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September  2011, 16(2): 669-685. doi: 10.3934/dcdsb.2011.16.669

Optimal regularity for $A$-harmonic type equations under the natural growth

Shenzhou Zheng 1, , Xueliang Zheng 2, and  Zhaosheng Feng 3,

1. 

Department of Mathematics, Beijing Jiaotong University, Beijing 100044
2. 

Department of Mathematics, Taizhou University, Linhai, Zhejiang 317000, China
3. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539

Received  January 2010 Revised  October 2010 Published  June 2011

In this paper we are concerned with a class of nonlinear degenerate elliptic equations under the natural growth. We show that each bounded weak solution of $A$-harmonic type equations under the natural growth belongs to local Hölder continuity based on a density lemma and the Moser-Nash's argument. Then we show that its weak solution is of optimal regularity with the Hölder exponent for any $\gamma$: $0\le \gamma<\kappa$, where $\kappa$ is the same as the Hölder's index for homogeneous $A$-harmonic equations.
Keywords: degenerate elliptic equation, weak solution, Interior Hölder continuity, Moser-Nash's argument, Hölder's index., A-harmonic equation.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35B45, 35D3.
Citation: Shenzhou Zheng, Xueliang Zheng, Zhaosheng Feng. Optimal regularity for $A$-harmonic type equations under the natural growth. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 669-685. doi: 10.3934/dcdsb.2011.16.669
References:
[1]

E. Acerbi and N. Fusco, Regularity of minimizers of non-quadratic functionals: the case 1$<$p$<$2,, J. Math. Anal. Appl., 140 (1989), 115.   Google Scholar

[2]

L. Budney and T. Iwaniec, Removability of singularities of $A$-harmonic functions,, Differential and Integral Equations, 12 (1999), 261.   Google Scholar

[3]

E. Dibenedetto, $C$$1+\alpha$ Local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[4]

S. Ding and D. Sylvester, Weak reverse Hölder inequalities and imbedding inequalities for solutions to the $A$-harmonic equation,, Nonlinear Anal., 51 (2002), 783.  doi: 10.1016/S0362-546X(01)00862-8.  Google Scholar

[5]

L. D'Onofrio and T. Iwaniec, The p-harmonic transform beyond its natural domain of definition,, Indiana Univ. Math. J., 53 (2004), 683.  doi: 10.1512/iumj.2004.53.2462.  Google Scholar

[6]

Z. Feng, S. Zheng and H. Lu, Green's function of nonlinear degenerate elliptic operators and its application to regularity,, Differential and Integral Equations, 21 (2008), 717.   Google Scholar

[7]

Z. Feng and Q. G. Meng, Exact solution for a two-dimensional kdv-burgers-type equation with nonlinear terms of any order,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 397.   Google Scholar

[8]

Z. Feng and G. Chen, Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth,, Discrete Contin. Dyn. Syst., 24 (2009), 763.  doi: 10.3934/dcds.2009.24.763.  Google Scholar

[9]

N. Fusco and J. Hutchinson, Partial regularity for minimizers of certain functionals having nonquadratic growth,, Ann. Mat. Pura. Appl., 155 (1989), 1.  doi: 10.1007/BF01765932.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Spinger-Verlag, (2001).   Google Scholar

[11]

M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,", Annals of Mathematics Studies, (1983).   Google Scholar

[12]

M. Giaquinta and G. Modica, Remark on the regularity of the minimizers of certain degenerate functionals,, Manuscripta Math., 57 (1986), 55.  doi: 10.1007/BF01172492.  Google Scholar

[13]

J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations,", Oxford University Press, (1993).   Google Scholar

[14]

J. Heinonen and T. Kilpeläinen, A-superharmonic functions and supersolutions of degenerate elliptic equations,, Ark. Mat., 26 (1988), 87.  doi: 10.1007/BF02386110.  Google Scholar

[15]

Q. Han and F. H. Lin, "Elliptic Partial Differential Equations,", American Mathematical Society, (1997).   Google Scholar

[16]

R. Hardt, F. H. Lin and L. Mou, Strong convergence of p-harmonic mappings, Longman Scientific and Technical,, Pitman Res. Notes Math. Ser. Harlow, 314 (1994), 58.   Google Scholar

[17]

T. Iwaniec and C. Sbordone, Weak minima of variational integrals,, J. Reine Angew Math., 454 (1994), 143.  doi: 10.1515/crll.1994.454.143.  Google Scholar

[18]

T. Kilpeläinen, p-Laplacian type equations involving measures,, Proceedings of the International Congress of Mathematicians, (2002), 167.   Google Scholar

[19]

T. Kilpeläinen and J. Malý, The wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137.  doi: 10.1007/BF02392793.  Google Scholar

[20]

P. Koskela and O. Martio, Removability theorems for solution of degenerate elliptic PDEs,, Ark. Mat., 31 (1993), 339.  doi: 10.1007/BF02559490.  Google Scholar

[21]

P. Lindqvist and O. Martio, Two theorems of N.Wiener for solutions of quasilinear elliptic equations,, Acta Math., 155 (1985), 153.  doi: 10.1007/BF02392541.  Google Scholar

[22]

Q. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations" (in chinese),, Science Press House, (1985).   Google Scholar

[23]

C. A. Nolder, Hardy-Littlewood theorems for $A$-harmonic tensors,, Illinois J. Math., 43 (1999), 613.   Google Scholar

[24]

Y. G. Reshetnyak, "Space Mappings with Bounded Distortion,", Amer. Math. Soc. (Translation of Math Monographs), 73 (1989).   Google Scholar

[25]

J. Serrin and H. H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Math., 189 (2002), 79.  doi: 10.1007/BF02392645.  Google Scholar

[26]

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.  doi: 10.1353/ajm.2002.0012.  Google Scholar

[27]

K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems,, Acta Math., 138 (1977), 219.  doi: 10.1007/BF02392316.  Google Scholar

[28]

S. Zheng, Regualrity results for the generalized Beltrami systems,, Acta Math. Sinica, 20 (2004), 193.  doi: 10.1007/s10114-003-0250-x.  Google Scholar

[29]

S. Zheng, Removable singularities of solutions of $A$-harmonic type equations,, Acta Math. Appl. Sinica, 20 (2004), 115.  doi: 10.1007/s10255-004-0154-2.  Google Scholar

[30]

S. Zheng, X. Zheng and Z. Feng, Regularity for a class of degenerate elliptic equations with discontinuous coefficients under natural growth,, J. Math. Anal. Appl., 346 (2008), 359.  doi: 10.1016/j.jmaa.2008.05.059.  Google Scholar

show all references

References:
[1]

E. Acerbi and N. Fusco, Regularity of minimizers of non-quadratic functionals: the case 1$<$p$<$2,, J. Math. Anal. Appl., 140 (1989), 115.   Google Scholar

[2]

L. Budney and T. Iwaniec, Removability of singularities of $A$-harmonic functions,, Differential and Integral Equations, 12 (1999), 261.   Google Scholar

[3]

E. Dibenedetto, $C$$1+\alpha$ Local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[4]

S. Ding and D. Sylvester, Weak reverse Hölder inequalities and imbedding inequalities for solutions to the $A$-harmonic equation,, Nonlinear Anal., 51 (2002), 783.  doi: 10.1016/S0362-546X(01)00862-8.  Google Scholar

[5]

L. D'Onofrio and T. Iwaniec, The p-harmonic transform beyond its natural domain of definition,, Indiana Univ. Math. J., 53 (2004), 683.  doi: 10.1512/iumj.2004.53.2462.  Google Scholar

[6]

Z. Feng, S. Zheng and H. Lu, Green's function of nonlinear degenerate elliptic operators and its application to regularity,, Differential and Integral Equations, 21 (2008), 717.   Google Scholar

[7]

Z. Feng and Q. G. Meng, Exact solution for a two-dimensional kdv-burgers-type equation with nonlinear terms of any order,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 397.   Google Scholar

[8]

Z. Feng and G. Chen, Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth,, Discrete Contin. Dyn. Syst., 24 (2009), 763.  doi: 10.3934/dcds.2009.24.763.  Google Scholar

[9]

N. Fusco and J. Hutchinson, Partial regularity for minimizers of certain functionals having nonquadratic growth,, Ann. Mat. Pura. Appl., 155 (1989), 1.  doi: 10.1007/BF01765932.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Spinger-Verlag, (2001).   Google Scholar

[11]

M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,", Annals of Mathematics Studies, (1983).   Google Scholar

[12]

M. Giaquinta and G. Modica, Remark on the regularity of the minimizers of certain degenerate functionals,, Manuscripta Math., 57 (1986), 55.  doi: 10.1007/BF01172492.  Google Scholar

[13]

J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations,", Oxford University Press, (1993).   Google Scholar

[14]

J. Heinonen and T. Kilpeläinen, A-superharmonic functions and supersolutions of degenerate elliptic equations,, Ark. Mat., 26 (1988), 87.  doi: 10.1007/BF02386110.  Google Scholar

[15]

Q. Han and F. H. Lin, "Elliptic Partial Differential Equations,", American Mathematical Society, (1997).   Google Scholar

[16]

R. Hardt, F. H. Lin and L. Mou, Strong convergence of p-harmonic mappings, Longman Scientific and Technical,, Pitman Res. Notes Math. Ser. Harlow, 314 (1994), 58.   Google Scholar

[17]

T. Iwaniec and C. Sbordone, Weak minima of variational integrals,, J. Reine Angew Math., 454 (1994), 143.  doi: 10.1515/crll.1994.454.143.  Google Scholar

[18]

T. Kilpeläinen, p-Laplacian type equations involving measures,, Proceedings of the International Congress of Mathematicians, (2002), 167.   Google Scholar

[19]

T. Kilpeläinen and J. Malý, The wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137.  doi: 10.1007/BF02392793.  Google Scholar

[20]

P. Koskela and O. Martio, Removability theorems for solution of degenerate elliptic PDEs,, Ark. Mat., 31 (1993), 339.  doi: 10.1007/BF02559490.  Google Scholar

[21]

P. Lindqvist and O. Martio, Two theorems of N.Wiener for solutions of quasilinear elliptic equations,, Acta Math., 155 (1985), 153.  doi: 10.1007/BF02392541.  Google Scholar

[22]

Q. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations" (in chinese),, Science Press House, (1985).   Google Scholar

[23]

C. A. Nolder, Hardy-Littlewood theorems for $A$-harmonic tensors,, Illinois J. Math., 43 (1999), 613.   Google Scholar

[24]

Y. G. Reshetnyak, "Space Mappings with Bounded Distortion,", Amer. Math. Soc. (Translation of Math Monographs), 73 (1989).   Google Scholar

[25]

J. Serrin and H. H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Math., 189 (2002), 79.  doi: 10.1007/BF02392645.  Google Scholar

[26]

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.  doi: 10.1353/ajm.2002.0012.  Google Scholar

[27]

K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems,, Acta Math., 138 (1977), 219.  doi: 10.1007/BF02392316.  Google Scholar

[28]

S. Zheng, Regualrity results for the generalized Beltrami systems,, Acta Math. Sinica, 20 (2004), 193.  doi: 10.1007/s10114-003-0250-x.  Google Scholar

[29]

S. Zheng, Removable singularities of solutions of $A$-harmonic type equations,, Acta Math. Appl. Sinica, 20 (2004), 115.  doi: 10.1007/s10255-004-0154-2.  Google Scholar

[30]

S. Zheng, X. Zheng and Z. Feng, Regularity for a class of degenerate elliptic equations with discontinuous coefficients under natural growth,, J. Math. Anal. Appl., 346 (2008), 359.  doi: 10.1016/j.jmaa.2008.05.059.  Google Scholar

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