RUDN mathematicians estimate the solution for the inverse problems of the Laplace differential equation

RUDN University mathematicians obtained an estimate of the solution of inverse problems of the Laplace differential equation. Having a set of values characterizing the Laplace map and the Dirichlet-to-Neumann terminal parameters, they were able to obtain conditions determining the type of the studied space. The obtained result allows us to describe the nature of physical space in the problems of electrostatics, electrodynamics, quantum mechanics, continuum physics and in the problem of studying the thermal conductivity of the medium. The article is published in the journal Inverse Problems & Imaging.

Functions are assigned to certain physical processes during solving physical problems by mathematical methods. For example, to describe the process of thermal conductivity, the Laplace differential equation ― Laplace map is necessary. These functions operate within a specific characteristic space that corresponds to the physical model of the phenomenon.

The experimental data of a physical problem can be a set of values that characterize the type of Laplace map for a particular space. Direct problems of the differential equation allow us to obtain this set of characteristic values in the known mathematical form of the Laplace map. In this case, the solution of inverse problems of these equations with a known set of characteristic values allows us to explore the mathematical form of space, and hence the physical process of the investigated problem.

Space is characterized by a metric ― the rule of distribution of elements in this space. Knowing the mathematical species of the space metric, it is possible to obtain the space structure of the physical model of the investigated problem. Thus, analyzing the data of neural activity (characteristic values), it is possible to obtain a metric of the neural network space, which will be associated with the physical model of this network.

It is known that the type of metric can be uniquely determined up to the level at which two metrics― g1 и g2 ― with a slightly different set of characteristic values become equivalent, therefore, the solution will not be the only one. In this case, if there are two metrics with the same set of characteristic values, a transformation, which translates one metric into another, is allowed.

The question to the authors: is it possible to tell about the smooth homeomorphism briefly and popularly?

For an adequate description of the type of the studied space, it is necessary to make sure that the found form of the metric is valid for solving the direct problem of differential equations. For such verification, the study of the uniqueness of the solution of inverse problems on an arbitrarily bounded area of space can serve as a tool.

After considering a given set of characteristic values on an arbitrarily fixed domain, the authors obtained a mathematical estimate of the uniqueness of the solution for the inverse problem in determining the metric of the Laplace operator.

Another type of input data for the inverse problem can be boundary conditions under which the behavior of the physical model is known. In this case, to obtain an estimate of the uniqueness of the solution of the inverse problem, the authors used data on Dirichlet-to-Neumann boundary conditions for the Laplace map on an arbitrary domain of the studied space.

The question to the authors: do your results allow you to approach the solution of the problem?

More information:
Oleg Yu. Imanuvilov et al. Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary, Inverse Problems & Imaging (2019). DOI: 10.3934/ipi.2019054

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