Hadamard’s example and solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation
-
Tynysbek S. Kalmenov
,
Alexander V. Rogovoy
and
Sergey I. Kabanikhin
Abstract
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. Hadamard’s 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 Hadamard’s 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.
Funding source: Ministry of Education and Science of the Republic of Kazakhstan
Award Identifier / Grant number: AP08856042
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 20-51-54004
Funding statement: This research has been funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant no. AP08856042). The work by S. I. Kabanikhin was supported by RFBR (Grant no. 20-51-54004).
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Articles in the same Issue
- Frontmatter
- Parameter identification for portfolio optimization with a slow stochastic factor
- Simultaneous inversion of a fractional order and a space source term in an anomalous diffusion model
- On the inverse gravimetry problem with minimal data
- On stable parameter estimation and short-term forecasting with quantified uncertainty with application to COVID-19 transmission
- Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems
- Convergence analysis of iteratively regularized Gauss–Newton method with frozen derivative in Banach spaces
- Identification of an unknown flexural rigidity of a cantilever Euler–Bernoulli beam from measured boundary deflection
- Hadamard’s example and solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation
- Ill-posed problems and the conjugate gradient method: Optimal convergence rates in the presence of discretization and modelling errors
- Simultaneous determination of mass density and flexural rigidity of the damped Euler–Bernoulli beam from two boundary measured outputs
