Abstract
Degenerate elliptic equations and anisotropic equations are considered. An estimate of the maximum of the modulus of a generalized solution of the Dirichlet problem is obtained in the domain Ω with zero on the boundary.
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References
E. de Giorgi, “Sulla differenziabilita e l'analicita delle estremali degli integrali multipli regulari,”Mem. Acad. Sci. Torino, Ser. 3-a,3, Pt. 1, 25–43 (1957).
J. Moser, “A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations,”Commun. Pure Appl. Math.,13, No. 3, 457–468 (1960).
J. Moser, “On Harnack's theorem for elliptic differential equations,”Commun. Pure Appl. Math.,14, No. 3, 577–591 (1961).
J. Nash, “Continuity of solutions of parabolic and elliptic equations,”Amer. J. Math.,80, No. 4, 931–954 (1958).
S. N. Kruzhkov, “A priori estimates for generalized solutions of elliptic and parabolic equations,”Dokl. Akad. Nauk SSSR,150, No. 4, 748–751 (1963).
S. N. Kruzhkov, “On some properties of solutions of elliptic equations,”Dokl. Akad. Nauk SSSR,150, No. 3, 470–473 (1963).
S. N. Kruzhkov, “A priori estimates and some properties of solutions of elliptic and parabolic equations,”Mat. Sb.,65, No. 4, 522–570 (1964).
S. N. Kruzhkov, “Boundary-value problems for degenerating second-order elliptic equations,”Mat. Sb.,77, No. 3, 299–334 (1968).
J. Serrin, “Local behavior of solutions of quasilinear equations,”Acta. Math.,111, No. 3–4, 247–302 (1964).
N. Trudinger, “Generalized solutions of quasilinear differential inequalities,”Bull. Amer. Math. Soc.,77, No. 4, 576–579 (1971).
N. Trudinger, “On the regularity of generalized solutions of linear, non-uniformly elliptic equations,”Arch. Rat. Mech. Anal.,42, No. 1, 50–62 (1971).
A. V. Ivanov, “Uniform Hölder estimates for generalized solutions of quasilinear parabolic equations admitting double degeneration,”Algebra Anal.,3, No. 2, 139–179 (1991).
A. V. Ivanov, “Quasilinear parabolic equations admitting double degeneration,”Algebra Anal.,4, No. 6, 114–130 (1992).
E. Fabes, C. Kenig, and R. Serapioni, “The local regularity of solutions of degenerate elliptic equations,”Commun. P. D. E.,7, 77–116 (1982).
S. Chanillo and R. Wheeden, “Harnack's inequality and mean value inequalities for solutions of degenerate elliptic equations,”Commun. P. D. E.,11 (10), 1111–1134 (1986).
F. Chiarenza, A. Rustichini, and R. Serapioni “The De Giorgi-Moser theorem for a class of degenerate non-uniformly elliptic equations,”Commun. P. D. E.,4 (5), 635–669 (1989).
I. M. Kolodii, “Imbedding theorems for the spaces\(\mathop {W_\beta ^1 }\limits^ \circ \left( {\bar \lambda \left( x \right),K_r } \right)\) and\(W_\beta ^1 \left( {\bar \lambda \left( x \right),K_r } \right)\), Kr),”Ukr. Mat. Zh.,34, No. 4, 450–457 (1987).
Lu Ven-Tuan, “On imbedding theorems for the spaces of functions with partial derivatives summable with various indices,”Vestn. Leningrad. Univ., No. 7, 23–27 (1961).
S. N. Kruzhkov and I. M. Kolodii, “On imbedding theory for anisotropic Sobolev spaces,”Usp. Mat. Nauk,38, Issue 2, 207–208 (1983).
S. N. Kruzhkov and A. G. Korolev, “On imbedding theory for anisotropic function spaces,”Dokl. Akad. Nauk SSSR,285, No. 5, 1054–1057 (1985).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 635–648, May, 1995.
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Kolodii, I.M. An estimate of the maximum of the modulus of generalized solutions of the Dirichlet problem for elliptic equations of divergent form. Ukr Math J 47, 733–748 (1995). https://doi.org/10.1007/BF01059047
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DOI: https://doi.org/10.1007/BF01059047