American Institute of Mathematical Sciences

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 & 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.  doi: 10.1006/jfan.1997.3175.  Google Scholar [2] E. Giusti, Direct Methods in the Calculus of Variations,, World Scientific, (2003).  doi: 10.1142/9789812795557.  Google Scholar [3] R. L. Johnson and C. J. Neugebauer, Properties of BMO functions whose reciprocals are also BMO,, Z. Anal. Anwendungen, 12 (1993), 3.   Google Scholar [4] D. Le, Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy,, Transactions of AMS, 365 (2013), 2723.  doi: 10.1090/S0002-9947-2012-05720-5.  Google Scholar [5] D. Le, Everywhere regularity of BMO weak solutions to uniform elliptic systems,, submitted., ().   Google Scholar [6] D. Le, L. Nguyen and T. Nguyen, Coexistence in Cross Diffusion systems,, Indiana Univ. J. Math., 56 (2007), 1749.   Google Scholar [7] J. Orobitg and C. Pérez, $A_p$ weights for nondoubling measures in $\RR^n$ and applications,, Transactions of AMS, 354 (2002), 2013.  doi: 10.1090/S0002-9947-02-02922-7.  Google Scholar [8] E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals,, Princeton Univ. Press, (1993).   Google Scholar

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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.  doi: 10.1006/jfan.1997.3175.  Google Scholar [2] E. Giusti, Direct Methods in the Calculus of Variations,, World Scientific, (2003).  doi: 10.1142/9789812795557.  Google Scholar [3] R. L. Johnson and C. J. Neugebauer, Properties of BMO functions whose reciprocals are also BMO,, Z. Anal. Anwendungen, 12 (1993), 3.   Google Scholar [4] D. Le, Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy,, Transactions of AMS, 365 (2013), 2723.  doi: 10.1090/S0002-9947-2012-05720-5.  Google Scholar [5] D. Le, Everywhere regularity of BMO weak solutions to uniform elliptic systems,, submitted., ().   Google Scholar [6] D. Le, L. Nguyen and T. Nguyen, Coexistence in Cross Diffusion systems,, Indiana Univ. J. Math., 56 (2007), 1749.   Google Scholar [7] J. Orobitg and C. Pérez, $A_p$ weights for nondoubling measures in $\RR^n$ and applications,, Transactions of AMS, 354 (2002), 2013.  doi: 10.1090/S0002-9947-02-02922-7.  Google Scholar [8] E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals,, Princeton Univ. Press, (1993).   Google Scholar
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