Geodesic
On the Spectrum of Well-Defined Restrictions and Extensions for the Laplace Operator
Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 507-516

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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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     author = {B. N. Biyarov},
     title = {On the {Spectrum} of {Well-Defined} {Restrictions} and {Extensions} for the {Laplace} {Operator}},
     journal = {Matemati\v{c}eskie zametki},
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B. N. Biyarov. On the Spectrum of Well-Defined Restrictions and Extensions for the Laplace Operator. Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 507-516. http://geodesic.mathdoc.fr/item/MZM_2014_95_4_a2/