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.   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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