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

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

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.
Citation: Shenzhou Zheng, Xueliang Zheng, Zhaosheng Feng. Optimal regularity for $A$-harmonic type equations under the natural growth. Discrete and 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), 115-135.

[2]

L. Budney and T. Iwaniec, Removability of singularities of $A$-harmonic functions, Differential and Integral Equations, 12 (1999), 261-274.

[3]

E. Dibenedetto, $C$$1+\alpha$ Local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5.

[4]

S. Ding and D. Sylvester, Weak reverse Hölder inequalities and imbedding inequalities for solutions to the $A$-harmonic equation, Nonlinear Anal., 51 (2002), 783-800. doi: 10.1016/S0362-546X(01)00862-8.

[5]

L. D'Onofrio and T. Iwaniec, The p-harmonic transform beyond its natural domain of definition, Indiana Univ. Math. J., 53 (2004), 683-718. doi: 10.1512/iumj.2004.53.2462.

[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), 717-741.

[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), 397-413.

[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), 763-780. doi: 10.3934/dcds.2009.24.763.

[9]

N. Fusco and J. Hutchinson, Partial regularity for minimizers of certain functionals having nonquadratic growth, Ann. Mat. Pura. Appl., 155 (1989), 1-24. doi: 10.1007/BF01765932.

[10]

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

[11]

M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Annals of Mathematics Studies, 105, Princeton Univ. Press, 1983.

[12]

M. Giaquinta and G. Modica, Remark on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math., 57 (1986), 55-99. doi: 10.1007/BF01172492.

[13]

J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations," Oxford University Press, New York, 1993.

[14]

J. Heinonen and T. Kilpeläinen, A-superharmonic functions and supersolutions of degenerate elliptic equations, Ark. Mat., 26 (1988), 87-105. doi: 10.1007/BF02386110.

[15]

Q. Han and F. H. Lin, "Elliptic Partial Differential Equations," American Mathematical Society, Providence, Rhode Island, 1997.

[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), 58-64.

[17]

T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew Math., 454 (1994), 143-161. doi: 10.1515/crll.1994.454.143.

[18]

T. Kilpeläinen, p-Laplacian type equations involving measures, Proceedings of the International Congress of Mathematicians, 167-176, Vol. III, Higher Ed. Press, Beijing, 2002.

[19]

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.

[20]

P. Koskela and O. Martio, Removability theorems for solution of degenerate elliptic PDEs, Ark. Mat., 31 (1993), 339-353. doi: 10.1007/BF02559490.

[21]

P. Lindqvist and O. Martio, Two theorems of N.Wiener for solutions of quasilinear elliptic equations, Acta Math., 155 (1985), 153-171. doi: 10.1007/BF02392541.

[22]

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

[23]

C. A. Nolder, Hardy-Littlewood theorems for $A$-harmonic tensors, Illinois J. Math., 43 (1999), 613-632.

[24]

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

[25]

J. Serrin and H. H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.

[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), 369-410. doi: 10.1353/ajm.2002.0012.

[27]

K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.

[28]

S. Zheng, Regualrity results for the generalized Beltrami systems, Acta Math. Sinica, 20 (2004), 193-205. doi: 10.1007/s10114-003-0250-x.

[29]

S. Zheng, Removable singularities of solutions of $A$-harmonic type equations, Acta Math. Appl. Sinica, 20 (2004), 115-122. doi: 10.1007/s10255-004-0154-2.

[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), 359-373. doi: 10.1016/j.jmaa.2008.05.059.

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), 115-135.

[2]

L. Budney and T. Iwaniec, Removability of singularities of $A$-harmonic functions, Differential and Integral Equations, 12 (1999), 261-274.

[3]

E. Dibenedetto, $C$$1+\alpha$ Local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5.

[4]

S. Ding and D. Sylvester, Weak reverse Hölder inequalities and imbedding inequalities for solutions to the $A$-harmonic equation, Nonlinear Anal., 51 (2002), 783-800. doi: 10.1016/S0362-546X(01)00862-8.

[5]

L. D'Onofrio and T. Iwaniec, The p-harmonic transform beyond its natural domain of definition, Indiana Univ. Math. J., 53 (2004), 683-718. doi: 10.1512/iumj.2004.53.2462.

[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), 717-741.

[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), 397-413.

[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), 763-780. doi: 10.3934/dcds.2009.24.763.

[9]

N. Fusco and J. Hutchinson, Partial regularity for minimizers of certain functionals having nonquadratic growth, Ann. Mat. Pura. Appl., 155 (1989), 1-24. doi: 10.1007/BF01765932.

[10]

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

[11]

M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Annals of Mathematics Studies, 105, Princeton Univ. Press, 1983.

[12]

M. Giaquinta and G. Modica, Remark on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math., 57 (1986), 55-99. doi: 10.1007/BF01172492.

[13]

J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations," Oxford University Press, New York, 1993.

[14]

J. Heinonen and T. Kilpeläinen, A-superharmonic functions and supersolutions of degenerate elliptic equations, Ark. Mat., 26 (1988), 87-105. doi: 10.1007/BF02386110.

[15]

Q. Han and F. H. Lin, "Elliptic Partial Differential Equations," American Mathematical Society, Providence, Rhode Island, 1997.

[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), 58-64.

[17]

T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew Math., 454 (1994), 143-161. doi: 10.1515/crll.1994.454.143.

[18]

T. Kilpeläinen, p-Laplacian type equations involving measures, Proceedings of the International Congress of Mathematicians, 167-176, Vol. III, Higher Ed. Press, Beijing, 2002.

[19]

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.

[20]

P. Koskela and O. Martio, Removability theorems for solution of degenerate elliptic PDEs, Ark. Mat., 31 (1993), 339-353. doi: 10.1007/BF02559490.

[21]

P. Lindqvist and O. Martio, Two theorems of N.Wiener for solutions of quasilinear elliptic equations, Acta Math., 155 (1985), 153-171. doi: 10.1007/BF02392541.

[22]

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

[23]

C. A. Nolder, Hardy-Littlewood theorems for $A$-harmonic tensors, Illinois J. Math., 43 (1999), 613-632.

[24]

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

[25]

J. Serrin and H. H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.

[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), 369-410. doi: 10.1353/ajm.2002.0012.

[27]

K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.

[28]

S. Zheng, Regualrity results for the generalized Beltrami systems, Acta Math. Sinica, 20 (2004), 193-205. doi: 10.1007/s10114-003-0250-x.

[29]

S. Zheng, Removable singularities of solutions of $A$-harmonic type equations, Acta Math. Appl. Sinica, 20 (2004), 115-122. doi: 10.1007/s10255-004-0154-2.

[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), 359-373. doi: 10.1016/j.jmaa.2008.05.059.

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