Abstract:
Boundary-value problems are considered for strongly elliptic functional-differential equations in bounded domains. In contrast to the case of elliptic differential equations, smoothness of generalized solutions of such problems can be violated in the interior of the domain and may be preserved only on some subdomains, and the symbol of a self-adjoint semibounded functional-differential operator can change sign. Both necessary and sufficient conditions are obtained for the validity of a Gårding-type inequality in algebraic form. Spectral properties of strongly elliptic functional-differential operators are studied, and theorems are proved on smoothness of generalized solutions in certain subdomains and on preservation of smoothness on the boundaries of neighbouring subdomains. Applications of these results are found to the theory of non-local elliptic problems, to the Kato square-root problem for an operator, to elasticity theory, and to problems in non-linear optics.
Bibliography: 137 titles.
Keywords:
elliptic functional-differential equations, spectral properties, smoothness of generalized solutions, non-local elliptic problems, Kato square-root problem, three-layer plates, non-linear optical systems with two-dimensional feedback.
This research was supported by the Russian Foundation for Basic Research (grant no. 14-01-00265) and the Programme "Leading Scientific Schools" (grant no. НШ-4479.2014.1).
Citation:
A. L. Skubachevskii, “Boundary-value problems for elliptic functional-differential equations and their applications”, Russian Math. Surveys, 71:5 (2016), 801–906
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