# 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. Google Scholar [2] M. Borsuk and V. Kondratiev, Elliptic boundary value problems of second order in piecewise smooth domains,, North-Holland Mathematical Library, (2006). Google Scholar [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. Google Scholar [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. Google Scholar [5] G. Di Fazio, $L^p$ estimates for divergence form elliptic equations with discontinuous coefficients,, Boll. Unione Mat. Ital. A, 10 (1996), 409. Google Scholar [6] D. Gilbarg andN. S. Trudinger, Elliptic partial differential equations of second order,, Reprint of the 1998 edition, (1998). Google Scholar [7] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations,, Academic Press, (1968). Google Scholar [8] N. S. Landkof, Foundations of modern potential theory,, Die Grundlehren der mathematischen Wissenschaften, (1972). Google Scholar [9] F. I. Mamedov, Regularity of solutions of linear and quasilinear equations of elliptic type in divergence form,, Math. Notes, 53 (1993), 50. Google Scholar [10] A. Maugeri, D. K. Palagachev and L. G. Softova, Elliptic and parabolic equations with discontinuous coefficients,, Mathematical Research, (2000). Google Scholar [11] C. Miranda, Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui,, Ann. Mat. Pura Appl., 63 (1963), 353. Google Scholar [12] S. G. Samko, Hypersingular integrals and their applications,, Analytical Methods and Special Functions, (2002). Google Scholar [13] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives. Theory and applications,, translated from the 1987 Russian original, (1987). Google Scholar [14] E. Stein, Singular integrals and differentiability properties of functions,, Princeton Mathematical Series, (1970). Google Scholar [15] G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili,, Ann. Mat. Pura Appl., 69 (1965), 285. Google Scholar

show all references

##### 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. Google Scholar [2] M. Borsuk and V. Kondratiev, Elliptic boundary value problems of second order in piecewise smooth domains,, North-Holland Mathematical Library, (2006). Google Scholar [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. Google Scholar [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. Google Scholar [5] G. Di Fazio, $L^p$ estimates for divergence form elliptic equations with discontinuous coefficients,, Boll. Unione Mat. Ital. A, 10 (1996), 409. Google Scholar [6] D. Gilbarg andN. S. Trudinger, Elliptic partial differential equations of second order,, Reprint of the 1998 edition, (1998). Google Scholar [7] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations,, Academic Press, (1968). Google Scholar [8] N. S. Landkof, Foundations of modern potential theory,, Die Grundlehren der mathematischen Wissenschaften, (1972). Google Scholar [9] F. I. Mamedov, Regularity of solutions of linear and quasilinear equations of elliptic type in divergence form,, Math. Notes, 53 (1993), 50. Google Scholar [10] A. Maugeri, D. K. Palagachev and L. G. Softova, Elliptic and parabolic equations with discontinuous coefficients,, Mathematical Research, (2000). Google Scholar [11] C. Miranda, Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui,, Ann. Mat. Pura Appl., 63 (1963), 353. Google Scholar [12] S. G. Samko, Hypersingular integrals and their applications,, Analytical Methods and Special Functions, (2002). Google Scholar [13] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives. Theory and applications,, translated from the 1987 Russian original, (1987). Google Scholar [14] E. Stein, Singular integrals and differentiability properties of functions,, Princeton Mathematical Series, (1970). Google Scholar [15] G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili,, Ann. Mat. Pura Appl., 69 (1965), 285. Google Scholar
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