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

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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

Abstract
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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

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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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