Hadamards example and solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation
Kalmenov T.S. Rogovoy A.V. Kabanikhin S.I.
1 December 2022De Gruyter Open Ltd
Journal of Inverse and Ill-Posed Problems
2022#30Issue 6891 - 904 pp.
In the theory of partial differential equations, an example constructed by J. Hadamard, which shows the instability of the solution of the Cauchy problem for the Laplace equation with respect to small changes in the initial data, is of great importance. Hadamards example served as the beginning of a systematic study of ill-posed problems in mathematical physics. On the other hand, the study of the Cauchy problem for the Laplace equation arises from problems of geophysics. At the same time, the question arises whether the Cauchy problem is correct for other elliptic equations including degenerate elliptic equations. We have constructed analogs of Hadamards example and established the incorrectness of the solution of the Cauchy problem for the Gellerstedt equation in two-dimensional and multidimensional cases. The condition of strong solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation in a cylindrical domain is found. The proof is based on the spectral properties of the Laplace operator and the properties of special functions.
Bessel functions , Gellerstedt equation , Mixed Cauchy problem , regular solution , solvability
Text of the article Перейти на текст статьи
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Institute of Computational Mathematics and Geophysics SB RAS, Novosibirsk, Russian Federation
Institute of Mathematics and Mathematical Modeling
Institute of Computational Mathematics and Geophysics SB RAS
10 лет помогаем публиковать статьи Международный издатель
Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026