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Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L∞ tensor coefficient under relaxed ellipticity condition
Pointwise gradient bounds for a class of very singular quasilinear elliptic equations
1. | Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam |
2. | Group of Analysis and Applied Mathematics, Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam |
A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations $ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $ is established via Wolff type potentials. It is worthwhile to note that the model case of $ \mathbb{A} $ here is the non-degenerate $ p $-Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278(5) (2020), 108391] to the very singular case $ 1<p \le \frac{3n-2}{2n-1} $, where the data $ \mu $ on right-hand side is assumed belonging to some classes that close to $ L^1 $. Moreover, a global pointwise estimate for gradient of weak solutions to such problem is also obtained under the additional assumption that $ \Omega $ is sufficiently flat in the Reifenberg sense.
References:
[1] |
E. Acerbi and N. Fusco,
Regularity for minimizers of non-quadratic functionals: The case 1 < p < 2, J. Math. Anal. Appl., 140 (1989), 115-135.
doi: 10.1016/0022-247X(89)90098-X. |
[2] |
B. Avelin, T. Kuusi and G. Mingione,
Nonlinear Calderón-Zygmund theory in the limiting case, Arch. Rational. Mech. Anal., 227 (2018), 663-714.
doi: 10.1007/s00205-017-1171-7. |
[3] |
P. Baroni,
Lorentz estimates for degenerate and singular evolutionary systems, J. Differential Equations, 255 (2013), 2927-2951.
doi: 10.1016/j.jde.2013.07.024. |
[4] |
P. Baroni and J. Habermann,
Elliptic interpolation estimates for non-standard growth operators, Ann. Acad. Sci. Fenn. Math., 39 (2014), 119-162.
doi: 10.5186/aasfm.2014.3915. |
[5] |
P. Benilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez,
An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV), 22 (1995), 241-273.
|
[6] |
V. Bögelein and J. Habermann,
Gradient estimates via non standard potentials and continuity, Ann. Acad. Sci. Fenn. Math., 35 (2010), 641-678.
doi: 10.5186/aasfm.2010.3541. |
[7] |
S.-S. Byun and L. Wang,
$L^p$-estimates for general nonlinear elliptic equations, Indiana Univ. Math. J., 56 (2007), 3193-3221.
doi: 10.1512/iumj.2007.56.3034. |
[8] |
S.-S. Byun and L. Wang,
Elliptic equations with BMO nonlinearity in Reifenberg domains, Adv. Math., 219 (2008), 1937-1971.
doi: 10.1016/j.aim.2008.07.016. |
[9] |
S.-S. Byun and Y. Youn,
Optimal gradient estimates via Riesz potentials for $p(\cdot)$-Laplacian type equations, Quart. J. Math., 68 (2017), 1071-1115.
doi: 10.1093/qmath/hax013. |
[10] |
A. Cianchi and S. Schwarzacher,
Potential estimates for the $p$-Laplace system with data in divergence form, J. Differential Equations, 265 (2018), 478-499.
doi: 10.1016/j.jde.2018.02.038. |
[11] |
U. Dini,
Sur la méthode des approximations successives pour les équations aux dérivées partielles du deuxième ordre, Acta Math., 25 (1902), 185-230.
doi: 10.1007/BF02419026. |
[12] |
F. Duzaar and G. Mingione,
Gradient estimates via linear and nonlinear potentials, J. Funct. Anal., 259 (2010), 2961-2998.
doi: 10.1016/j.jfa.2010.08.006. |
[13] |
F. Duzaar and G. Mingione,
Gradient continuity estimates, Calc. Var. and PDE, 39 (2010), 379-418.
doi: 10.1007/s00526-010-0314-6. |
[14] |
F. Duzaar and G. Mingione,
Gradient estimates via non-linear potentials, Amer. J. Math., 133 (2011), 1093-1149.
doi: 10.1353/ajm.2011.0023. |
[15] |
F. W. Gehring,
The $L^p$ integrability of partial derivatives of a quasiconformal mapping, Bull. Amer. Math. Soc., 79 (1973), 465-466.
doi: 10.1090/S0002-9904-1973-13218-5. |
[16] |
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, 2003.
doi: 10.1142/9789812795557. |
[17] |
Q. Han and F. Lin, Elliptic Partial Differential Equations, 2nd edn. American Mathematical Society, Providence, RI, 2011. |
[18] |
C. Kenig and T. Toro, Poisson kernel characterization of Reifenberg flat chord arc domains, Ann. Sci. Éc. Norm. Supér., 36 (2003), 323–401.
doi: 10.1016/S0012-9593(03)00012-0. |
[19] |
T. Kilpeläinen and J. Malý,
Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV), 19 (1992), 591-613.
|
[20] |
T. Kilpeläinen and J. Malý,
The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161.
doi: 10.1007/BF02392793. |
[21] |
R. Korte and T. Kuusi,
A note on the Wolff potential estimate for solutions to elliptic equations involving measures, Adv. Calc. Var., 3 (2010), 99-113.
doi: 10.1515/ACV.2010.005. |
[22] |
T. Kuusi and G. Mingione,
Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.
doi: 10.1016/j.jfa.2012.02.018. |
[23] |
T. Kuusi and G. Mingione,
Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal., 207 (2013), 215-246.
doi: 10.1007/s00205-012-0562-z. |
[24] |
T. Kuusi and G. Mingione,
The Wolff gradient bound for degenerate parabolic equations, J. Eur. Math. Soc., 16 (2014), 835-892.
doi: 10.4171/JEMS/449. |
[25] |
T. Kuusi and G. Mingione,
Riesz potentials and nonlinear parabolic equations, Arch. Ration. Mech. Anal., 212 (2014), 727-780.
doi: 10.1007/s00205-013-0695-8. |
[26] |
T. Kuusi and G. Mingione,
Guide to nonlinear potential estimates, Bull. Math. Sci., 4 (2014), 1-82.
doi: 10.1007/s13373-013-0048-9. |
[27] |
T. Kuusi and G. Mingione,
Vectorial nonlinear potential theory, J. Eur. Math. Soc., 20 (2018), 929-1004.
doi: 10.4171/JEMS/780. |
[28] |
D. Labutin,
Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49.
doi: 10.1215/S0012-7094-02-11111-9. |
[29] |
A. Lemenant, E. Milakis and L. V. Spinolo,
On the extension property of reifenberg flat domains, Ann. Acad. Sci. Fenn. Math., 39 (2014), 51-71.
doi: 10.5186/aasfm.2014.3907. |
[30] |
G. Lieberman,
Higher regularity for nonlinear oblique derivative problems in Lipschitz domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (V), 1 (2002), 111-152.
|
[31] |
P. Lindqvist, Notes on the $p$-Laplace equation, Univ. Jyväskylä, Report, 102 (2006). |
[32] |
T. Mengesha and N. C. Phuc,
Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations, 250 (2011), 1485-2507.
doi: 10.1016/j.jde.2010.11.009. |
[33] |
G. Mingione,
The Calderón-Zygmund theory for elliptic problems with measure data, Ann. Scuola. Norm. Super. Pisa Cl. Sci. (V), 6 (2007), 195-261.
|
[34] |
G. Mingione,
Gradient estimates below the duality exponent, Math. Ann., 346 (2010), 571-627.
doi: 10.1007/s00208-009-0411-z. |
[35] |
G. Mingione,
Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459-486.
doi: 10.4171/JEMS/258. |
[36] |
Q.-H. Nguyen, Gradient estimates for singular quasilinear elliptic equations with measure data, preprint, arXiv: 1705.07440. |
[37] |
Q.-H. Nguyen and N. C. Phuc,
Good-$\lambda$ and Muckenhoupt-Wheeden type bounds, with applications to quasilinear elliptic equations with gradient power source terms and measure data, Math. Ann., 374 (2019), 67-98.
doi: 10.1007/s00208-018-1744-2. |
[38] |
Q.-H. Nguyen and N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal., 278 (2020), 108391, 35pp.
doi: 10.1016/j.jfa.2019.108391. |
[39] |
Q.-H. Nguyen and N. C. Phuc, Existence and regularity estimates for quasilinear equations with measure data: The case $1 < p \le \frac{3n-2}{2n-1}$, preprint, arXiv: 2003.03725. |
[40] |
T.-N. Nguyen and M.-P. Tran, Lorentz improving estimates for the $p$-Laplace equations with mixed data, Nonlinear Anal., 200 (2020), 111960, 23pp.
doi: 10.1016/j.na.2020.111960. |
[41] |
T.-N. Nguyen and M.-P. Tran, Level-set inequalities on fractional maximal distribution functions and applications to regularity theory, J. Funct. Anal., 280 (2021), 108797, 47pp.
doi: 10.1016/j.jfa.2020.108797. |
[42] |
E. Reifenberg,
Solutions of the plateau problem for $m$-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92.
doi: 10.1007/BF02547186. |
[43] |
M. E. Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81 (2000), Providence, RI: American Mathematical Society.
doi: 10.1090/surv/081. |
[44] |
M.-P. Tran,
Good-$\lambda$ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal., 178 (2019), 266-281.
doi: 10.1016/j.na.2018.08.001. |
[45] |
M.-P. Tran and T.-N. Nguyen, Lorentz-Morrey global bounds for singular quasilinear elliptic equations with measure data, Commun. Contemp. Math., 22 (2020), 1950033, 30pp.
doi: 10.1142/S0219199719500330. |
[46] |
M.-P. Tran and T.-N. Nguyen,
New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data, J. Differential Equations, 268 (2020), 1427-1462.
doi: 10.1016/j.jde.2019.08.052. |
[47] |
M.-P. Tran and T.-N. Nguyen, Global gradient estimates for very singular nonlinear elliptic equations with measure data, preprint, arXiv: 1909.06991. |
[48] |
N. S. Trudinger and X. J. Wang,
On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math., 124 (2002), 369-410.
doi: 10.1353/ajm.2002.0012. |
[49] |
N. S. Trudinger and X. J. Wang,
Quasilinear elliptic equations with signed measure data, Disc. Cont. Dyn. Systems A, 23 (2009), 477-494.
doi: 10.3934/dcds.2009.23.477. |
show all references
References:
[1] |
E. Acerbi and N. Fusco,
Regularity for minimizers of non-quadratic functionals: The case 1 < p < 2, J. Math. Anal. Appl., 140 (1989), 115-135.
doi: 10.1016/0022-247X(89)90098-X. |
[2] |
B. Avelin, T. Kuusi and G. Mingione,
Nonlinear Calderón-Zygmund theory in the limiting case, Arch. Rational. Mech. Anal., 227 (2018), 663-714.
doi: 10.1007/s00205-017-1171-7. |
[3] |
P. Baroni,
Lorentz estimates for degenerate and singular evolutionary systems, J. Differential Equations, 255 (2013), 2927-2951.
doi: 10.1016/j.jde.2013.07.024. |
[4] |
P. Baroni and J. Habermann,
Elliptic interpolation estimates for non-standard growth operators, Ann. Acad. Sci. Fenn. Math., 39 (2014), 119-162.
doi: 10.5186/aasfm.2014.3915. |
[5] |
P. Benilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez,
An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV), 22 (1995), 241-273.
|
[6] |
V. Bögelein and J. Habermann,
Gradient estimates via non standard potentials and continuity, Ann. Acad. Sci. Fenn. Math., 35 (2010), 641-678.
doi: 10.5186/aasfm.2010.3541. |
[7] |
S.-S. Byun and L. Wang,
$L^p$-estimates for general nonlinear elliptic equations, Indiana Univ. Math. J., 56 (2007), 3193-3221.
doi: 10.1512/iumj.2007.56.3034. |
[8] |
S.-S. Byun and L. Wang,
Elliptic equations with BMO nonlinearity in Reifenberg domains, Adv. Math., 219 (2008), 1937-1971.
doi: 10.1016/j.aim.2008.07.016. |
[9] |
S.-S. Byun and Y. Youn,
Optimal gradient estimates via Riesz potentials for $p(\cdot)$-Laplacian type equations, Quart. J. Math., 68 (2017), 1071-1115.
doi: 10.1093/qmath/hax013. |
[10] |
A. Cianchi and S. Schwarzacher,
Potential estimates for the $p$-Laplace system with data in divergence form, J. Differential Equations, 265 (2018), 478-499.
doi: 10.1016/j.jde.2018.02.038. |
[11] |
U. Dini,
Sur la méthode des approximations successives pour les équations aux dérivées partielles du deuxième ordre, Acta Math., 25 (1902), 185-230.
doi: 10.1007/BF02419026. |
[12] |
F. Duzaar and G. Mingione,
Gradient estimates via linear and nonlinear potentials, J. Funct. Anal., 259 (2010), 2961-2998.
doi: 10.1016/j.jfa.2010.08.006. |
[13] |
F. Duzaar and G. Mingione,
Gradient continuity estimates, Calc. Var. and PDE, 39 (2010), 379-418.
doi: 10.1007/s00526-010-0314-6. |
[14] |
F. Duzaar and G. Mingione,
Gradient estimates via non-linear potentials, Amer. J. Math., 133 (2011), 1093-1149.
doi: 10.1353/ajm.2011.0023. |
[15] |
F. W. Gehring,
The $L^p$ integrability of partial derivatives of a quasiconformal mapping, Bull. Amer. Math. Soc., 79 (1973), 465-466.
doi: 10.1090/S0002-9904-1973-13218-5. |
[16] |
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, 2003.
doi: 10.1142/9789812795557. |
[17] |
Q. Han and F. Lin, Elliptic Partial Differential Equations, 2nd edn. American Mathematical Society, Providence, RI, 2011. |
[18] |
C. Kenig and T. Toro, Poisson kernel characterization of Reifenberg flat chord arc domains, Ann. Sci. Éc. Norm. Supér., 36 (2003), 323–401.
doi: 10.1016/S0012-9593(03)00012-0. |
[19] |
T. Kilpeläinen and J. Malý,
Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV), 19 (1992), 591-613.
|
[20] |
T. Kilpeläinen and J. Malý,
The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161.
doi: 10.1007/BF02392793. |
[21] |
R. Korte and T. Kuusi,
A note on the Wolff potential estimate for solutions to elliptic equations involving measures, Adv. Calc. Var., 3 (2010), 99-113.
doi: 10.1515/ACV.2010.005. |
[22] |
T. Kuusi and G. Mingione,
Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.
doi: 10.1016/j.jfa.2012.02.018. |
[23] |
T. Kuusi and G. Mingione,
Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal., 207 (2013), 215-246.
doi: 10.1007/s00205-012-0562-z. |
[24] |
T. Kuusi and G. Mingione,
The Wolff gradient bound for degenerate parabolic equations, J. Eur. Math. Soc., 16 (2014), 835-892.
doi: 10.4171/JEMS/449. |
[25] |
T. Kuusi and G. Mingione,
Riesz potentials and nonlinear parabolic equations, Arch. Ration. Mech. Anal., 212 (2014), 727-780.
doi: 10.1007/s00205-013-0695-8. |
[26] |
T. Kuusi and G. Mingione,
Guide to nonlinear potential estimates, Bull. Math. Sci., 4 (2014), 1-82.
doi: 10.1007/s13373-013-0048-9. |
[27] |
T. Kuusi and G. Mingione,
Vectorial nonlinear potential theory, J. Eur. Math. Soc., 20 (2018), 929-1004.
doi: 10.4171/JEMS/780. |
[28] |
D. Labutin,
Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49.
doi: 10.1215/S0012-7094-02-11111-9. |
[29] |
A. Lemenant, E. Milakis and L. V. Spinolo,
On the extension property of reifenberg flat domains, Ann. Acad. Sci. Fenn. Math., 39 (2014), 51-71.
doi: 10.5186/aasfm.2014.3907. |
[30] |
G. Lieberman,
Higher regularity for nonlinear oblique derivative problems in Lipschitz domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (V), 1 (2002), 111-152.
|
[31] |
P. Lindqvist, Notes on the $p$-Laplace equation, Univ. Jyväskylä, Report, 102 (2006). |
[32] |
T. Mengesha and N. C. Phuc,
Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations, 250 (2011), 1485-2507.
doi: 10.1016/j.jde.2010.11.009. |
[33] |
G. Mingione,
The Calderón-Zygmund theory for elliptic problems with measure data, Ann. Scuola. Norm. Super. Pisa Cl. Sci. (V), 6 (2007), 195-261.
|
[34] |
G. Mingione,
Gradient estimates below the duality exponent, Math. Ann., 346 (2010), 571-627.
doi: 10.1007/s00208-009-0411-z. |
[35] |
G. Mingione,
Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459-486.
doi: 10.4171/JEMS/258. |
[36] |
Q.-H. Nguyen, Gradient estimates for singular quasilinear elliptic equations with measure data, preprint, arXiv: 1705.07440. |
[37] |
Q.-H. Nguyen and N. C. Phuc,
Good-$\lambda$ and Muckenhoupt-Wheeden type bounds, with applications to quasilinear elliptic equations with gradient power source terms and measure data, Math. Ann., 374 (2019), 67-98.
doi: 10.1007/s00208-018-1744-2. |
[38] |
Q.-H. Nguyen and N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal., 278 (2020), 108391, 35pp.
doi: 10.1016/j.jfa.2019.108391. |
[39] |
Q.-H. Nguyen and N. C. Phuc, Existence and regularity estimates for quasilinear equations with measure data: The case $1 < p \le \frac{3n-2}{2n-1}$, preprint, arXiv: 2003.03725. |
[40] |
T.-N. Nguyen and M.-P. Tran, Lorentz improving estimates for the $p$-Laplace equations with mixed data, Nonlinear Anal., 200 (2020), 111960, 23pp.
doi: 10.1016/j.na.2020.111960. |
[41] |
T.-N. Nguyen and M.-P. Tran, Level-set inequalities on fractional maximal distribution functions and applications to regularity theory, J. Funct. Anal., 280 (2021), 108797, 47pp.
doi: 10.1016/j.jfa.2020.108797. |
[42] |
E. Reifenberg,
Solutions of the plateau problem for $m$-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92.
doi: 10.1007/BF02547186. |
[43] |
M. E. Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81 (2000), Providence, RI: American Mathematical Society.
doi: 10.1090/surv/081. |
[44] |
M.-P. Tran,
Good-$\lambda$ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal., 178 (2019), 266-281.
doi: 10.1016/j.na.2018.08.001. |
[45] |
M.-P. Tran and T.-N. Nguyen, Lorentz-Morrey global bounds for singular quasilinear elliptic equations with measure data, Commun. Contemp. Math., 22 (2020), 1950033, 30pp.
doi: 10.1142/S0219199719500330. |
[46] |
M.-P. Tran and T.-N. Nguyen,
New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data, J. Differential Equations, 268 (2020), 1427-1462.
doi: 10.1016/j.jde.2019.08.052. |
[47] |
M.-P. Tran and T.-N. Nguyen, Global gradient estimates for very singular nonlinear elliptic equations with measure data, preprint, arXiv: 1909.06991. |
[48] |
N. S. Trudinger and X. J. Wang,
On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math., 124 (2002), 369-410.
doi: 10.1353/ajm.2002.0012. |
[49] |
N. S. Trudinger and X. J. Wang,
Quasilinear elliptic equations with signed measure data, Disc. Cont. Dyn. Systems A, 23 (2009), 477-494.
doi: 10.3934/dcds.2009.23.477. |
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