December  2014, 19(10): 3245-3265. doi: 10.3934/dcdsb.2014.19.3245

On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems

1. 

Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, United States

Received  July 2013 Revised  October 2013 Published  October 2014

This paper studies Holder continuity of weak solutions to strongly coupled elliptic systems. We do not assume that the solutions are bounded but BMO and the ellipticity constants can be unbounded.
Citation: Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245
References:
[1]

B. Franchi, C. Perez and R. L. Wheeden, Self-Improving Properties of John Nirenberg and Poincaré Inequalities on Spaces of Homogeneous Type, J. Functional Analysis, 153 (1998), 108-146. doi: 10.1006/jfan.1997.3175.

[2]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003. doi: 10.1142/9789812795557.

[3]

R. L. Johnson and C. J. Neugebauer, Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwendungen, 12 (1993), 3-11.

[4]

D. Le, Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy, Transactions of AMS, 365 (2013), 2723-2753. doi: 10.1090/S0002-9947-2012-05720-5.

[5]

D. Le, Everywhere regularity of BMO weak solutions to uniform elliptic systems, submitted.

[6]

D. Le, L. Nguyen and T. Nguyen, Coexistence in Cross Diffusion systems, Indiana Univ. J. Math., 56 (2007), 1749-1791.

[7]

J. Orobitg and C. Pérez, $A_p$ weights for nondoubling measures in $\RR^n$ and applications, Transactions of AMS, 354 (2002), 2013-2033. doi: 10.1090/S0002-9947-02-02922-7.

[8]

E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.

show all references

References:
[1]

B. Franchi, C. Perez and R. L. Wheeden, Self-Improving Properties of John Nirenberg and Poincaré Inequalities on Spaces of Homogeneous Type, J. Functional Analysis, 153 (1998), 108-146. doi: 10.1006/jfan.1997.3175.

[2]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003. doi: 10.1142/9789812795557.

[3]

R. L. Johnson and C. J. Neugebauer, Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwendungen, 12 (1993), 3-11.

[4]

D. Le, Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy, Transactions of AMS, 365 (2013), 2723-2753. doi: 10.1090/S0002-9947-2012-05720-5.

[5]

D. Le, Everywhere regularity of BMO weak solutions to uniform elliptic systems, submitted.

[6]

D. Le, L. Nguyen and T. Nguyen, Coexistence in Cross Diffusion systems, Indiana Univ. J. Math., 56 (2007), 1749-1791.

[7]

J. Orobitg and C. Pérez, $A_p$ weights for nondoubling measures in $\RR^n$ and applications, Transactions of AMS, 354 (2002), 2013-2033. doi: 10.1090/S0002-9947-02-02922-7.

[8]

E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.

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