# American Institute of Mathematical Sciences

2015, 2015(special): 793-800. doi: 10.3934/proc.2015.0793

## Potential estimates and applications to elliptic equations

 1 Mathematics and Mechanics Institute, Nat. Acad. Sci. of Azerbaijan, Az1001, 10, Istiglaliyyat str, Baku, Azerbaidjan 2 Dipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy, Italy

Received  September 2014 Revised  March 2015 Published  November 2015

In this paper we prove a potential type estimate for the solutions of some classes of Dirichlet problems associated to certain non divergence structure elliptic equations with smooth datum. As a consequence of our potential bound, we can get an a priori estimate for the solutions of the same kind of Dirichlet problem, but with less regular datum.
Citation: Farman Mamedov, Sara Monsurrò, Maria Transirico. Potential estimates and applications to elliptic equations. Conference Publications, 2015, 2015 (special) : 793-800. doi: 10.3934/proc.2015.0793
##### References:
 [1] R. A. Amanov and F. I. Mamedov, On the regularity of the solutions of degenerate elliptic equations in divergence form, Math. Notes, 83 (2008), 3-13. [2] M. Borsuk and V. Kondratiev, Elliptic boundary value problems of second order in piecewise smooth domains, North-Holland Mathematical Library, Elsevier, 2006. [3] F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat., 40 (1991), 149-168. [4] F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with $VMO$ coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853. [5] G. Di Fazio, $L^p$ estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Unione Mat. Ital. A, 10 (1996), 409-420. [6] D. Gilbarg andN. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [7] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. [8] N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. [9] F. I. Mamedov, Regularity of solutions of linear and quasilinear equations of elliptic type in divergence form, Math. Notes, 53 (1993), 50-58. [10] A. Maugeri, D. K. Palagachev and L. G. Softova, Elliptic and parabolic equations with discontinuous coefficients, Mathematical Research, 109, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000. [11] C. Miranda, Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Ann. Mat. Pura Appl., 63 (1963), 353-386. [12] S. G. Samko, Hypersingular integrals and their applications, Analytical Methods and Special Functions, 5. Taylor & Francis, Ltd., London, 2002. [13] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives. Theory and applications, translated from the 1987 Russian original, Gordon and Breach Science Publishers, Yverdon, 1993. [14] E. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J., 1970. [15] G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304.

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##### References:
 [1] R. A. Amanov and F. I. Mamedov, On the regularity of the solutions of degenerate elliptic equations in divergence form, Math. Notes, 83 (2008), 3-13. [2] M. Borsuk and V. Kondratiev, Elliptic boundary value problems of second order in piecewise smooth domains, North-Holland Mathematical Library, Elsevier, 2006. [3] F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat., 40 (1991), 149-168. [4] F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with $VMO$ coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853. [5] G. Di Fazio, $L^p$ estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Unione Mat. Ital. A, 10 (1996), 409-420. [6] D. Gilbarg andN. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [7] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. [8] N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. [9] F. I. Mamedov, Regularity of solutions of linear and quasilinear equations of elliptic type in divergence form, Math. Notes, 53 (1993), 50-58. [10] A. Maugeri, D. K. Palagachev and L. G. Softova, Elliptic and parabolic equations with discontinuous coefficients, Mathematical Research, 109, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000. [11] C. Miranda, Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Ann. Mat. Pura Appl., 63 (1963), 353-386. [12] S. G. Samko, Hypersingular integrals and their applications, Analytical Methods and Special Functions, 5. Taylor & Francis, Ltd., London, 2002. [13] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives. Theory and applications, translated from the 1987 Russian original, Gordon and Breach Science Publishers, Yverdon, 1993. [14] E. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J., 1970. [15] G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304.
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