On Some Problems Generated by a Sesquilinear Form
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ON SOME PROBLEMS GENERATED BY A SESQUILINEAR FORM N. D. Kopachevskii and A. R. Yakubova
UDC 517.984.5
Abstract. Based on the generalized Green formula for a sesquilinear nonsymmetric form for the Laplace operator, we consider spectral nonself-adjoint problems. Several such problems are similar to classical ones; others arise in problems of hydrodynamics and diffraction and in problems with surface dissipation of energy. Properties of solutions of such problems are considered. Also, we study initialboundary value problems generating the considered spectral problems and prove correct solvability theorems for such problems on any interval of time.
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract Green Formulae for Triples of Hilbert Spaces and Sesquilinear Forms . . . . . . 1.1. Green formulae for triples of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Sesquilinear bounded forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Abstract Green formula for sesquilinear forms . . . . . . . . . . . . . . . . . . . . . . Boundary-Value Problems Generated by Sesquilinear Nonsymmetric Forms Based on the Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The Green formula for the unperturbed problem . . . . . . . . . . . . . . . . . . . . . 2.2. The Green formula for the perturbed problem . . . . . . . . . . . . . . . . . . . . . . 2.3. Boundary-value problems generated by nonsymmetric forms . . . . . . . . . . . . . . Spectral Problems Generated by Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . 3.1. The spectral Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The Neumann–Newton spectral problem . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The Steklov spectral problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The Stefan spectral problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Other classes of perturbed spectral problems . . . . . . . . . . . . . . . . . . . . . . . On Initial-Boundary Value Problems Generated by Sesquilinear Forms and Generating Spectral Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Perturbed classical initial-boundary value problems . . . . . . . . . . . . . . . . . . . 4.2. Nonclassical initial-boundary value problems of mathematical physics . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction The present paper is a detailed presentation of the authors’ conference talks [26, 27] (see [21, Chap. 6] as well). The authors are inspired to investigate spectral and initial-boundary value problems in Lipschitz domains by Agranovich’s works [1–4] and his lectures at the annual Crimean Autumn Mathematical School-Symposium.
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