Analysis of Cauchy data in ill-posed continuation problems from a part of the boundary

Analysis of Cauchy data in ill-posed continuation problems from a part of the boundary

(Russian, English abstract)

Kabanikhin S. I., Shishlenin M. A.
Siberian Electronic Mathematical Reports, 22, 2, pp. A255-A269 (2025)

УДК 519.63,519.64 
DOI: 10.33048/semi.2025.22.A16  
MSC 65N21,65R32


Abstract:

The paper analyzes the ill-posedness of the Cauchy problem with data on a part of the boundary for partial differential equations. The variants of the boundaries of the region (time-like, characteristic, space-like) are considered, in which the Cauchy data on the part of the boundary are interconnected. The conditions are obtained, the violation of which leads to instability of the continuation problem.

Keywords: ill-posed problem, continuation problems, Cauchy problem, Laplace equation, Helmholtz equation, wave equation, Dirichlet and Neumann data