American Institute of Mathematical Sciences

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

Potential estimates and applications to elliptic equations

Farman Mamedov 1, , Sara Monsurrò 2, and  Maria Transirico 2,

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.
Keywords: a priori bounds, Elliptic equations, potential estimates..
Mathematics Subject Classification: Primary: 35J25, 35B45; Secondary: 31B1.
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.

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), 3-13.

[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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