On the absence of Volterra correct restrictions and extensions of the Laplace operator

TL;DR

This paper proves that there are no Volterra correct restrictions or extensions of the Laplace operator in the unit disk, highlighting fundamental limitations in solving elliptic equations with Volterra-type operators.

Contribution

It establishes the non-existence of Volterra correct restrictions and extensions for the Laplace operator, a novel theoretical result in elliptic operator theory.

Findings

No Volterra correct restrictions of the maximal Laplace operator exist.

No Volterra correct extensions of the minimal Laplace operator exist.

The result applies specifically to the Laplace operator in the unit disk.

Abstract

At the beginning of the last century J. Hadamard constructed the well-known example illustrating the incorrectness of the Cauchy problem for elliptic-type equations. If the Cauchy problem for some differential equation is correct, then it is usually a Volterra problem, i.e., the inverse operator is a Volterra operator. At present, not a single Volterra correct restriction or extension for elliptic-type equations is known. In the present paper, we prove the absence of Volterra correct restrictions of the maximal operator $\widehat{L}$ and Volterra correct extensions of the minimal operator $L_0$ generated by the Laplace operator in $L_2(\Omega)$, where $\Omega$ is the unit disk.