Abstract
Unlike the case of elliptic differential equations, generalized solutions of elliptic differential-difference equations may be nonsmooth on an entire domainQ, only preserving smoothness on certain subdomainsQ r ⊂Q. The conditions are considered under which the generalized solutions of the third boundary-value problem remain smooth on the boundaries of the neighboring subdomainsQ r .
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References
A. Skubachevskii, “The first boundary-value problem for strongly elliptic differential-difference equations,”J. Differ. Equal.,63, No. 3, 332–361 (1986).
A. L. Skubachevskii and E. L. Tsvetkov, “The second boundary-value problem for the elliptic differential-difference equations,”Differents. Uravn.,25, No. 10, 1766–1776 (1989).
G. A. Kamenskii and A. D. Myshkis, “On the statement of boundary-value problems for differential equations with deviating argument and several higher terms,”Differents. Uravn.,10, No. 3, 409–418 (1974).
A. L. Skubachevskii, “Eigenvalues and eigenfunctions of some nonlocal boundary-value problems,”Differents. Uravn.,25, No. 1, 127–136 (1989).
G. G. Onanov and A. L. Skubachevskii, “Differential equations with deviating arguments in stationary problems in the mechanics of deformable bodies,”Prikl. Mekh.,15, No. 5, 39–47 (1979).
V. P. Mikhailov,Partial Differential Equations [in Russian], Nauka, Moscow (1983).
J. L. Lions and E. Magenes,Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris (1970).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1140–1150, August, 1993.
The author is grateful to A. D. Myshkis and A. L. Skubachevaskii for their kind attention to the work and valuable advice.
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Tsvetkov, E.L. Smoothness of generalized solutions of the third boundary-value problem for an elliptic differential-difference equation. Ukr Math J 45, 1272–1284 (1993). https://doi.org/10.1007/BF01070975
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DOI: https://doi.org/10.1007/BF01070975