We invite you to the joint scientific seminar of the Institute of Mechanical Engineering SB RAS, ICM&MG SB RAS, the Department of Mathematical Methods of Geophysics of the MMF NSU and the Mathematical Center in Academgorodok 鈥淢athematical problems of geophysics鈥�.
The next seminar will be devoted to inverse problems for elliptic equations and Sobolev type equations. The seminar will take place on Thursday, September 14, at 13:00 (MSK+4).
Speaker - A. V. Velisevich (senior lecturer at the Department of Mathematical Analysis and Differential Equations of the Siberian Federal University). Based on the materials of the dissertation for the degree of Ph.D. Scientific supervisor: A. Sh. Lyubanova, Ph.D., Siberian Federal University
Three inverse problems of finding the unknown function u and the unknown minor coefficient k in the elliptic equation are considered
Mu + kr(u) = -div(M(x) grad u) + m(x)u + kr(u) = f,
(two of them are for a linear equation with r(u) = u) with boundary data of various types and an integral redefinition condition at the boundary of the study area. The conditions for stabilization of a strong solution of the inverse problem for a Sobolev type equation to the solution of one of these problems are also studied. The M operator is assumed to be strongly elliptic and self-adjoint.The main results of the work are theorems on the existence and uniqueness of a strong generalized solution to the original problems, as well as sufficient conditions for the continuous dependence of the solutions of these problems on the original data. In addition, the main results include sufficient conditions for stabilization of a strong solution of the inverse problem for a Sobolev type equation to a strong solution of the corresponding stationary inverse problem for an elliptic equation with an integral overdetermination condition on the boundary.
Existence and uniqueness are proved by a method, the essence of which is to continue the data from the boundary to the region and reduce the inverse problem to an operator equation Ak = k of the second kind, for an unknown coefficient k .
Practical interest in these problems is due to the fact that in numerous applications the coefficients of the original equation characterize the physical properties of the medium: permeability, thermal conductivity, and so on. In the problems considered, the unknown is the absorption coefficient.
