Abstract:
The study of the spectral properties of operators generated by differential equations of hyperbolic or parabolic type with Cauchy initial data involve, as a rule, Volterra boundary-value problems that are well posed. But Hadamard's example shows that the Cauchy problem for the Laplace equation is ill posed. At present, not a single Volterra well-defined restriction or extension for elliptic-type equations is known. Thus, the following question arises: Does there exist a Volterra well-defined restriction of a maximal operator $\widehat{L}$ or a Volterra well-defined extension of a minimal operator $L_0$ generated by the Laplace operator? In the present paper, for a wide class of well-defined restrictions of the maximal operator $\widehat{L}$ and of well-defined extensions of the minimal operator $L_0$ generated by the Laplace operator, we prove a theorem stating that they cannot be Volterra.
Keywords:Laplace operator, maximal (minimal) operator, Volterra operator, Volterra well-defined restrictions and extensions of operators, Hilbert space, elliptic operator, Poisson operator.
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