American Institute of Mathematical Sciences

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doi: 10.3934/dcds.2021043

Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

Minh-Phuong Tran 1, and  Thanh-Nhan Nguyen 2,,

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

* Corresponding author: Thanh-Nhan Nguyen

Received  October 2020 Published  March 2021

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.

Keywords: Singular quasilinear elliptic equations, measure data, pointwise gradient estimates, Wolff's potential.
Mathematics Subject Classification: Primary: 35J62; 35J75; 35J92.
Citation: Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021043
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

F. Duzaar and G. Mingione, Gradient continuity estimates, Calc. Var. and PDE, 39 (2010), 379-418.  doi: 10.1007/s00526-010-0314-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, 2003. doi: 10.1142/9789812795557.  Google Scholar

[17]

Q. Han and F. Lin, Elliptic Partial Differential Equations, 2nd edn. American Mathematical Society, Providence, RI, 2011.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.  doi: 10.1016/j.jfa.2012.02.018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

T. Kuusi and G. Mingione, Vectorial nonlinear potential theory, J. Eur. Math. Soc., 20 (2018), 929-1004.  doi: 10.4171/JEMS/780.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[31]

P. Lindqvist, Notes on the $p$-Laplace equation, Univ. Jyväskylä, Report, 102 (2006). Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[34]

G. Mingione, Gradient estimates below the duality exponent, Math. Ann., 346 (2010), 571-627.  doi: 10.1007/s00208-009-0411-z.  Google Scholar

[35]

G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459-486.  doi: 10.4171/JEMS/258.  Google Scholar

[36]

Q.-H. Nguyen, Gradient estimates for singular quasilinear elliptic equations with measure data, preprint, arXiv: 1705.07440. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

M.-P. Tran and T.-N. Nguyen, Global gradient estimates for very singular nonlinear elliptic equations with measure data, preprint, arXiv: 1909.06991. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

F. Duzaar and G. Mingione, Gradient continuity estimates, Calc. Var. and PDE, 39 (2010), 379-418.  doi: 10.1007/s00526-010-0314-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, 2003. doi: 10.1142/9789812795557.  Google Scholar

[17]

Q. Han and F. Lin, Elliptic Partial Differential Equations, 2nd edn. American Mathematical Society, Providence, RI, 2011.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.  doi: 10.1016/j.jfa.2012.02.018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

T. Kuusi and G. Mingione, Vectorial nonlinear potential theory, J. Eur. Math. Soc., 20 (2018), 929-1004.  doi: 10.4171/JEMS/780.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[31]

P. Lindqvist, Notes on the $p$-Laplace equation, Univ. Jyväskylä, Report, 102 (2006). Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[34]

G. Mingione, Gradient estimates below the duality exponent, Math. Ann., 346 (2010), 571-627.  doi: 10.1007/s00208-009-0411-z.  Google Scholar

[35]

G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459-486.  doi: 10.4171/JEMS/258.  Google Scholar

[36]

Q.-H. Nguyen, Gradient estimates for singular quasilinear elliptic equations with measure data, preprint, arXiv: 1705.07440. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

M.-P. Tran and T.-N. Nguyen, Global gradient estimates for very singular nonlinear elliptic equations with measure data, preprint, arXiv: 1909.06991. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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