On the uniqueness theorems for transmissions problems related to models of elasticity, diffusion and electrocardiography

TL;DR

This paper extends the inverse problem of electrocardiography by formulating a generalized transmission problem for elliptic and parabolic equations, establishing conditions for solution uniqueness, and connecting it to elasticity theory and diffusion models.

Contribution

It introduces a generalized transmission problem framework with uniqueness theorems, applicable to electrocardiography, elasticity, and diffusion models, broadening the scope of inverse problem analysis.

Findings

Established uniqueness theorems for the transmission problem.

Connected the transmission problem to elasticity theory in composite media.

Proved a uniqueness theorem for an evolutionary transmission problem.

Abstract

We consider a generalization of the inverse problem of the electrocardiography in the framework of the theory of elliptic and parabolic differential operators. More precisely, starting with the standard bidomain mathematical model related to the problem of the reconstruction of the transmembrane potential in the myocardium from known body surface potentials we formulate a more general transmission problem for elliptic and parabolic equations in the Sobolev type spaces and describe conditions, providing uniqueness theorems for its solutions. Next, the new transmission problem is interpreted in the framework of the elasticity theory applied to composite media. Finally, we prove a uniqueness theorem for an evolutionary transmission problem that can be easily adopted to many models involving the diffusion type equations.