Abstract
The nontrivial solvability of the homogeneous Dirichlet problem in a unit disk \( K\subset {{\mathbb{R}}^2} \) in positive Sobolev spaces is studied for a typeless differential equation of arbitrary even order 2m, m ≥ 2, with constant complex-valued coefficients and homogeneous symbol. The detailed proofs of the criteria of nontrivial solvability of the problem are given in various cases that form a complete picture. The example considered by A. V. Bitsadze is generalized for the equations of arbitrary even order 2m, m ≥ 2.
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References
R. A. Aleksandryan, “On the Dirichlet problem for the equation of a string and on the completeness of a system of functions in a disk,” Dokl. AN SSSR, 73, No. 5, 869–872 (1950).
R. A. Aleksandryan, “Spectral properties of operators generated by systems of differential equations of the Sobolev type,” Trudy Mosk. Mat. Obshch., 5, 455–505 (1960).
V. I. Arnol’d, “Small denominators. I,” Izv. AN SSSR, Ser. Mat., 25, No. 1, 21–86 (1961).
A. O. Babayan, “On unique solvability of the Dirichlet problem for a fourth-order properly elliptic equation,” Izv. Nats. Akad. Nauk Armen., 34, No. 5, 5–18 (1999).
A. O. Babayan, “ The Dirichlet problem for properly elliptic equations in a unit disk,” Izv. NAN Arm. SSR. Matem., 38, No. 6, 39–48 (2003).
H. Bateman and A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953–1955, Vols. 1–3.
Yu. M. Berezansky, Expansions in Eigenfunctions of Self-Adjoint Operators, Amer. Math. Soc., Providence, RI, 1969.
A. V. Bitsadze, “On the uniqueness of the solution of the Dirichlet problem for elliptic equations with partial derivatives,” Uspekhi Mat. Nauk, 3, (1948) No. 6, 211–212.
A. V. Bitsadze, Equations of the Mixed Type, Pergamon Press, New York, 1964.
A. V. Bitsadze, Some Classes of Partial Differential Equations, Gordon and Breach, New York, 1988.
D. Burgin and R. Duffin, “The Dirichlet problem for the vibrating string equations,” Bull. Am. Math. Soc., 45, 851–858 (1939).
V. P. Burskii, Methods of Study of Boundary-Value Problems for General Differential Equations [in Russian], Naukova Dumka, Kiev, 2002.
V. P. Burskii, “On the breakdown of the uniqueness of a solution of the Dirichlet problem for elliptic systems in a disk,” Mat. Zam., 48, No. 3, 32–36 (1990).
V. P. Burskii, “The boundary properties of L2-solutions of linear differential equations and the equation-domain duality,” Dokl. AN SSSR, 309, No. 5, 1036–1039 (1989).
V. P. Burskii and E. A. Buryachenko, “Some questions of the nontrivial solvability of a homogeneous Dirichlet problem for linear equations of arbitrary even order in a disk,” Mat. Zam., 74, No. 4, 1032–1043 (2005).
E. A. Buryachenko, “On the uniqueness of solutions of the Dirichlet problem in a disk for differential fourth-order equations in degenerate cases,” Nelin. Gran. Zad., 10, 44–49 (2000).
E. A. Buryachenko, “The conditions of nontrivial solvability of a homogeneous Dirichlet problem for equations of arbitrary even order in the case of multiple characteristics possessing no slope angles,” Ukr. Mat. Zh., 62, No. 5, 591–603 (2010).
K. O. Buryachenko, “Uniqueness of the solution of the Dirichlet problem for equations of arbitrary even order in the case of multiple characteristics possessing no slope angles,” Dopov. NAN Ukr., No. 1, 7–12 (2011).
F. John, “The Dirichlet problem for a hyperbolic equation,” Amer. J. Math., 63, No. 1, 141–154 (1941).
Ya. B. Lopatinskii, “On a means of reduction of boundary-value problems for systems of differential equations of the elliptic type to regular integral equations,” Ukr. Mat. Zh., 5, No. 2, 123–151 (1953).
B. I. Ptashnik, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev, 1984.
A. P. Soldatov, “On the first and second boundary-value problems for elliptic systems on a plane,” Diff. Uravn., 39, No. 5, 674–786 (2003).
N. E. Tovmasyan, “Efficient methods of solution of the Dirichlet problem for elliptic systems of second-order differential equations with constant coefficients in regions bounded by an ellipse,” Diff. Uravn., 5, No. 1, 60–71 (1969).
N. E. Tovmasyan, “New statements and studies of the first, second, and third boundary-value problems for strongly coupled elliptic systems of second-order differential equations with constant coefficients,” Izv. AN Arm. SSR. Matem., 3, No. 6, 497–521 (1968).
I. N. Vekua, New Methods for Solving Elliptic Equations, Wiley, New York, 1967.
M. I. Vishik, “On strongly elliptic systems of differential equations,” Mat. Sborn., 29, Issue 3, 615–677 (1951).
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Translated from Russian by V. V. Kukhtin
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 9, No. 4, pp. 477–514, October–November, 2012.
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Burskii, V.P., Buryachenko, K.A. On the breakdown of the uniqueness of a solution of the Dirichlet problem for typeless differential equations of arbitrary even order in a disk. J Math Sci 190, 539–566 (2013). https://doi.org/10.1007/s10958-013-1270-4
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DOI: https://doi.org/10.1007/s10958-013-1270-4