Literature Cited
G. M. Verzhbinskii and V. G. Maz'ya, “On the asymptotic behavior of the solutions of the Dirichlet problem close to a nonregular boundary,” Dokl. Akad. Nauk SSSR,176, No. 3, 498–501 (1967).
T. Kato, “Growth properties of solutions of the reduced wave equation with a variable coefficient,” Comm. Pure Appl. Math.,12, 403–425 (1959).
J. Mařik and M. Ráb, “Asymptotische Eigenschaften von Lösungen der Differentialgleichung y″=A(x)y im nichtoszillatorischen Fall,” Czechoslovak Math. J.,10, (85), No. 4 501–522 (1960).
A. Yu. Levin, “Classification of nonoscillatory cases for the equation\(x + p(\dot x) + q(t)x = 0\) with q(t) of constant sign,” Dokl. Akad. Nauk SSSR,171, No. 5, 1037–1040 (1966).
J. Mařik and M. Ráb, “Nichtoszillatorische lineare Differentialgleichungen 2 Ordnung,” Czechoslovak Math. J.,13, (88), No. 2, 209–225 (1963).
P. Hartman and A. Wintner, “The asymptotic arcus variation of solutions of real linear differential equations of second order,” Amer. J. Math.,70, No. 1, 1–11 (1948).
P. Hartman and A. Wintner, “Asymptotic integrations of linear differential equations,” Amer. J. Math.,77, No. 1, 45–86 (1955).
A. Dinghas, “Wachstumprobleme harmonischer und verwandter Funktionen in En,” Ann. Acad. Sci. Fenn.,A1, 250/8 (1958).
T. Carleman, “Sur une integralité differentielle dans la théorie des fonctions analytiques,” Compt. Rend. Acad. Sci. Paris,196, 995–997 (1933).
P. Fife, “Growth and decay properties of solutions of second order elliptic equations,” Ann. Scuola Norm. Super. Pisa, s. 3,20, No. 4, 675–701 (1966).
A. Huber, “Ein räumliches Analogon des Wiman-Heinsschen Satzes,” Studies in Math. Anal. and Rel. Topics, Stanford, No. 4, 152–155 (1962).
M. Heins, “On a notion of convexity connected with a method of Carleman,” J. Analyse Math.,7, 53–77 (1959).
A. Dinghas, “Über einige Konvexitätsfragen bei partiellen Differentialgleichungen vom Strumschen Typus,” Math. Ann., 155, No. 5, 397–421 (1964).
M. Tsuji, “A theorem on the majoration of harmonic measure and its applications,” Tohoku Math. J.,3, 13–22 (1951).
K. Haliste, “Estimates of harmonic measures,” Arkiv. Mat.,6, No. 1, 1–30 (1965).
E. M. Landis, “Some problems of the qualitative theory of elliptic equations of the second order (case of many independent variables),” Usp. Matem. Nauk,18, No. 1, 3–62 (1963).
V. G. Maz'ya, “On the regularity on the boundary of solutions of elliptic equations and conformal mapping,” Dokl. Akad. Nauk SSSR,152, No. 6, 1297–1300 (1963).
V. G. Maz'ya, “On the behavior close to the boundary of the solution of the Dirichlet problem for an elliptic equation of the second order in divergent form,” Matem. Zametki,2, No. 2, 209–220 (1967).
G. N. Blokhina, “Theorems of Phragmen-Lindeloftype for linear elliptic equations of the second order,” Dokl. Akad. Nauk SSSR,162, 727 (1965).
J. Moser, “A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations,” Comm. Pure and Appl. Math.,13, No. 3, 457–468 (1960).
W. Littman, G. Stampacchia, and H. F. Woinberger, “Regular points for elliptic equations with discontinuous coefficients,” Ann. Scuola Norm. Super. Pisa, s. 3,17 1–2, 45–79 (1963).
J. Deny, J. L. Lions, “Les espaces du type de Beppo Levi,” Ann. Inst. Fourier,5, 305–370 (1955).
R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York (1962).
L. Payne, G. Polya, and H. F. Weinberger, “On the ratio of consecutive eigenvalues,” J. Math. and Phys.,35, No. 3, 289–298 (1956).
H. Royden, “The growth of a fundamental solution of an elliptic divergence structure equation.” Studies in Math. Anal. and Rel. Topics, Stanford, 333–340 (1962).
S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions,” Comm. Pure Appl. Math.,12, 623–727 (1959).
A. G. Vitushkin, On Multidimensional Variations [in Russian], Moscow (1955).
H. Federer, “Curvature measures,” Trans. Amer. Math. Soc.,93, 418–491 (1959).
S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon, New York (1965).
J. Serrin and H. F. Weinberger, “Isolated singularities of solutions of linear elliptic equations,” Amer. J. Math.,88, No. 1, 258–272 (1966).
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Translated from Sibirskii Matematicheskii Zhurnal, Vol.12, No. 6, pp. 1217–1249, November–December, 1971.
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Verzhbinskii, G.M., Maz'ya, V.G. Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary. I. Sib Math J 12, 874–899 (1971). https://doi.org/10.1007/BF00966531
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DOI: https://doi.org/10.1007/BF00966531