
M. Giaquinta and G. Modica,
Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 185-208.
|
[20] |
M. Giaquinta and M. Struwe,
On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z., 179 (1982), 437-451.
doi: 10.1007/BF01215058. |
[21] |
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Company, Tuck Link, Singapore, 2003.
doi: 10.1142/9789812795557. |
[22] |
P. Hajłasz, P. Koskela and H. Tuominen,
Sobolev embeddings, extensions and measure density condition, J. Funct. Anal., 254 (2008), 1217-1234.
doi: 10.1016/j.jfa.2007.11.020. |
[23] |
L. I. Hedberg,
Two approximation problems in function spaces, Ark. Mat., 16 (1978), 51-81.
doi: 10.1007/BF02385982. |
[24] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Oxford University Press, New York, 1993.
![]() ![]() |
[25] |
T. Kilpeläinen and P. Koskela,
Global integrability of the gradients of solutions to partial differential equations, Nonlinear Anal., 23 (1994), 899-909.
doi: 10.1016/0362-546X(94)90127-9. |
[26] |
J. Kinnunen and J. L. Lewis,
Higher integrability for parabolic systems of $p$-Laplacian type, Duke Math. J., 102 (2000), 253-271.
doi: 10.1215/S0012-7094-00-10223-2. |
[27] |
J. Kinnunen and J. Lewis,
Very weak solutions of parabolic systems of p-Laplacian type, Ark. Mat., 40 (2002), 105-132.
doi: 10.1007/BF02384505. |
[28] |
J. Kinnunen and P. Lindqvist,
Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl., 185 (2006), 411-435.
doi: 10.1007/s10231-005-0160-x. |
[29] |
J. Kinnunen and M. Parviainen,
Stability for degenerate parabolic equations, Adv. Cal. Var., 3 (2010), 29-48.
doi: 10.1515/ACV.2010.002. |
[30] |
J. L. Lewis,
Uniformly fat sets, Trans. Am. Math. Soc., 308 (1988), 177-196.
doi: 10.2307/2000957. |
[31] |
N. G. Meyers and A. Elcrat,
Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J., 42 (1975), 121-136.
|
[32] |
M. Parviainen,
Global gradient estimates for degenerate parabolic equations in nonsmooth domains, Ann. Mat. Pura Appl., 188 (2009), 333-358.
doi: 10.1007/s10231-008-0079-0. |
[33] |
M. Parviainen,
Reverse Hölder inequalities for singular parabolic equations near the boundary, J. Differential Equations, 246 (2009), 512-540.
doi: 10.1016/j.jde.2008.06.013. |
[34] |
S. Schwarzacher,
Hölder-Zygmund estimates for degenerate parabolic systems, J. Differential Equations, 256 (2014), 2423-2448.
doi: 10.1016/j.jde.2014.01.009. |
show all references
References:
[1] |
K. Adimurthi and S. Byun,
Boundary Higher Integrability for Very Weak Solutions of Quasilinear Parabolic Equations, J. Math. Pures Appl., 121 (2019), 244-285.
doi: 10.1016/j.matpur.2018.06.005. |
[2] |
V. Bögelein,
Higher integrability for weak solutions of higher order degenerate parabolic systems, Ann. Acad. Sci. Fenn. Math., 33 (2008), 387-412.
|
[3] |
V. Bögelein, F. Duzaar, J. Kinnunen and C. Scheven, Higher integrability for doubly nonlinear parabolic systems, Preprint., |
[4] |
V. Bögelein, F. Duzaar, R. Korte and C. Scheven,
The higher integrability of weak solutions of porous medium systems, Adv. Nonlinear Anal., 8 (2019), 1004-1034.
doi: 10.1515/anona-2017-0270. |
[5] |
V. Bögelein, F. Duzaar and P. Marcellini,
Parabolic systems with $p, q$-growth: a variational approach, Arch. Ration. Mech. Anal., 210 (2013), 219-267.
doi: 10.1007/s00205-013-0646-4. |
[6] |
V. Bögelein, F. Duzaar and C. Scheven, Higher integrability for the singular porous medium system, Preprint, 2018. |
[7] |
V. Bögelein and M. Parviainen,
Self-improving property of nonlinear higher order parabolic systems near the boundary, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 21-54.
doi: 10.1007/s00030-009-0038-5. |
[8] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Universitytext xv, 387, New York, NY, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[9] |
E. DiBenedetto and A. Friedman, Hölder estimates for non-linear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1–22. 1985.
doi: 10.1515/crll.1985.357.1. |
[10] |
E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, 2011.
doi: 10.1007/978-1-4614-1584-8. |
[11] |
L. Diening, P. Kaplický and S. Schwarzacher,
BMO estimates for the $p$-Laplacian, Nonlinear Anal., 75 (2012), 637-650.
doi: 10.1016/j.na.2011.08.065. |
[12] |
L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
doi: 10.1007/978-1-4614-1584-8. |
[13] |
H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969. |
[14] |
F. W. Gehring,
The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277.
doi: 10.1007/BF02392268. |
[15] |
U. Gianazza and S. Schwarzacher,
Self-improving property of degenerate parabolic equations of porous medium-type, Amer. J. Math., 141 (2019), 399-446.
doi: 10.1353/ajm.2019.0009. |
[16] |
U. Gianazza and S. Schwarzacher, Self-improving property of the fast diffusion equation, J. Funct. Anal., 277 (2019), 108291.
doi: 10.1016/j.jfa.2019.108291. |
[17] |
M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, 1983.
![]() ![]() |
[18] |
M. Giaquinta and G. Modica,
Regularity results for some classes of higher order nonlinear elliptic systems, J. Reine Ang. Math., 311/312 (1979), 145-169.
|
[19] |
M. Giaquinta and G. Modica,
Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 185-208.
|
[20] |
M. Giaquinta and M. Struwe,
On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z., 179 (1982), 437-451.
doi: 10.1007/BF01215058. |
[21] |
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Company, Tuck Link, Singapore, 2003.
doi: 10.1142/9789812795557. |
[22] |
P. Hajłasz, P. Koskela and H. Tuominen,
Sobolev embeddings, extensions and measure density condition, J. Funct. Anal., 254 (2008), 1217-1234.
doi: 10.1016/j.jfa.2007.11.020. |
[23] |
L. I. Hedberg,
Two approximation problems in function spaces, Ark. Mat., 16 (1978), 51-81.
doi: 10.1007/BF02385982. |
[24] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Oxford University Press, New York, 1993.
![]() ![]() |
[25] |
T. Kilpeläinen and P. Koskela,
Global integrability of the gradients of solutions to partial differential equations, Nonlinear Anal., 23 (1994), 899-909.
doi: 10.1016/0362-546X(94)90127-9. |
[26] |
J. Kinnunen and J. L. Lewis,
Higher integrability for parabolic systems of $p$-Laplacian type, Duke Math. J., 102 (2000), 253-271.
doi: 10.1215/S0012-7094-00-10223-2. |
[27] |
J. Kinnunen and J. Lewis,
Very weak solutions of parabolic systems of p-Laplacian type, Ark. Mat., 40 (2002), 105-132.
doi: 10.1007/BF02384505. |
[28] |
J. Kinnunen and P. Lindqvist,
Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl., 185 (2006), 411-435.
doi: 10.1007/s10231-005-0160-x. |
[29] |
J. Kinnunen and M. Parviainen,
Stability for degenerate parabolic equations, Adv. Cal. Var., 3 (2010), 29-48.
doi: 10.1515/ACV.2010.002. |
[30] |
J. L. Lewis,
Uniformly fat sets, Trans. Am. Math. Soc., 308 (1988), 177-196.
doi: 10.2307/2000957. |
[31] |
N. G. Meyers and A. Elcrat,
Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J., 42 (1975), 121-136.
|
[32] |
M. Parviainen,
Global gradient estimates for degenerate parabolic equations in nonsmooth domains, Ann. Mat. Pura Appl., 188 (2009), 333-358.
doi: 10.1007/s10231-008-0079-0. |
[33] |
M. Parviainen,
Reverse Hölder inequalities for singular parabolic equations near the boundary, J. Differential Equations, 246 (2009), 512-540.
doi: 10.1016/j.jde.2008.06.013. |
[34] |
S. Schwarzacher,
Hölder-Zygmund estimates for degenerate parabolic systems, J. Differential Equations, 256 (2014), 2423-2448.
doi: 10.1016/j.jde.2014.01.009. |

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