Cauchy Problem for Dynamic Elasticity Equations

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IAL DIFFERENTIAL EQUATIONS

Cauchy Problem for Dynamic Elasticity Equations O. I. Makhmudov1∗ and I. E. Niyozov1∗∗ 1

Samarkand State University, Samarkand, 140104 Uzbekistan e-mail: ∗ [email protected], ∗∗ [email protected]

Received March 31, 2020; revised June 8, 2020; accepted June 26, 2020

Abstract—We consider the problem on the analytic continuation of the solution of the system of vibration equations in elasticity theory in a spatial domain based on the values of the solution and the stresses on part of the boundary of this domain, i.e., a Cauchy problem. The problem is ill posed. If the part of the domain on which the Cauchy data are given is real analytic, then the problem has a local solution by the Cauchy–Kovalevskaya theorem. The special structure of the vibration equation is used to obtain explicit global solvability conditions and construct approximate solutions. DOI: 10.1134/S0012266120090037

INTRODUCTION We consider the problem on the analytic continuation of the solution of a system of equations describing the elastic-vibrational state of a medium in a spatial domain based on the values of the solution and the stresses on part of the boundary of the domain, i.e., a Cauchy problem. The Cauchy problem for elliptic equations was the subject of study throughout the 20th century and still continues to attract researchers’ attention. Just as the general analytic continuation problem, the Cauchy problem for elliptic equations is ill posed. Applied demands have given an impetus to the development of special methods for ill-posed Cauchy problems. Such problems arise in hydrodynamics, signal transmission theory, tomography, and geological exploration. Tikhonov, Lavrent’ev, and others developed the concept of conditionally well-posed problems [1]. It is fundamentally important that they not only derived solvability conditions for such problems but also obtained formulas representing the solutions in the form of series. This permits constructing reasonable approximate solutions by summing an appropriate finite segment of the series. The main technique used in the present paper is the series expansion of elements of a suitable space in homogeneous harmonic functions forming a basis on the sphere. The specific form of the domain guarantees the existence of this basis. The derivation of Carleman formulas reconstructing a function in the domain from the function values on some subset of the boundary is based on the Lavrent’ev–Mergelyan method, in which one approximates the kernel of the integral representation on the complement of this subset with respect to the boundary. The present authors [2–8] used Sh.Ya. Yarmukhamedov’s construction to construct explicit Carleman formulas for special classes of domains as well as a regularized solution of the Cauchy problem. The present study is based on the integral representation method. In addition, we use functional analysis tools such as the theory of Hilbert spaces as well as the technique of doubly orthogonal bases to represent the solution. Consider a medium in a dom