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Dirichlet Problem for a Class of Quasilinear Elliptic Equations

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Abstract

In this paper, the Dirichlet problem for quasilinear elliptic equations is studied. New a priori estimates of the solution and its gradient are obtained. These estimates are derived without any assumptions on the smoothness of the coefficients and the right-hand side of the equation. Moreover, an arbitrary growth of the right-hand side with respect to the gradient of the solution is assumed. On the basis of the resulting estimates, existence theorems are proved.

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REFERENCES

  1. O.A. Ladyzhenskaya and N.N. Ural tseva,Linear and Quasilinear Equations of Elliptic Type [in Russian],Nauka, Moscow,1973.

    Google Scholar 

  2. D. Gilbarg and N.S. Trudinger,Elliptic Partial Differential Equations of Second Order,Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],224.Springer-Verlag, Berlin-New York,1983.Russian translation:Nauka,Moscow,1989.

    Google Scholar 

  3. N.V. Krylov,Nonlinear Elliptic and Parabolic Equations of Second Order [in Russian ],Nauka, Moscow, 1985.

    Google Scholar 

  4. S.N. Bernstein,On the Equations of the Variational Calculus [in Russian ],Collected Works,vol.3, Akad.Nauk SSSR, Moscow,1960.

    Google Scholar 

  5. H. Amman and M.G. Crandall,"On some existence theorems for semilinear elliptic equations,"Indiana Univ.Math.J.,27 (1978), no.5,779–790.

    Google Scholar 

  6. S.I. Pokhozhaev,"On the equation of the form Δu =f(x, u, ∇u),"Mat.Sb.[Math.USSR-Sb.],113 (155) (1980),no.1 (10),324–338.

    Google Scholar 

  7. Al.S. Tersenov,"Estimate of the solution of the Dirichlet problem for parabolic equations and appli-cations,"J.Math.Anal.Appl.,273 (2002),no.1,206–216.

    Google Scholar 

  8. S.N. Kruzhkov,"Quasilinear parabolic equations and systems with two independent variables,"Trudy Sem.Petrovsk.(1979),no.5,217–271.

  9. V.L. Kamynin,"A priori estimates and global solvability of quasilinear parabolic equations,"Vest ni k Moskov.Univ.Ser.I Mat.Mekh.[Moscow Univ.Math.Bull.](1981),no.1,33–38.

  10. V.L. Kamynin,"A priori estimates of the solutions of quasilinear parabolic equations in the plane and their applications,"Differentsial nye Uravneniya [Differential Equations ],19 (1983),no.5,590–598.

    Google Scholar 

  11. N.V. Khusnutdinova,"On the boundedness conditions for the gradient of the solution of degenerate parabolic equations,"Dynamika Sploshn.Sredy,71 (1985),120–129.

    Google Scholar 

  12. Al.S. Tersenov,"On quasilinear non-uniformly elliptic equations in some nonconvex domains,"Comm. Partial Differential Equations,23 (1998),no.11,2165–2186.

    Google Scholar 

  13. Al.S. Tersenov,"On quasilinear non-uniformly parabolic equations in general form,"J.Differential Equations,142 (1998),no.1,263–276.

    Google Scholar 

  14. Al.S. Tersenov and Ar.S. Tersenov,"The Cauchy problem for a class of quasilinear parabolic equa-tions,"Ann.Mat.Pura Appl.,182 (2003),no.3,325–336.

    Google Scholar 

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  1. University of Crete, Greece

    Al. S. Tersenov

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  1. Al. S. Tersenov
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Tersenov, A.S. Dirichlet Problem for a Class of Quasilinear Elliptic Equations. Mathematical Notes 76, 546–557 (2004). https://doi.org/10.1023/B:MATN.0000043484.27246.b3

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  • DOI: https://doi.org/10.1023/B:MATN.0000043484.27246.b3

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