On the Spectrum of Well-Defined Restrictions and Extensions for the Laplace Operator

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.